Real and complex multiplication on K3 surfaces via period integration
Andreas-Stephan Elsenhans Jörg Jahnel










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theoTheorem[section]
lem[theo]Lemma
prop[theo]Proposition
coro[theo]Corollary

definition
defi[theo]Definition
term[theo]Terminology
algo[theo]Algorithm
mconstr[theo]Main construction
constr[theo]Construction
strat[theo]Strategy
ttt[theo]

*nota_allgNotation
*comp_allgComputations

*proo_periodsProof of Theorem <ref>
*proo_indepProof of Theorem <ref>

*assAssumptions

remark
nota[theo]Notation
ex[theo]Example
exs[theo]Examples
rem[theo]Remark
rems[theo]Remarks
cau[theo]Caution




































































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Real and complex multiplication via period integration]Real and complex multiplication on
K3
surfaces via period integration





Institut für Mathematik
 Universität Würzburg
 Emil-Fischer-Straße 30
 D-97074 Würzburg
 Germany
stephan.elsenhans@mathematik.uni-wuerzburg.de
https://www.mathematik.uni-wuerzburg.de/computeralgebra/team/elsenhans-stephan-prof-dr/

Jörg Jahnel]Jörg Jahnel


jahnel@mathematik.uni-siegen.de
http://www.uni-math.gwdg.de/jahnel







[2010]14J28; Secondary 14J15, 14F25, 65D30, 65D10


We report on a new approach, as well as some related experiments, to construct families of
K3 surfaces
having real or complex multiplication. The approach is based on an explicit description of the transcendental part of the cohomology in a topological way, using topological tori. Fundamental ideas include considering the period space of marked
K3 surfaces,
determining the periods by numerical integration, as well as tracing the modular curve by a numerical continuation method.



[
    [
    October 24, 2021
====================

empty



§ INTRODUCTION


The endomorphism algebra of an elliptic curve
X
over

is isomorphic either
to 
or to an order in an imaginary quadratic field. The latter phenomenon is called complex multiplication. The theory of CM elliptic curves is very rich, cf. <cit.> or <cit.>, and their construction in an analytic setting is classical.

The whole theory generalises to higher dimensions, most obviously to abelian varieties. There are, however, natural generalisations of elliptic curves, other than abelian varieties. One such kind is provided by the surfaces of
type K3.
Indeed, elliptic curves may be characterised as being the curves with trivial canonical class, a property shared by
K3 surfaces.

Since K3 surfaces
do not carry a natural group structure, the endomorphism algebra needs to be defined cohomologically.
For X
a
K3 surface,
H := H^2(X, )
is a pure

Hodge structure of
dimension 22,
being the direct sum of the image
of X ⊗_
under the Chern class homomorphism and its orthogonal
complement T,
the transcendental part
of H.
The endomorphism algebra
_(T)
in the category of pure

Hodge
structures is either
,
or a totally real number field, or a CM field <cit.>, cf. Proposition <ref>.i).
If E ⫌
is totally real then
X
is said to have real multiplication (RM).
If E
is CM then one speaks of complex multiplication (CM).

For example, the Kummer surface
X := (E_1 × E_2)
attached to the product of two elliptic curves has complex multiplication if one of the elliptic curves has. In this case, CM is actually caused by an endomorphism
of X.
In <ref> to <ref>, we give examples of
K3 surfaces
of lower Picard ranks having CM even due to an automorphism.

On the other hand, a Kummer surface does not inherit real multiplication from the underlying abelian surface <cit.>. Nonetheless, the existence of
K3 surfaces
having real multiplication has been established for quite some time by analytic, i.e. Hodge-theoretic, means. It is known, for instance, that
K3 surfaces
having RM
by E
exist for every real quadratic
field E.
The same is true for every cubic field that is totally real. Note though that these methods do not provide explicit equations for the surfaces found to exist. Cf. the work <cit.> of B. van Geemen, in particular <cit.>.



  §.§.§ Dimensions of the RM and CM loci

Theorem <ref> is similar in spirit to van Gee­men's result. It covers, however, the CM case, too, and provides exact formulae for the dimensions of the RM and CM loci in period space.

These loci are, strictly speaking, not even closed sets. The reason is that there is a semicontinuity phenomenon. If the general fibre of a family has RM or CM by a
field K
then each special fibre has RM or CM by a field
containing K.
Cf. Corollary <ref>.



  §.§.§ Our terminology concerning RM/CM

In order to cope with that situation, we use a particular terminology.

If
_(T) ⊇ K
then we say that the transcendental part
T ⊂ H^2(X, )
is acted upon
by K.
Thus, if
T
is acted upon by
K ⫌
then
X
has RM or CM by a field that
contains K.



  §.§.§ Semicontinuity of
_(T)
in families

Semicontinuity may thus be formulated as follows. If the general fibre of a family
q→ Y
of complex
K3 surfaces
has RM or CM by a field
K
then, for every special
fibre _t,
the transcendental part
T_t ⊂ H^2(_t, )
is acted upon
by K.
Moreover, as Corollary <ref> shows, there is a countable union
V ⊂ Y
of analytic subsets such that
_t
actually has RM or CM
by K
if and only if
t ∈ Y ∖ V.



  §.§.§ Periods

By a marked
K3 surface, we mean a complex
K3 surface
X
together with an isomorphism
i^22→ H^2(X, ),
i.e. equipped with a distinguished basis
(c^1, …, c^22)
of
H^2(X, ).





A marked
K3 surface
(X,i)
gives rise to the period point

    Π_X,i := ((c^1, [ω]) : ⋯ : (c^22, [ω])) ∈^21()  .

Here,
[ω] ∈ H^2(X, )
denotes the class of a nowhere vanishing holomorphic
(2,0)-form.

The form
ω
is unique up to a constant factor
λ∈^*.
Having chosen a particular such form, one may consider the period vector
((c^1, [ω]), …, (c^22, [ω])) ∈^22,
which is usually denoted by
Π_X,i,
as well. The marking
i
is often suppressed from the notation, when there seems to be no danger of confusion. The coordinates of the period vector are usually referred to as periods.

-1
Moreover, a marking
i^22→ H^2(X, ), e_k ↦ c^k,
induces a perfect symmetric bilinear pairing
on ^22,
the pull-back of the cup product pairing. On the other hand, let such a symmetric bilinear
pairing κ
be given
on ^22.
Then a classical result (cf. <cit.>) states that the period points of all marked
K3 surfaces
that induce
κ
on ^22
form a (possibly void) open subset
Ω_κ
of the quadric
Q_κ
in ^21(),
defined
by κ.



  §.§.§ RM in the situation of Picard
rank 16

In a sufficiently general family
q→ Y
of
K3 surfaces
that is generically of geometric Picard
rank 16,
and not containing an isotrivial subfamily, the surfaces that are acted upon by a real quadratic field form families over base
curves C ⊂ Y.
For arbitrary Picard rank, number fields of larger degree, or CM instead of RM, the dimensions of the base varieties occurring are known, too. Cf. Theorem <ref>, shown below.

In coincidence with this, several explicit

families of RM surfaces, but also isolated examples, have been found. There are two families, for which RM by
(√(2))
and (√(5)),
respectively, is proven. Cf. Examples <ref> and <ref> or <cit.>. For conjectural examples, we refer to <cit.>, cf. Remarks <ref>.

On the side of the period space, the geometry of the corresponding base curves
Π(C) ⊂^21()
is very simple. Concretely, the restricted period space
Ω_κ,16⊂Ω_κ,
taking into account the fact that a fixed

subspace of
H^2(X, )
is contained in the algebraic part, is an open subset of a quadric
in ^5() ⊂^21().
And Π(C) ⊂^5()
is just the intersection
of Ω_κ,16
with a projective subspace of
codimension 3.



  §.§.§ Explicit construction of tori representing transcendental cohomology classes

Unfortunately, the period map
Π Y →Ω_κ,16
is not too explicit. In order to deal with this problem, in Section <ref>, we describe a particular family of
K3 surfaces,
which is the one we work with.

For the surfaces in this family, we give a topological construction of topological tori, representing a generating system of the transcendental part of the cohomology. I.e., of continuous mappings
α T^2 → X,
for
T^2
the
2-dimensional
torus. This construction is fundamental to the whole approach.


In what follows, we just write torus to mean a topological torus. The reader should, however, keep in mind that
α
needs not even to be injective.


The tori we construct are not just continuous, but
almost C^1,
cf. Definition <ref>. This gives us a link to the de Rham cohomology theory, so that the coordinates
(c_k, [ω])
of the period vector turn out to be improper integrals, cf. Theorem <ref>.

This makes it possible to compute, for a single point
t ∈ Y,
i.e. a single
surface X = _t,
the period point
Π(t) = Π__t
by numerical integration. In addition, the directional derivatives of
Π
at t
may be calculated.

We describe the construction of the tori in the course of Section <ref>. The technical details are treated in the final section.


There is a completely different approach to the calculation of periods, due to  E. C. Sertöz <cit.>. Sertöz's method is based on an understanding of the periods under deformation. I.e., he computes the directional derivatives of
Π
exactly, something we do not do, and derives a partial differential equation for
Π
from the outcome. This works for nonsingular hypersurfaces, which includes
K3 surfaces
of
degree 4,
while our method is, at least at this moment, limited to double covers
of ^2,
ramified in a union of six lines. Thus, a fair comparison seems to be difficult to undertake.




  §.§.§ The base curve
C
parametrising RM surfaces–Numerical continuation

Given an isolated example as mentioned above, numerical continuation methods (cf. <cit.>) may apply in order to extend to example to a

family. One determines a list of further surfaces in the same family, the period points of which lie
on Π(C)
at high precision. Then, using numerical linear algebra (in particular the singular value decomposition), one can recover algebraic equations with small coefficients defining a curve that numerically contains the parameters of the surfaces found.

This approach should work rather generally. We illustrate it at one particular example, which is a

family of
K3 surfaces
of
degree 2,
for which we report strong evidence for real multiplication
by (√(13)).
The generic geometric Picard rank is
16.
Cf. Example <ref> for the isolated example that was known to us before and Example <ref> the

family found.


We follow standard mathematical conventions and use standard notation. In particular,



   ∙ 
The standard basis of the free
R-module
R^n
is denoted by
(e_1, …, e_n).

   ∙ 
When
f M →^n
is a map from a set
M
to ^n,
we denote by
f_1, …, f_n
the components
of f.
I.e.,
f = (f_1, …, f_n).

   ∙ 
When
K
is a field, we identify
K = ^1(K)
with a subset
of ^1(K),
according to the embedding
K →^1(K),
x ↦ (1:x).
Moreover, we let
∞ := (0:1).

   ∙ 
For positive integers
m < n,
we denote by
^m() ⊂^n()
the particular
­men­sional
projective subspace
{(x_0:⋯:x_n) ∈^n() | x_m+1 = ⋯ = x_n = 0}
of ^n().

   ∙ 
For
l_1,…,l_r
global sections of an invertible sheaf
∈ X
on a scheme
X,
we let
V(l_1,…,l_r)
be the common vanishing locus of
l_1,…,l_r.

   ∙ 
When
f S → X
is a continuous map between topological spaces, we denote by
f^* H^i(X, ) → H^i(S, )
the induced homomorphism in cohomology and by
f_* H_i(S, ) → H_i(X, )
that in homology. Moreover, as in <cit.>, we use the notation
f_#π_1(S,·) →π_1(X,·)
for the induced homomorphism between the fundamental groups.

   ∙ 
When
f S → X
is a continuous map between equidimensional, oriented, compact manifolds, we let
f_! H^i(S, ) → H^i+ X- S(X, )
denote the (cohomological) transfer map. For a definition, cf. <cit.>.









   ∙ 
By a manifold, we always mean a smooth
(i.e.C^∞)
manifold.

   ∙ 
When
X
is a complex
K3 surface,
we usually identify the Picard group
X
with its image in
H^2(X, ),
under the Chern class homomorphism.




All computations are done with magma <cit.> on one core of an Intel i7-7700 processor running at 3.6 GHz.




§ K3 SURFACES




  §.§.§ The algebraic setting

Let k
be an algebraically closed field. Then
K3 surfaces
form one of the types of surfaces
over k,
according to the Enriques–Kodaira classification of surfaces <cit.>. 
This classification is due to the Italian school of Algebraic Geometry, cf. <cit.>, and today called the Enriques–Kodaira classification. More recent treatments are given in <cit.>, <cit.>, and <cit.>.
By a surface, here, one means a
,
connected, regular scheme that is projective
over k.

More concretely,
K3 surfaces
form one of the four types of surfaces of Kodaira dimension
0
and, among them, one of the two types of surfaces having trivial canonical class. The other such type are abelian surfaces.
A surface
X
is characterised (<cit.> and <cit.>) to
be K3
by the properties that



  * K_X ≅_X
and

  * π_1^(X,·) = 0.


Moreover, H^2_(X, _l)
is free of
rank 22
and, of course,
H^4_(X, _l) ≅ H^0_(X, _l) ≅_l
as well as
H^3_(X, _l) ≅ H^1_(X, _l) = 0
[under the assumption that
l ≠ k].




  * The degree of a
K3 surface
embedded in ordinary projective space is automatically even. There are examples in every even degree
d ≥ 4
<cit.>.



  * Smooth quartics
in ^3
are
K3 surfaces
of
degree 4.

  * Smooth complete intersections of a quadric and a cubic
in ^4
are
K3 surfaces
of
degree 6.

  * Smooth complete intersections of three quadrics
in ^5
are
K3 surfaces
of
degree 8.



  * Similarly, double covers
of ^2
ramified in a smooth sextic curve are
K3 surfaces
of
degree 2.

  * In each of the cases mentioned in a) and b), isolated singular points may be allowed, as long as all of them are of type
ADE <cit.>.
Then a
K3 surface
is obtained as the minimal desingularisation of the surface described.





  §.§.§ The setting of complex manifolds


There is a theory, parallel to the classification just mentioned, for complex surfaces. I.e., for

compact complex manifolds. In this setting, a
K3 surface
is a complex surface such that



  * there is a global holomorphic

ω∈Ω^2,0(X)
without zeros (or poles), and

  * π_1(X,·) = 0.


A complex
K3 surface
is automatically a Kähler manifold <cit.>. In general, however, it is not projective so that this theory does not exactly mirror the algebraic theory of
K3 surfaces
in the case
that k =,
but is more general.

All
K3 surfaces
X
are mutually diffeomorphic <cit.>.
Moreover,
H^2(X, )
is free of
rank 22
and, of course,
H^4(X, ) ≅ H^0(X, ) ≅
as well as
H^3(X, ) ≅ H^1(X, ) = 0.
Consequently,
H_2(X, ) ≅^22, too.



§ THE PERIOD SPACE




  §.§.§ Explicit cohomology classes

A complex
K3 surface
X
has a natural orientation, defined by the complex structure. Consider a compact, oriented

S,
together with a continuous map
α S → X.
Then α
defines a cohomology class

    c_α := α_!(1) ∈ H^2(X, )  ,

for
1 ∈ H^0(S, )
the canonical generator.



We call the class
c_α∈ H^2(X, )
the cohomology class given by
S,
together with the continuous map
α S → X.



The class
c_α∈ H^2(X, )
is the same as the Poincaré dual of the homology class of the image of
S
in X.
Cf. Lemma <ref>.b), below.





  * Note that the continuous map
α S → X
might be the embedding of a divisor, the embedding of a non-holomorphic submanifold, or not even an embedding.

  * In the particular case that
i_D D → X
is the embedding of a divisor, we prefer the more classical notation
[D] := c_i_D∈ H^2(X, ).









  * There is the canonical isomorphism

    ι H^2(X, ) → H_2(X, )  ,    u ↦ u ∩ z_X  ,

for
z_X ∈ H_4(X, )
the fundamental class.

  * Let α S → X
be as above. Then,
for u ∈ H^2(X, )
arbitrary,

    ⟨ u ∪ c_α, z_X ⟩ = ⟨α^*(u), z_S ⟩ ,

where
z_S ∈ H_2(S, )
denotes the fundamental class
of S.
Moreover,

    ι(c_α) = α_*(z_S)  .



Proof.
a)
This is a version of Poincaré duality <cit.>.

b)
The cohomological transfer map
α_!
is characterised <cit.> by the property that
α_!(a) ∩ z_X = α_* (a ∩ z_S),
for every
a ∈ H^0(S, ).
The second assertion is just a particular case of this.

For the first one, the claim is simply
(u ∪ c_α) ∩ z_X = α_*(α^*(u) ∩ z_S).
But the term on the left hand side is
u ∩ (c_α∩ z_X) = u ∩ι(c_α) = u ∩α_*(z_S),
due to , while that to the right is
u ∩α_*(z_S),
by .






  * In what follows, we generally prefer cohomology versus homology, although there are situations, in which homology classes appear to be more natural. For instance, as
K3 surfaces
are simply connected, the Hurewicz isomorphism theorem <cit.> shows that
π_2(X,·) ≅ H_2(X,).
Thus, every class in 
H_2(X,),
and hence every class in
H^2(X, ),
may be represented by a spheroid
S^2 → X.
Therefore, it can be represented by a torus
α T^2 → X, too.

  * Lemma <ref>.a) does not hold for singular surfaces. Cf. Proposition <ref>, below.

  * There is the cup product pairing
(· ,·) := ⟨·∪·, z_X⟩ H^2(X, ) × H^2(X, ) →,
which is symmetric, bilinear, and perfect. The cup product pairing bilinearly extends to a pairing

    (· ,·) H^2(X, ) × H^2(X, ) → .





  §.§.§ Marked
K3 surfaces



By a marked
K3 surface,
we mean a complex
K3 surface
together with an isomorphism
i^22⟶ H^2(X, ).






  * The marking
i^22→ H^2(X, )
determines the cohomology classes

    c^k := i(e_k) ∈ H^2(X, )  ,

for k = 1,…,22,
which form a basis
of H^2(X, ).

  * There is the dual basis
(c_1, …, c_22)
of
H^2(X, ),
given by the condition that
(c_k, c^j) = δ_kj,
for
( . ,. ) H^2(X, ) × H^2(X, ) →
the cup product pairing.

We denote the isomorphism
^22→ H^2(X, ), e_k → c_k,
by .

  * Moreover, the marking defines a perfect, symmetric pairing
on ^22,
namely the pull-back of the cup product pairing via
.



Given any cohomology class
u ∈ H^2(X, ),
there is the tautological decomposition

    u = (c^1, u) c_1 + ⋯ + (c^22, u) c_22 ,

of u
with respect
to i.

For instance, this applies to the distinguished cohomology class
[ω] ∈ H^2(X, )
of the
K3 surface
X,
which is uniquely determined up to scaling. Here and in what follows, we denote
by ω
a nowhere vanishing holomorphic

on X.


A marked
K3 surface
(X,i)
gives rise to a point

    Π_X,i := ((c^1, [ω]) : ⋯ : (c^22, [ω])) ∈^21()  ,

which is called the period point of
(X,i).




Let X
be a complex
K3 surface.
Then the period point
Π_X,i
is completely determined by the composition
^22i⟶ H^2(X, ) ↠ H^2(X, )/ X.

Proof.
Suppose that
i
and i'
are two markings
on X
differing only by elements
in X.
I.e., that
c'^k - c^k ∈ X,
for
k = 1,…,22.
Then

    (c'^k, [ω]) - (c^k, [ω]) = (c'^k - c^k, [ω]) = 0  ,

since
c'^k - c^k ∈ X
is of type
(1,1),
while
[ω]
is of type
(2,0).
The assertion follows.



[I. R. Shafarevich et al.]

Let κ
be a perfect, symmetric pairing
on ^22.
Denote the
-bilinear
pairing induced
by κ
on ^22,
by κ
as well.



  * Let
(X_1,i_1)
and
(X_2,i_2)
be marked
K3 surfaces
defining the pairing
κ
on ^22
and having the same period point.
Then
(X_1,i_1)
and
(X_2,i_2)
are isomorphic.

  * The set
Ω_κ
of the period points of all marked
K3 surfaces
defining the
pairing κ
is possibly empty. Otherwise, one has

    Ω_κ = {(x_1:⋯:x_22) ∈^21() |κ(x,x) = 0, κ(x,x) > 0 } .

This is an open subset of the quadric
Q_κ⊂^21(),
defined
by κ.


Proof.
This is <cit.>, together with <cit.>. Cf. <cit.>.





  §.§.§ The relative situation


Let q→ Y
be a holomorphic submersion of complex manifolds and suppose that every fibre is a
K3 surface.
Then the higher direct image sheaf
R^2 q_*
on Y
is locally constant of
dimension 22.
Moreover, due to Grauert's Theorem (<cit.>), one has a natural base change isomorphism
ι_t^* (R^2 q_* )_t ≅⟶ H^2(_t, ),
for
every t ∈ Y.



By a family of marked
K3 surfaces,
one means a holomorphic submersion
q→ Y,
every fibre of which is a 
K3 surface,
together with a system
= (i_t)_t∈ Y
of markings satisfying the following conditions.



  * For each
t∈ Y,
i_t
is a marking on the
fibre _t.

  * For every
v ∈^22,
t ↦ (ι_t^*)^-1∘ i_t(v) ∈ (R^2 q_* )_t
is a global section
of R^2 q_*.
I.e., the markings
i_t
trivialise
R^2 q_* globally.



























Let q (, ) → Y
be a family of marked
K3 surfaces.
Then the period mapping
Π Y →^21(),
t ↦Π__t, i_t,
is holomorphic.

Proof.
This is easily verified by a direct calculation, cf. <cit.>.





  §.§.§ The restricted period space




Let
(X,i)
be a marked
K3 surface,
x = Π_X,i∈^21()
its period point, and
κ
the perfect, symmetric pairing
on ^22
defined
by (X,i).
Denote the pairing induced
on ^22
by κ,
too. Then

    ^-1 X = (x)^⊥∩^22 ,

the orthogonal complement being taken
in ^22
with respect
to κ.

Proof.
One has
X ≅{u ∈ H^2(X, ) | u ∪ [ω] = 0},
according to the Lefschetz theorem on
 <cit.>.
Moreover,

    ⟨ u ∪ [ω], z_X⟩ = ⟨(∑_k (c^k, u) c_k) ∪(∑_j x_j c_j), z_X⟩   =   κ((c^k, u)_k=1,…,22, (x_j)_j=1,…,22) 
       =   κ((c^k, u)_k=1,…,22, x)  .

Thus,
^-1 X = {v ∈^22|κ(v,x) = 0} = (x)^⊥∩^22,
as required.





Let r ∈{1,…,20}
and
κ
a perfect, symmetric pairing
on ^22.
Denote the pairing induced
on ^22
by κ,
too. Then the set
Ω_κ,r
of the period points of all marked
K3 surfaces
(X, i)
such that



  * the cohomology classes
c^22-r+1, …, c^22∈ H^2(X, )
are algebraic, i.e. contained in
X ⊂ H^2(X, ), and

  * the pairing
on ^22
induced
by (X,i)
is
exactly κ,


is either void or one has

    Ω_κ,r = {(x_1:⋯:x_22-r:0:⋯:0) ∈^21() |κ(x,x) = 0, κ(x,x) > 0 } .

This is an open subset of a quadric
Q_κ,r⊂^21-r() ⊂^21().

Proof.
In comparison with Theorem <ref>, 
c^22-r+1, …, c^22∈ H^2(X, )
being algebraic is the only additional condition. By Lemma <ref>, this means precisely that

    x ⊥^-1(c^22-r+1),  …,  x ⊥^-1(c^22)

with respect to
κ.
I.e., that
(x) ⊥(c^22-r+1, …, c^22)
with respect to the cup product pairing. The latter is equivalent to
(x) ∈(c_1, …, c_22-r),
which exactly means that
x ∈(e_1,…,e_22-r).




Let
q (, ) → Y
be a family of marked
K3 surfaces
of the kind that
i_t(e_22-r+1), …, i_t(e_22)
are algebraic, for every
t ∈ Y.
Then we call

    Π Y →^21-r()  ,    t ↦Π__t, _t := ((i_t(e_1), [ω]) : ⋯ : (i_t(e_22-r), [ω]))  ,

the restricted period map. In the case that
Y
is a point, we speak of the restricted period point.


[A basis of the transcendental part induces a class of markings]


Let r ∈{1,…,20},
X
a complex
K3 surface,
P ⊆ X
a subgroup that is co­tor­sion-free, and
ι^22-r→ H^2(X, )/P
an isomorphism.



  * Then



  i) 
lifting ι
to a homomorphism
^22-r→ H^2(X, )
and
  ii) 
extending that to a homomorphism
i^22→ H^2(X, )
by choosing an arbitrary basis
(u_22-r+1, …, u_22)
of P
and
putting i(e_k) := u_k,
for
k = 22-r+1, …, 22,


provides a
marking i
on X.

  * The
marking i
yields a restricted period point
Π_X ∈^21-r().

  * The restricted period point
Π_X ∈^21-r()
in independent of the choices made in steps a.i) and a.ii).


Proof.
The only assertion in a) is that
i is
an isomorphism, which is clear. b) follows, as the assumptions imply that
i(e_22-r+1), …, i_t(e_22)
are algebraic. Finally, c) is a direct consequence of Lemma <ref>.





  §.§.§ Periods as integrals


Suppose that, on a marked
K3 surface
(X,i),
the cohomology class
c^j
is given by a compact, oriented

S,
together with a continuous map
α S → X.
Then the
j-th
period is

    (c^j, [ω]) = (c_α, [ω]) = ⟨ c_α∪ [ω], z_X⟩ = ⟨α^*([ω]), z_S⟩ ,

by Lemma <ref>.b). For smooth
α,
this is simply
∫_S α^*([ω]).
Thus, one might have the idea to avoid the topological machinery and to consider the periods just as integrals of the nowhere vanishing
(2,0)-form.

While we indeed use a representation as an integral for computational purposes, cf. Theorem <ref> below, such a representation is not well suitable as a definition. One problem is that, in our situation, the map
α
is usually non-smooth. Also, and more crucially, the application of the periods we have in mind is towards real multiplication, which is defined in terms of cohomology.



§ EXPLICIT DESCRIPTION OF TRANSCENDENTAL COHOMOLOGY CLASSES





  §.§.§ The family


We consider double covers
of ^2_,
ramified in a union of six lines. These are given by
X' w^2 = l_1 ⋯ l_6,
for
l_1, …, l_6
linear forms in three variables. We assume that no point of the plane is contained in three lines. Then there are 15 singular points, each of which is isolated and conical, i.e. of
type A_1.
The minimal desingularisation
X
of X'
is hence a
K3 surface.

Concretely, the
K3 surface
X
is obtained
from X'
by blowing up the 15 singular points. This yields 15 exceptional curves, each of which is a
.



In particular,
X ≥ 16.





  * For a = 1, …, 6,
we denote the coefficients of the linear form
l_a
by
A_a1,
A_a2,
and A_a3.
I.e., l_a = A_a1 x + A_a2 y + A_a3 z.

  * We write
x_ij := V(l_i,l_j) ∈^2(),
for
1 ≤ i, j ≤ 6,
for the 15 double points of the branch locus. Accordingly, we denote the 15 exceptional curves by
E_ij,
1 ≤ i, j ≤ 6.

  * We use the notation
π X →^2()
for the natural map. By definition,
π
factors
via X',

    π X ⟶ X' π_X'⟶^2()   .

Similarly, as is well-known,
π
factors via the blowing-up
of ^2()
in the 15 double points of the branch locus,

    π X π'⟶_x_12,…,x_56(^2()) ⟶^2()  .

The map
π' is
a double cover, ramified in the union of six mutually disjoint projective lines.

  * Moreover, X
carries the natural involution
ζ X → X,
which is induced by the map
ζ' X' → X',
(w; x:y:z) ↦ (-w; x:y:z).

  * We put

    P := V ∩ H^2(X, )  ,

for
V ⊂ H^2(X, )
the subvector space spanned by the classes
[E_ij],
for
1 ≤ i,j ≤ 6,
together with
π^*[l],
the inverse image of a general line
on ^2.

According to this definition,
P ⊂ H^2(X, )
is a saturated sublattice of
rank 16.
Let us note explicitly that
P
is generated by the classes
[E_ij]
and π^*[l]
only up to finite index.






  * It is a well-known fact from projective geometry that configurations of six lines
in ^2
have exactly four moduli. Indeed, one might normalise the six lines under the operation of
(^2_) = _3()
in such a way that
X'
is given by an equation of the form

    X' = X'_(a_0,b_0,c_0,d_0) w^2 = xyz(x+y+z)(x+a_0y+b_0z)(x+c_0y+d_0z)  .



  * The restricted period space corresponding to a system of generators
of P
is an open subset of a
quadric Q ⊂^5,
which is in perfect coincidence with the fact that geometrically there are four moduli.





Let X
be a
K3 surface
of the family described.



  * Then the
involution ζ
acts on
P ⊗_⊂ H^2(X, )
as the identity map and on
P^⊥⊂ H^2(X, )
as the multiplication
by (-1).

  * In particular,
ζ
operates
on H^2(X, )/P
as the multiplication
by (-1).


Proof.
a)
The
involution ζ
has a disjoint union of six projective lines as its fixed point set. The topological Euler characteristic of such a configuration amounts
to 12.
Therefore, the Lefschetz trace formula <cit.>, cf. <cit.>, shows that
ζ |_H^2(X, ) = 10.
In other words, the
eigenvalue 1
has
multiplicity 16,
while the
eigenvalue (-1)
occurs with
multiplicity 6.

On the other hand,
ζ clearly
fixes
P
pointwise, which shows that indeed
P ⊗_
is the eigenspace
for 1.
Finally, the operation
of ζ
is compatible with the cup product pairing. Thus,
for x ∈ P
and
c ∈ H^2(X, )
in the eigenspace
for (-1),
one finds

    (x,c) = (ζ^*x, ζ^*c) = (x,-c) = -(x,c)

and hence
c ∈ P^⊥.
This completes the proof of a).

b)
directly follows from a).





Let Y'
be a complex space that is equidimensional of dimension two and has an isolated singular point
y ∈ Y'
of
type A_1,
i.e. an ordinary double point. Denote by
b Y → Y'
the blowing-up of
Y'
at y
and by
E ⊂ Y
the exceptional curve.

Then the homomorphism
b_* H_2(Y,) → H_2(Y',)
is surjective with kernel
_(ι([E])).
I.e.,
b_*
induces a natural isomorphism

    H_2(Y,)/_(ι([E])) ≅⟶ H_2(Y',)  .


Proof.
There is the following commutative diagram of Mayer-Vietoris exact sequences

    2.7mm
    H_2(U(E) ∖ E, ) @->[r] @=[d]     H_2(Y ∖ E,) ⊕ H_2(U(E),) @->[r]@->[d]     H_2(Y,) @->[r]@->[d]^b_*    H_1(U(E) ∖ E, ) @=[d]  .
    
    H_2(U(y) ∖{y}, ) @->[r]    H_2(Y' ∖{y},) ⊕ H_2(U(y),) @->[r]    H_2(Y',) @->[r]    H_1(U(y) ∖{y}), )  .


Here, we write
U(y)
for the intersection
of Y'
with a tiny ball around the point
y
and
U(E) := b^-1(U(y))
for its preimage
in Y.
Clearly, the natural map

    b |_Y ∖ E Y ∖ E → Y' ∖{y}

is a homeomorphism. Moreover, one readily sees that
U(y)
is homotopy equivalent to an affine quadratic cone, and hence contractible, while
U(E)
is of the homotopy type
of ^1() ≅ S^2.
Finally,
U(y) ∖{y}
has the homotopy type of a complex conic, i.e. that of
^1() ≅ S^2, too.
In particular, we find that
H_1(U(y) ∖{y}, ) = 0.
The commutative diagram therefore takes the form below,

    H' @->[r] @=[d]     H”⊕ι([E]) ·@->[r]@->>[d]     H_2(Y,) @->[r]@->[d]^b_*    0  .
    
    H' @->[r]    H”@->[r]    H_2(Y',) @->[r]    0  .


The assertion follows from this by a simple diagram chase.





Let X
be a
K3 surface
of the family described and
X'
the underlying double cover
of ^2().



  * Then the homomorphism
_* H_2(X,) → H_2(X',)
is surjective with kernel
_(ι([E_12]), …, ι([E_56])).
I.e.,
_*
induces a natural isomorphism

    H_2(X,)/_(ι([E_12]), …, ι([E_56])) ≅⟶ H_2(X',)  .


  * In particular, the natural homomorphism
H^2(X, ) ↠ H^2(X, )/P
factors via

    H^2(X, ) ι⟶ H_2(X,) _*⟶ H_2(X',)  .



Proof.
Assertion a) follows applying Lemma <ref> repeatedly, 15 times in total.
b) is a direct consequence of a).




Proposition <ref> shows, in particular, that
H_2(X',)
has torsion. Indeed, the cohomology class
[E_12] + ⋯ + [E_16] + 5π^*[l] = [÷(G)] ∈ H^2(X, ),
for
G := l_2 ⋯ l_6 - l_1^5,
is divisible
by 2.
The reason is that,
modulo G,
the equation
w^2 = l_1 ⋯ l_6
of the surface goes over into
w^2 = l_1^6.
Thus, the divisor
÷(G)
splits into two components that are interchanged under the
involution ζ
and the claim follows from Lemma <ref>.a). Consequently,
[E_13] + ⋯ + [E_16] - ([E_23] + ⋯ + [E_26]) ∈ H^2(X, )
is divisible
by 2,
too, and the same is true for its image
under ι
in H_2(X,).

The nontrivial torsion classes
in H_2(X',)
are certainly worth being considered more closely, but, in this article, we do not have any applications for them.




Let α T^2 → X'
be a torus and
α, α T^2 → X
continuous lifts
of α.
Then c_α - c_α∈ P ⊂ H^2(X, )
(cf. the notation introduced in <ref>.v).

Proof.
According to Proposition <ref>.b), it is sufficient to verify that

    _*(ι(c_α)) = _*(ι(c_α)) ∈ H_2(X',)  .

But this is clear, since
ι(c_α) = α_*(z_T^2),
by Lemma <ref>.b), and therefore

    _*(ι(c_α)) = _*(α_*(z_T^2)) = (∘α)_*(z_T^2) = α_*(z_T^2)  ,

while exactly the same holds for
_*(ι(c_α)).









  §.§.§ Assumptions on the branch locus


[on the branch locus]


  * We assume once and for all that the branch locus is the union of six real lines
V(l_1), …, V(l_6)
[no three of which have a point in common]
in ^2_.
I.e., that
A_a1, A_a2, A_a3∈,
for a = 1, …, 6.

  * We suppose that any two of the vectors

    (A_11, A_12), …, (A_61, A_62)

are linearly independent. I.e., it is assumed that the open chart given
by “z=1”, which is the one we are going to work with, contains the 15 double points
x_ij,
for
1 ≤ i, j ≤ 6.

-

The linear forms
l_1, …, l_6
then go over into the affine-linear maps
l'_1, …, l'_6,

    l_a'(x,y) = l_a(x,y,1)    =    A_a1 x + A_a2 y + A_a3
       =:   l_a(x,y) + A_a3 .




  §.§.§ Particular
1-manifolds
in ^2()


The starting point of our construction are compact, connected

Γ,
embedded
into ^2(),
of the behaviour indicated in the two figures below. Observe that, in either case, the

meets the branch locus
V(l_1 ⋯ l_6)
only in its double points.






We call a

as in Figure <ref> a deformed line and one of the type indicated in Figure <ref> a curve encircling a triangle.

We allow as well compact

that are of a similar shape as the one in Figure <ref>, but encircle a quadrangle or pentagon instead of a triangle. Note here that every compact, connected

is diffeomorphic
to S^1.




  §.§.§ Technical assumptions on the



Consider a single compact

in ^2()
of one of the kinds described. We may certainly assume that it is specified in parametrised form.

In the case of a deformed line, the affine part allows a parametrisation given by a
C^1-map

    γ→^2 ⊂^2()  .

Similarly, a curve encircling a polygon may be parametrised by a
C^1-map

    γ^1() →^2 ⊂^2()  .

In this situation, we denote the restriction
γ|_
by γ.

For reasons of simplicity of the presentation, let us make the following assumptions on the

we consider, as well as on their parametrisations.

[on the
1-manifolds]


  * 
In the case that
Γ
is a curve encircling a polygon, the parametrisation is of the kind that
γ(∞)
is not a branch point.

  * 
Suppose that the branch
point x_ij
is met by the
­i­fold Γ,
and let
x_ij = γ(t_ij).
Then the parametrisation is linear
near t_ij.
More precisely,

    γ(t) = (t-t_ij)b_ij + x_ij ,

for some
b_ij∈^2
and t ∈ [t_ij-1, t_ij+1].
We require, moreover, that

    b_ij∉(A_i2, -A_i1) ∪(A_j2, -A_j1)  .

I.e., that the direction
of Γ
near
x_ij
differs from the directions of the lines
V(l_i)
and V(l_j)
of the branch locus that meet
at x_ij.

  * 
In addition, for all branch points that are met
by Γ,
the intervals
[t_ij-1, t_ij+1]
are mutually disjoint.

  * 
Finally, for a deformed line, we assume that the map
γ
is of the shape 

    γ(t) = tb + γ_0(t)

for all
t ∈,
where
b ∈^2 ∖{(0,0)}
is a certain vector and the

γ_0→^2
has compact support. We require, moreover, that

    b ∉(A_12, -A_11) ∪⋯∪(A_62, -A_61)  .

I.e., that the direction
of Γ
near infinity differs from the directions of the lines
V(l_1), …, V(l_6)
forming the branch locus.







  * We let
x_i_1,j_1, …, x_i_n,j_n∈ V(l_1 ⋯ l_6) ⊂^2(),
for
n = 3,
4,
or 5,
be the points of the branch locus that are met
by Γ.

  * Moreover, we denote by
t_1 = t_i_1,j_1, …, t_n = t_i_n,j_n∈
the parameters of these points. I.e. the reals such that
γ(t_1) = x_i_1,j_1,
…,
γ(t_n) = x_i_n,j_n.




  §.§.§ The main construction—2-tori
from
1-manifolds



Write := ^1() ×^1() ⊃× = ^2,
for the obvious compactification. This is our standard
.



By a liftable torus, in this article, we mean a torus
α_∙→ X'
that lifts through the blowing-down map
X → X'
between the
K3 surface
X
and the singular double cover
X'
of ^2.
I.e., we require the existence of a continuous map
α→ X
to the
K3 surface,
such that
∘α = α_∙.


[
tori from
]




  * There are two liftable tori, i.e. continuous maps

    α_Γ, α_Γ⟶ X'  ,

which differ only by the
involution ζ,
associated with each deformed
line Γ,
as above.

  * Similarly, there are two liftable tori

    α_Γ,b, α_Γ,b⟶ X'  ,

differing only by the
involution ζ,
associated with each
curve Γ
encircling a polygon, together with a vector

    b ∈^2 ∖( (A_12, -A_11) ∪⋯∪(A_62, -A_61) )  .






The tori are obtained by the following purely topological construction.



  * (Adding an imaginary direction)
Extend the map
γ→^2 ⊂^2()
to a continuous map in two variables by putting

    γ'^2 ⟶^2 ⊂^2(),      (t,u) ↦γ(t) +  ub  .

Here, in the case of a deformed line,
the vector b
is taken from A.<ref>, while, in the case of a curve encircling a polygon,
b is
part of the input data.

The map
γ'
allows a continuous prolongation

    α'→^2()  .



  * (Lifting to the double cover)
The map
α'
allows a continuous lift

    α”→ X'

to the double cover. There is, in fact, a second such lift,
α”,
differing
from α”
by the
involution ζ.

  * (Widening the holes)
Use a self-map
Ψ→
that induces a homeomorphism

    Ψ |_∖( U_1/2(t_1,0)∪…∪U_1/2(t_n,0))∖( U_1/2(t_1,0)∪…∪U_1/2(t_n,0)) ≅⟶∖{(t_1,0), …, (t_n,0)}

and sends
U_1/2(t_i,0)
constantly to
(t_i,0),
for i=1,…,n,
to define

    α”' := α”∘Ψ→ X'  .

This is the liftable torus
α_Γ
or
α_Γ,b desired.






  * The cohomology class
c_α∈ H^2(X, )
is dependent on the lift
α→ X
chosen. Two different lifts lead to cohomology classes differing by a summand
from P.
I.e., the image
c_α∈ H^2(X, )/P
depends only on the torus
α_Γ
or
α_Γ,b→ X'
lifted. In fact,
c_α
depends only on the homotopy class of
α_Γ
or
α_Γ,b, respectively. This is a direct consequence of Corollary <ref>.

Moreover, by Lemma <ref>.b), the classes
c_α
and
c_α∈ H^2(X, )/P
only differ by sign.

  * Step iii) in the construction above could be eliminated at the cost of a somewhat less elegant description of the class
c_α∈ H^2(X, )/P,
which works as follows.

Take the homology class
α”_*(z_) ∈ H_2(X',)
and put
c_α∈ H^2(X, )/P
to be its image, according to the homomorphism described in Lemma <ref>.b).

  * The two theorems below are the principal results on the construction just described. As their proofs are a bit technical, we postpone them to the final section. The same applies to the claims made in <ref>. Cf., in particular, Propositions <ref> and <ref>.



[Independence
of b
in the case of a curve encircling a polygon]

Let Γ
be a curve encircling a polygon and

    b, b∈^2 ∖( (A_12, -A_11) ∪…∪(A_62, -A_61) )

any two vectors. Moreover, let
α→ X
be any continuous lift
of α_Γ,b→ X'
and
α→ X
any continuous lift
of α_Γ,b→ X'.




Then c_α∈ H^2(X, )/P
coincides either with
c_α
or with
-c_α∈ H^2(X, )/P.


[Periods as improper integrals]

Let X
be the
K3 surface
obtained as the minimal desingularisation of the double cover
X' w^2 = l_1⋯ l_6,
for
l_1, …, l_6
real linear forms. Assume that no three of the linear forms have a zero in common.
On X,
fix the global holomorphic
 ω,
determined by formula (<ref>), below.

Let, moreover,
Γ
be a compact

in ^2()
being a deformed line or a curve encircling a polygon. We assume that
Γ
is parametrised by
γ→^2 ⊂^2()
and that Assumptions A.i) to A.iv) are fulfilled.



  * Let Γ
be a deformed line. Then, for every continuous lift
α→ X
of the torus
α_Γ→ X',

    (c_α, [ω]) = ∫_-∞^∞∫_-∞^∞γ̇_̇1̇(t) b_2 - γ̇_̇2̇(t) b_1/√(l_1(γ(t)+ ub) ⋯ l_6(γ(t)+ ub)) du dt  ,

where the vector
b ∈^2 ∖(A_12, -A_11) ∪…∪(A_62, -A_61)
is taken from A.iv).

  * Let b ∈^2 ∖(A_12, -A_11) ∪…∪(A_62, -A_61)
be any vector and
Γ
a curve encircling a polygon.


Then, for every continuous lift
α→ X
of the torus
α_Γ,b→ X',

    (c_α, [ω]) = ∫_-∞^∞∫_-∞^∞γ̇_̇1̇(t) b_2 - γ̇_̇2̇(t) b_1/√(l_1(γ(t)+ ub) ⋯ l_6(γ(t)+ ub)) du dt  .



In either case, the square root in the integrand is meant to be the same as that in the
form ω.





  * In the case of a deformed line, the outer integral in (<ref>) is actually proper. Indeed, formulae (<ref>) and (<ref>) together imply that the
map α',
and hence the
lift α”,
too, is holomorphic near every point
(t,u) ∈
with t ∉γ_0.
Furthermore, since
ω is
a

on X
and

is of real dimension two, holomorphicity enforces the pull-back
(α” |_^2 ∖{_1,…,_n} )^* ω'
to be the null form in a neighbourhood
of (t,u).
Finally, the integrand in (<ref>) exactly corresponds to this differential form. Thus, the outer integral actually ranges only
over γ_0. 

  * On the other hand, a curve encircling a polygon is a curve
in ^2
of finite length. It is convenient to reparametrise it by a compact interval, so that in this case the outer integral in (<ref>) becomes proper, too.





  §.§.§ The global holomorphic differential form



Let X
be the minimal desingularisation of the double cover
X' w^2 = l_1⋯ l_6,
for
l_1, …, l_6
linear forms in the variables
x, y,
and z.
Assume that no three of the six linear forms have a zero in common.

Then, for any linear
form l
that is not just a scalar multiple of
x
or y
and defines a nonsingular curve
on X',

    ω' := d(x/l) ∧ d(y/l)/w/l^3

is a differential form
on X',
whose pull-back
ω
to X
is a global holomorphic
(2,0)-form
without zeroes or poles.

Proof.
On the affine plane, given by
l ≠ 0,
the functions
ξ := x/l
and
η := y/l
form a system of coordinates. In particular, there are functions
λ_1, …, λ_6
of total
degree 1
such that
λ_i(ξ, η) = l_i(x/l, y/l, z/l),
for i = 1, …, 6.
The corresponding affine part of the double
cover X'
is thus given by
ψ^2 = λ_1 ⋯λ_6,
for ψ := w/l^3.

Hence, ω' = dξ∧ dη / ψ.
This immediately shows that, outside
V(l_1 ⋯ l_6),
which is the branch locus, and
V(l),
which is not on our chart,
ω
has neither zeroes, nor poles.

Next, consider a point that lies on exactly one of the lines
V(λ_1), …, V(λ_6).
Then at least one of the partial derivatives
∂(λ_1 ⋯λ_6)/∂ξ
and
∂(λ_1 ⋯λ_6)/∂η
is nonzero. Without restriction, assume that
∂(λ_1 ⋯λ_6)/∂η≠ 0.
Then ψ
and ξ
form a local system of coordinates. Moreover, the equation
ψ^2 = λ_1 ⋯λ_6
yields
2ψ dψ = ∂(λ_1 ⋯λ_6)/∂ξ dξ + ∂(λ_1 ⋯λ_6)/∂η dη,
which shows that
2ψ dξ∧ dψ = ∂(λ_1 ⋯λ_6)/∂η dξ∧ dη.
Thus, dξ∧ dη
is indeed divisible
by ψ
and the quotient
ω' = dξ∧ dη / ψ
is nonzero near the point considered.

Finally, let us write
W := ÷ω∈ X.
The calculations above prove that
W
has the form

    W = n [L] + ∑_1≤ i<j≤ 6 n_ij [E_ij]  ,

for L := V(l).
We have to show that, in fact, all the coefficients vanish.

For this, let us note that
W is,
by construction, a canonical divisor. Thus, the adjunction formula shows that
(W+[E_ij])[E_ij] = 2g_E_ij-2 = -2,
i.e.W[E_ij] = 0,
for 1 ≤ i < j ≤ 6.
Consequently,
n_ij = -1/2 W[E_ij] = 0.

Moreover, L
is a double cover of a projective line ramified over six points, and hence its genus
is g_L = 2.
Here, adjunction shows that
(W+[L])[L] = 2g_L-2 = 2.
As [L][L] = 2,
this yields
W[L] = 0
and
n = 1/2 W[L] = 0.
The assertion follows.





§ EXPLICIT DESCRIPTION OF TRANSCENDENTAL COHOMOLOGY CLASSES—
COMPUTATION OF PERIODS




  §.§.§ The main algorithms



For clarity of the presentation, let us present our main algorithms in a somewhat idealised setting.

Imagine for a moment that one could calculate

integrals exactly, as complex numbers. Then, given an arbitrary
K3 surface
X = X_(a_0,b_0,c_0,d_0)
(cf. formula (<ref>)),
we could run the following algorithm.


[Explicit description of the transcendental part of
H^2(X, )
— Idealised setting]




  * Set up a list of

in ^2()
consisting of



  ∙ 
for each triangle, quadrangle, or pentagon formed in the affine plane given by
“z=1”
by the lines of the branch locus, one curve encircling it,

  ∙ 
through each triple of double points of the branch locus that may be connected by a deformed line within the affine plane given by
“z=1”,
one deformed line.



  * For every

listed, construct a torus
α→ X,
as described in <ref>.

  * 
For every torus
α
constructed, calculate
(c_α, [ω]) ∈
using Theorem <ref>, formulae (<ref>) and (<ref>).

  * 
Detect all

dependencies among these numbers, and select six (linear combinations of) tori
α_1, …, α_6
that yield a

for the complex numbers obtained.

In the case that the numbers
(c_α, [ω])
form a

space of dimension less than six, do the calculations on a surface
X_(a,b,c,d),
for a quadruple
(a,b,c,d)
near
(a_0,b_0,c_0,d_0).

  * 
Detect the

dependency among the products
(c_α_i, [ω]) · (c_α_j, [ω]),
for
1 ≤ i ≤ j ≤ 6.
I.e., determine the projective quadric
Q_κ,16⊂^5()
in the restricted period space.

Again, do this on a neighbouring surface that is generic, if necessary.

  * Determine the quadratic form
κ
exactly, not only up to scaling, from the fact that the self-product is known for a deformed line. (Cf. Lemma <ref>.)

  * Verify that
c_α_1,…,c_α_6
indeed form a generating system of
H^2(X,)/P,
not only up to finite index. (Cf. Theorem <ref>.)






  * By Theorem <ref>, the result that
c_α_1,…,c_α_6
generate
H^2(X,)/P
implies that these elements give rise to a class of markings
on X,
for which the numbers
(c_α_i, [ω])
are indeed the coordinates of the period vector.

  * In practice, of course, in step <ref>, one has to use numerical integration methods working at a finite precision. Moreover, instead of the exact linear algebra in step <ref>, the singular value decomposition is playing the main role.

  * Algorithm <ref> below gives a more thorough description of step <ref>.




Furthermore, given a
K3 surface
X = X_(a_0,b_0,c_0,d_0)
having real multiplication (cf. Section <ref>), one could run the algorithm below.


[Tracing the modular curve — Idealised setting]




  * Detect the linear relations between the coordinates of the period points encoding real multiplication. Cf. Theorem <ref> for details.

  * Using Newton iteration, find surfaces
X_(a,b,c,d),
for quadruples
(a,b,c,d)
near
(a_0,b_0,c_0,d_0),
the coordinates of the period points of which fulfil the same relation, not exactly but at high precision.




If one had enough of these quadruples (exactly) then it would just be linear algebra to find an algebraic curve through these quadruples. Again, the singular value decomposition plays the main role in our adaption of Algorithm <ref> to practice.




  §.§.§ Numerical integration



In order to numerically calculate the
i­men­sional
integrals occurring in Theorem <ref>, our strategy is roughly as follows.



  * We decompose the domain of the outer integral into finitely many intervals
[a_k-1, t_k]
and
[t_k, a_k],
each of which contains none of the
t_1, …, t_n
as an inner point and exactly one as an endpoint. For integration, each such interval is treated individually, the results being added together at the very end.

  * For each concrete interval, the inner integral needs to be computed only over
[0,∞) ⊂.
Indeed, the sign change
u ↦ (-u)
transforms the integrand into its complex conjugate or minus its complex conjugate, depending on the sign of
(l_1 ⋯ l_6) ∘γ
on the interval.

  * We decompose the two-dimensional domains of integration emerging into three subdomains each, as indicated in Figure <ref> below.

For each subdomain, we determine the integral using Fubini's Theorem in a rather naive manner. The inner integrals, for the computation, are taken along the line segments and rays, as shown in the figure.

  * To compute the inner integrals of the integral over the triangle, we use the Gauß-Legendre method.

For the computation of the inner integrals that are improper, we decompose the ray into finitely many intervals
I_1,…,I_N
of lengths increasing by a factor of
|I_i+1|/|I_i| = q = 3/2
and the remaining ray. The substitution
u' := 1/u
transforms the integral over the latter into a definite one. We then use the Gauß-Legendre method for each of the individual integrals and add the results together.



Thereby, we work with adaptive stepsizes, as follows. First, we parametrise the ray. This results in a hyperelliptic integral of the form

    ∫_u_0^∞1/√((a_1 i u + b_1) ⋯ (a_6 i u + 
    b_6)) du  ,

for
u_0 ≥ 0
and a_1,…,a_6,b_1,…,b_6 ∈.
Note that all singularities of the integrand are located on the imaginary line, while the domain of integration is a ray contained in the reals.

We compute the smallest and the largest among the absolute values of all the singularities of the integrand. Denote these by
S
and M.

The first interval
I_1,
starting
at u_0,
is then taken to be of length
max(u_0, S).
We then increase the lengths of the intervals by the factor of
q = 3/2
in each step, until we reach a value of
u_1 > 4M.
Then, on the remaining ray, coordinates are changed as described above.

The aim of this strategy is to make sure that, for each interval of integration, the ratio between its length and the distance to the next singularity is bounded by a constant. This suffices to make sure that the Gauss-Legendre method provides the expected precision.

For example, applying the change of coordinates, the final ray is turned into the interval
[0,1/u_1],
whereas the smallest absolute value of a singularity becomes
1/M.
As 1/u_1 < 1/4M,
the resulting integral can be computed numerically without any further 
splitting of the
interval [0,1/u_1].

  * For the outer integrals, we again use the Gauß-Legendre method.






  * One might want to choose the degrees for the Gauß-Legendre method differently for the two levels, the computations of the inner and the outer integrals. However, we never did so.

  * Our experience is that the approximations obtained start to get useful for
degree 30
[i.e. order 60],
and that the calculations start to become slow at
degree 100,
while
degree 300
appears to be some kind of acceptance limit for usability in practice.

  * To illustrate this, let us present some data related to Example <ref>, below. Working on a suitably chosen affine chart, there are four deformed lines,
four curves encircling a triangle, and six curves encircling a quadrangle occurring. The running times to compute the corresponding 14 integrals at various precisions, as well as the largest relative errors in comparison with degree 200, are summarised in the table below. We did the numerical calculations in floats of as many decimal digits as the degree chosen.







  §.§.§ Transcendental
2-cocycles


Given a concrete
K3 surface
of the type aforementioned, we usually apply the main construction <ref> to all curves encircling a polygon and all deformed lines occurring on a fixed affine chart. This pro­vides by far more than six representatives of classes
in H^2(X, ) / P.

[Determining the cup product
on P^⊥⊗_,
up to scaling]



Let X = X_(a_0,b_0,c_0,d_0)
be a
K3 surface
given as the minimal desingularisation of a double cover of the form (<ref>), for
a_0, b_0, c_0, d_0 ∈.
Moreover, let
α_1, …, α_n→ X
be tori of the kind that the classes
c_α_1, …, c_α_n∈ H^2(X, )/P
form a system of generators.



  * Choose open neighbourhoods
≅ U(a_0) ∋ a_0,
…,
≅ U(d_0) ∋ d_0
in such a way that, for every
(a,b,c,d) ∈ U := U(a_0) ×⋯× U(d_0),
no three of the resulting six lines
in ^2_
have a point in common. Then the cohomology classes
c_α_1, …, c_α_n
uniquely extend to the whole family
→ U
of the
K3 surfaces
X_(a,b,c,d).

Moreover, choose
N
surfaces
X_1, …, X_N
at random from the family and write down the corresponding holomorphic

ω_1, …, ω_N.
(We usually work with
N=50.)

  * 
Set up the matrix
M := (c_α_j, [ω_i])_1≤ i≤ N, 1≤ j≤ n
using numerical integration and calculate the singular value decomposition
of M.
Six singular values should be numerically nonzero. The others give rise to linear relations among the cohomology classes
c_α_1, …, c_α_n.

  * Choose a basis
c_1, …, c_6
of the free

spanned by
c_α_1, …, c_α_n
modulo the relations found.

  * Build from
M
the

F := (c_j_1, [ω_i]) (c_j_2, [ω_i])_1≤ i≤ N, 1≤ j_1 ≤ j_2 ≤ 6.
Then determine an approximate solution of the corresponding homogeneous linear system of equations, using the QR-factorisation
of F.
I.e., detect the one linear relation between the 15 products that is approximately fulfilled for
all i.
This solution vector describes the symmetric, bilinear pairing desired.






  * As it relies only on the projective quadric
Q_κ,16⊂^5()
defined by the symmetric, bilinear 
form κ
(cf. Corollary <ref>, in particular formula (<ref>)) and not on
κ itself,
our method is inherently limited to determining the cup product pairing,
or κ,
up to scaling.

  * Algorithm <ref> works with the restricted period space. Therefore, it detects only the restriction of the cup product pairing
to P^⊥⊗_⊂ H^2(X, ).

  * Every class in
H^2(X, ) / P
has a unique representative
in P^⊥⊗_,
the orthogonal projection

    H^2(X, ) / P ⟶ P^⊥⊗_

being injective. (Note that
P ⊂ H^2(X, )
is a sublattice not of discriminant
±1,
so the image
of 
indeed contains non-integral classes.) Thus, we actually compute the pairing
((·), (·)) H^2(X, ) × H^2(X, ) →,
up to scaling.


  * The algorithm yields the matrix describing the cup product pairing, up to scaling, with respect to
(c^1, …, c^6),
but provides the dual basis
(c_1, …, c_6).
The corresponding matrices are, in fact, inverse to each other. Indeed,

    ∑_k (u, c_k) (c^k, v) = ( u, ∑_k (c^k, v) c_k ) = (u, v)

and therefore
∑_k (c_i, c_k) (c^k, c^j) = δ_ij.




Often, one may detect the cup product pairing (up to scaling) relying only on a single surface (instead
of N = 50),
using an LLL-based approach. Indeed, it seems plausible that the cohomology classes of the tori constructed are rather small within the transcendental lattice. Thus, between their periods, being irrational numbers known at high precision, there should be no further small quadratic relations besides that coming from the cup product pairing. 




Let α→ X
be a torus constructed from a deformed line, as in Figure <ref>.
Then ((c_α), (c_α)) = -1.

Proof.
The expression
((c_α), (c_α))
is purely cohomological, and therefore homotopy invariant. Up to homotopy, the three double points may be assumed to be collinear. Without restriction, the deformed
line Γ
is then actually a line, parametrised by
γ(t) = tb,
i.e.γ_0(t) ≡ 0.

In this situation, the main construction <ref> yields that

    α' |_^2 = γ' ^2 ⟶^2 ⊂^2()

is given by
(t,u) ↦ (t +  u)b.
Thus, α' |_^2
is a holomorphic embedding onto an affine line contained
in ^2().
In other words, the torus
α',
as described in (<ref>) and (<ref>), parametrises, generically one-to-one, the complex projective line
g' ⊂^2()
through the three double points.

The inverse image
of g'
in X'
splits into two complex algebraic curves both being rational as the branch locus consists entirely of double points. Thus, the lift
α”→ X'
parametrises, generically one-to-one, one such
component g”.
And
α_Γ→ X
parametrises the strict transform
g ⊂ X
of g”.
Consequently,
c_α = [g].

One has
[g]^2 = -2,
[g] ·π^*[l] = 1,
and
[g][E_1] = [g][E_2] = [g][E_3] = 1,
for
E_1,
E_2,
and E_3
the exceptional points over the double points met
by Γ.
Clearly, [g]
is perpendicular to the classes of the three other exceptional curves. As for algebraic classes, the cup product pairing coincides with the intersection pairing <cit.>, using the facts that
[E_i]^2 = -2
and
π^*[l] ·π^*[l] = 2,
one finds that
(c_α) = D,
for
D
the


    D := [g] - 1/2π^*[l] + 1/2 [E_1] + 1/2 [E_2] + 1/2 [E_3]  .

Finally,
D^2 = -1
is easily seen by a direct calculation.




We use the observation made in Lemma <ref> for scaling.




Let X
be the minimal desingularisation of a double cover
of ^2_,
ramified over a union of six real lines, such that no three of them have a point in common. Then the classes of the tori, as described in Proposition <ref> below, always generate the whole
of H^2(X, ) / P.

Proof (depending on numerical integration).
There are four essentially distinct configurations of six real lines
in ^2,
no three of which have a point in common <cit.>. We made an experiment for each case.




Our approach to find example configurations for each of the four types was invariant-theoretic and used only a small proportion of the theory developed in <cit.>. Some details are as follows.

The admissible configurations of six real lines correspond under projective duality to arrangements of six real points
(x_1:y_1:z_1), …, (x_6:y_6:z_6) ∈^2,
no three of which are collinear. I.e., such that each of the twenty determinants

    (
    [ x_i_1 x_i_2 x_i_3; y_i_1 y_i_2 y_i_3; z_i_1 z_i_2 z_i_3 ]) ,

for
1 ≤ i_1 < i_2 < i_3 ≤ 6,
is nonzero.

Thus, the vector of signs in
(±1)^10
of the ten products

    (
    [ x_i_1 x_i_2 x_i_3; y_i_1 y_i_2 y_i_3; z_i_1 z_i_2 z_i_3 ])
    ·(
    [ x_i_4 x_i_5 x_i_6; y_i_4 y_i_5 y_i_6; z_i_4 z_i_5 z_i_6 ]) ,

for
{i_1,…,i_6} = {1,…,6},
i_1 < i_2 < i_3,
i_4 < i_5 < i_6,
and
i_1 < i_4
is an invariant of the configuration, up to the action
of S_6
permuting the six points and up to that
of
(/2)^6,
by sign changes. However, the latter group acts via the summation character
+ (/2)^6 →/2.
I.e., there is either no change at all or all ten signs are changed simultaneously.

Working with a few explicit example configurations, one readily finds four distinct types, having invariant vectors as follows.







  * We constructed our example configurations systematically. We assumed without restriction that the first four points are
(1:0:0),
(0:1:0),
(0:0:1),
and (1:1:1).
The complement
in ^2()
of the six lines through these points breaks into
6·5/2 + 1 - 4·2·1/2 = 12
chambers. Thus, for the choice of the fifth pint, there are only twelve essentially different cases. Moreover, each time, the complement of the ten lines through the five points obtained is a disjoint union of
10·9/2 + 1 - 5·3·2/2 = 31
chambers <cit.>. This means that all cases are covered by
31 · 12 = 372
explicit configurations.

Calculating the invariants, we produced a partition of these 372 configurations into four subsets. We then checked that two configurations in the same subset were indeed always equivalent to each other, under the operation
of S_6,
followed by a renormalisation of the first four points to
(1:0:0),
(0:1:0),
(0:0:1),
and (1:1:1),
plus a naive relocation of points within the same chamber.

  * This means that our approach actually provides a new proof for the fact that there are exactly four types of configurations of six real lines
in ^2,
no three of which have a point in common. Our proof is essentially computational and thus rather different from the one presented in <cit.>.

  * The group
S_6,
of course, does not give rise to all permutations
in S_10,
but to a subgroup
of S_10
of index
5040.
We ignored the information that is perhaps carried by this subgroup and just took care of the multisets that are associated with the invariant vectors.

  * Plotting an example of each type and counting triangles, quadrangles, pentagons, and hexagons, respectively, one finds that the four types correspond, in this order, to the types O, I, II, and III in the terminology of M. Yoshida <cit.>.


[4]



§ REAL AND COMPLEX MULTIPLICATION. HODGE STRUCTURES





  §.§.§ The concepts


[Deligne]


  * A (pure
-)
Hodge structure of weight
i
is a finite dimensional
-vector
space
V
together with a decomposition

    V_ := V ⊗_ = H^0,i⊕ H^1,i-1⊕…⊕ H^i,0  ,

having the property that
H^m,n = H^n,m
for every
m,n ∈_≥0
such that
m+n=i.

A morphism
f V → V'
of (pure
-)
Hodge structures is a
-linear
map such that
f_ V_→ V'_
respects the decompositions.

  * A polarisation on a pure
-Hodge
structure
V
of even weight is a nondegenerate symmetric bilinear form
(·,·) V × V →
such that its
-bilinear
extension
(·,·) V_× V_→
satisfies the two conditions below.



   ∙ 
One has
(x,y) = 0
for all
x ∈ H^m,n
and
y ∈ H^m',n'
such that
m ≠ n'.

   ∙ 
The inequality
^m-n (x,x) > 0
is true for every
0 ≠ x ∈ H^m,n.





The weight
i
Hodge structures form an abelian category <cit.>. Furthermore, the subcategory of polarisable Hodge structures is semisimple <cit.>. I.e., every sub-Hodge structure of a polarisable Hodge structure is a direct summand.

Thus, every polarisable Hodge structure can be written as a direct sum of indecomposable subobjects, which are called primitive Hodge structures.


[Zarhin]
A Hodge structure of
K3 type
is a primitive polarisable Hodge structure of

such
that _ H^2,0 = 1.



Let X
be a compact complex manifold that is Kähler.



  * Then
H^j(X,)
is naturally a polarisable pure

structure of
weight j.

  * For X
a
K3 surface,
the transcendental part
T := ( X ⊗_)^⊥⊂ H^2(X, )
is a Hodge structure of
K3 type.
Indeed, if
T
would split off a direct
summand T'
then, without restriction, one had
T'^2,0 = 0.
But this yields, according to the Lefschetz
(1,1)-Theorem,
T'
that must be contained in the algebraic part, a contradiction.





Let T
be a Hodge structure of
K3 type.



  * Then
E := _(T)
is either a totally real field or a CM field.

  * Suppose that
T
is equipped with a polarisation
(·,·).
Then every
φ∈ E
operates as a self-adjoint mapping, i.e.(φ(x), y) = (x, φ(y))
for
ι
the identity map in the case that
E
is totally real and the complex conjugation in the case that
E
is a CM field.


Proof.
Assertion i) is <cit.>, while assertion ii) is <cit.>.





  §.§.§ Real and complex multiplication–Our terminology


[RM and CM]
Let T ⊂ H^2(X, )
be the transcendental part of the cohomology of a complex
K3 surface X.



  * If
_(T) ⫌
is a totally real field then
X
is said to have real multiplication.

  * If _(T)
is CM then we speak of complex multiplication.




For complex multiplication, some authors, e.g. L. Taelman <cit.>, require, in addition, that
__(T) T = 1.
However, for our purposes, the definition above appears to be more practical.



It is a rather common phenomenon that only a subfield of the endomorphism field
_(T)
is known. Moreover, if the generic member of a family of
K3 surfaces
has real or complex multiplication by a certain field
K
then for particular members the endomorphism field may well be larger, cf. Corollary <ref>, below.

We introduce the terminology below in order to cope with such situations.


[T' ⊇ T
being acted upon by some field]



Let X
be a
K3 surface
and
T' ⊆ H^2(X, )
be a subvector space such that
T' ⊇ T,
for
T ⊂ H^2(X, )
the transcendental part. We say that
T'
is acted upon by a
field K,
if there is a ring homomorphism
K →_(T')
taking
1
to 𝕀_T'.





  * In the situation of Definition <ref>,
T'
is automatically a sub-Hodge structure
of H^2(X, ).

  * If T'
is acted upon by
K
then
T
is clearly acted upon
by K.

Moreover, if T
is acted upon
by K ⫌
then
X
has real or complex multiplication by
K
or an extension field
of K.





  §.§.§ Arithmetic effects caused by real or complex multiplication



Let
X
be an algebraic
K3 surface
defined over a
field k
that is finitely generated
over .
Then we say that
X
has real multiplication or complex multiplication by a number
field E
if, for a certain embedding
k ↪
of fields, the complex
K3 surface
X() = (X ×_ k)() has.



This is independent of the choice of an embedding
k ↪
<cit.>.
The endomorphism field itself is independent, for
_ T < 7
<cit.> or when
k
is primary <cit.>.


For a
K3 surface
over a number field, the properties of RM and CM cause a particular arithmetic behaviour. We summarise the known effects below.


Let X
be a
K3 surface
over 
and
p
a prime number. We say that
X
has good reduction
at p,
if there exists a proper model

of X
over ,
the reduction
X_p := _p
modulo p
of which is again a
K3 surface.

In this case,
for f ∈,
we let
χ_p^f∈[Z]
be the characteristic polynomial of
^f ∈(_p/_p)
on
H^2_((X_p)__p, _l(1))
<cit.>.

We factorise
χ_p^f
completely in the form

    χ_p^f(Z) = χ^_p^f(Z) ·∏_i=1^d (Z - ζ_k_i^e_i)  ,

for
k_1,…,k_d ∈.
I.e. in such a way that
χ^_p^f∈[Z]
does not have any further zeroes being roots of unity.

According to the Tate conjecture <cit.>,
χ^_p^f
is the characteristic polynomial of
^f
on the transcendental part of
H^2_(X__p, _l(1)).
Let us note, in particular, that
χ^_p^f = 22 -  X__p.






Let
X
be a

over .

Suppose that
X
has real or complex multiplication by a number
field E.
Let, moreover,
p
be a prime of good reduction.



  * Then (X_p)__p-  X()
is divisible
by [E:].

  * Assume that
E ⊇(√(δ)),
for a non-square
δ∈^*.


  * Then one of the following is true.
Either χ^_p
splits over
(√(δ)).
Or, for a certain
f > 0,
the polynomial
χ^_p^f
is a square.

  * If the
prime  p
is inert
in (√(δ))
then the reduction
X_p
is non-ordinary. I.e., one has that
#X_p(_p) ≡ 1  p.



Proof.
A)
This result appears between the lines of F. Charles' famous article <cit.>, but was not explicitly stated. A proof is provided in <cit.>.

B)
Assertion B.a) is shown in <cit.>, while B.b) is <cit.>.







Let a,D ∈
be such that
(a,D) = 1
and X
a K3 surface
over .
Suppose that
#X_p(_p) ≡ 1  p
for every good
prime p ≡ a  D.
Then X
has real or complex multiplication.

Proof.
This is <cit.>.




A relative version of Proposition <ref> is established in <cit.>.




  §.§.§ Some known examples I




Let f_4
be a homogeneous quartic form in three variables that defines a regular curve
C ⊂^2.
Then the fourfold cover
X w^4 = f_4(x,y,z)
ramified
at C
is a
K3 surface
of
degree 4
and Picard rank at
least 8.
It has an automorphism, given by

    I (w:x:y:z) ↦ ( w:x:y:z)  .

Moreover, there is a

subvector space
T' ⊂ H^2(X, )
containing T
that is acted upon
by (√(-1)).
I.e., X
has complex multiplication by
(√(-1))
or by a field that properly contains
(√(-1)).

Proof.
The automorphism
J := I ∘ I
is of
order 2
and has the
curve C
as its fixed point set. The genus
of C
is 3,
hence the topological Euler characteristic is equal
to (-4).
Therefore, the Lefschetz trace formula <cit.>, cf. <cit.>, shows that
J |_H^2(X, ) = -6.
In other words,
J |_H^2(X, )
has the eigenvalue
1
with
multiplicity 8,
while the eigenvalue
(-1)
occurs with multiplicity 14.

On the other hand,
X
is a double cover of the degree two del Pezzo
surface X',
given by
w^2 = f_4(x,y,z).
One has
X' ≅^8
and the natural homomorphism
π^* X' → X
is an injection, as it doubles all intersection numbers.
Thus, J
acts nontrivially, with only eigenvalue
(-1),
on the 
i­men­sional
orthogonal complement of
(π^* H^2(X', ) → H^2(X, )).





Let f_2
and f_3
be homogeneous forms in four variables such that the subscheme
X ⊂^4,
given by
f_2(x,y,z,u) = 0
and v^3 = f_3(x,y,z,u)
is regular.
Then X
is a
K3 surface
of
degree 6.
It has an automorphism, given by

    I (v:x:y:z:u) ↦ (ζ_3 v:x:y:z:u)  .

Moreover, there is a

subvector space
T' ⊂ H^2(X, )
containing T
that is acted upon
by (√(-3)).

Proof.
The automorphism
I
is of
order 3
and has the curve

    C   f_2(x,y,z,u) = f_3(x,y,z,u) = 0  ,  v=0  ,

as its fixed point set. This is a canonical curve of
genus 4,
such that the Lefschetz trace formula yields
J |_H^2(X, ) = -8.
In other words, the eigenvalue
1
has
multiplicity 2,
while the eigenvalues
ζ_3
and ζ_3^-1
both occur with
multiplicity 10.

On the other hand,
X
is a threefold cover of the space quadric, given by
X' f_2(x,y,z,u) = 0.
One has
X' ≅^2
and the natural homomorphism
X' → X
is an injection, tripling all intersection numbers.
Thus, J
acts nontrivially, with eigenvalues only
ζ_3
and ζ_3,
on the 
i­men­sional
orthogonal complement of the image of
H^2(X', )
in H^2(X, ).





Let f_6
be a homogeneous sextic form in three variables that defines a regular curve
C ⊂^2.
Then the double cover
X w^2 = f_6(x,y,z)
ramified
at C
is a
K3 surface
of
degree 2.

Suppose that
f_6(x,y,z) = ζ_3 f_6(y,z,x).
Then X
has an automorphism, given by

    I (w, x:y:z)    ↦ (ζ_6 w, y:z:x)  .

Moreover, there is a

subvector space
T' ⊂ H^2(X, )
containing T
that is acted upon
by (√(-3)).
I.e., X
has complex multiplication either by
(√(-3))
or by a field that contains
(√(-3)).
Furthermore, X ≥ 8.

Proof.
The automorphism
J := I ∘ I ∘ I
of X
is of
order 2
and has the
curve C
as its fixed point set. The genus
of C
is 10,
hence the topological Euler characteristic is equal
to (-18).
Therefore, the Lefschetz trace formula shows that
J |_H^2(X, ) = -20.
This means that
J
acts nontrivially, with only eigenvalue
(-1),
on the 
i­men­sional
orthogonal complement of the inverse image of a general line on
^2.
Consequently, I
acts on this space with eigenvalues at most
(-1),
ζ_6,
and ζ_6^-1.

On the other hand, the fixed point set
of I
is
{(0, 1:1:1), (0, 1:ζ_6^2:ζ_6^4), 0, 1:ζ_6^4:ζ_6^2)},
which yields that
I |_H^2(X, ) = 1.
Consequently, each of the eigenvalues
(-1),
ζ_6,
and ζ_6^-1
occurs with
multiplicity 7.
The direct sum of the eigenspaces for
ζ_6,
and ζ_6^-1
has the property required. Finally, the assertion on the Picard rank follows from <cit.>.




Let f_5
be a homogeneous quintic form in three variables that defines a regular curve
C ⊂^2.
Suppose that
C ∩ V(z)
consists of five distinct points.

Then the minimal
desingularisation X
of the double cover
X' w^2 = z f_5(x,y,z)
is a
K3 surface
of Picard rank at
least 6.

Suppose that
f_5(x,y,-z) = f_5(x,y,z).
Then X
has an automorphism, given by

    I (w, x:y:z) ↦ (i w, x:y:-z)   .

Moreover, there is a

subvector space
T' ⊂ H^2(X, )
containing T
that is acted upon
by (√(-1)).
I.e., X
has complex multiplication either by
(√(-1))
or by a field that contains
(√(-1)).

Proof.
Here, the automorphism
J := I ∘ I
of X
is of
order 2.
Its fixed point set is the union of the strict transforms of
C
and V(z)
in ^2,
blown up in the five points
of C ∩ V(z).
Note that this is a disjoint union. As the genera of
C
and V(z)
are
6
and 0,
the topological Euler characteristic
of C ∪ V(z)
is equal
to (-10) + 2 = (-8).
Thus, the Lefschetz trace formula implies
J |_H^2(X, ) = -10.
This means that
J
acts nontrivially, with only eigenvalue
(-1),
on the 
i­men­sional
orthogonal complement of the span of the five exceptional curves and the inverse image of a general line on
^2.



More generally, when a
K3 surface
X
has an automorphism of finite order that operates nontrivially
on H^2,0(X)
then
X
has complex multiplication. More precisely,
T ⊂ H^2(X, )
is acted upon by a cyclotomic field. Such
K3 surfaces
have been intensively studied, there is even a classification for the case that the order is a prime number. The interested reader is advised to consult the article <cit.> and the references given therein.




  §.§.§ Some known examples II


Real multiplication tends to be more complicated than complex multiplication. In particular, no examples are known, in which real multiplication occurs due to an automorphism.

Instead, the examples presented below were found searching excessively for
K3 surfaces
showing the arithmetic abnormalities predicted by Proposition <ref>.

[A family of
K3 surfaces
for which RM
by (√(2))
is established]

Let
q→ B,
for B :=[T,1/T(T^2-2)(T^2+2)(T^2-4T+2)(T^2+4T+2)] ⊂^1_,
be the family of
K3 surfaces,
the fibre
at t ∈ B
of which is the minimal desingularisation of the double cover
of ^2,
given by

    w^2     =    
     [(1/8 t^2 -1/2 t +1/4)y^2 + (t^2 - 2t + 2)yz + (t^2 - 4t + 2)z^2] 
          
     [(1/8 t^2 +1/2 t +1/4)x^2 + (t^2 + 2t + 2)xz + (t^2 + 4t + 2)z^2] [2x^2 + (t^2 + 2)xy + t^2y^2]   .



  * Then the geometric generic fibre
_η
of q
is of Picard
rank 16.

  * For every
θ∈ B(),
the transcendental part
T ⊂ H^2(_θ(), )
of the cohomology of the fibre
_θ()
of the holomorphic submersion
q()() → B()
is acted upon
by (√(2)).

  * Let the complex point
θ∈ B()
be of the kind that the fibre
_θ()
of q()
has Picard
rank 16.
Then
_θ()
has real multiplication
by (√(2)).


Proof.
This is <cit.>. The family was presented in <cit.> for the first time.



For θ∈,
the field of definition of the Picard group is
(√(2)), too.


[A family of
K3 surfaces
for which RM
by (√(5))
is established]

Let
q→ B,
for
B :=[T,1/(T-1)(T^4-T^3+T^2-T+1)] ⊂^1_,
be the family of
K3 surfaces,
the fibre
at t ∈ B
of which is the minimal desingularisation of the double cover
of ^2,
given by

    w^2    =    y(x - 2(t-1)y - tz) 
           (x^4 + x^3y - x^3z + x^2y^2 - 2x^2yz + x^2z^2 + xy^3 - 3xy^2z - 2xyz^2 - xz^3 + y^4 
          + y^3z + y^2z^2 + yz^3 + z^4)   .




  * Then the generic fibre
_η
of q
is of geometric Picard
rank 16.

  * For every
θ∈ B(),
the transcendental part
T ⊂ H^2(_θ(), )
of the cohomology of the fibre
_θ
is acted upon
by (√(5)).

  * Let the complex point
θ∈ B()
be of the kind that the fibre
_θ
has geometric Picard
rank 16.
Then
_θ
has real multiplication
by (√(5)).


Proof.
This is <cit.>.



Here, for
θ∈,
the field of definition of the Picard group is
(ζ_5).






  * Several further families have been found, for the generic members of which there is strong evidence for real multiplication by specific fields, including
(√(2)),
(√(3)),
and (√(5)).
Furthermore, there are isolated examples that conjecturally have complex multiplication by
(√(-1), ζ_7 + ζ_7^-1),
(√(-1), ζ_9 + ζ_9^-1),
and
L(√(-1)),
respectively, for
L ⊂(ζ_19)
the unique cubic subfield. Explicit equations are given in <cit.>.

For each of these surfaces, it has been verified that the arithmetic effects caused by real or complex multiplication (cf. Proposition <ref>) occur for all primes
p < 500.
This required fast algorithms for point counting on algebraic surfaces over finite fields, cf. <cit.> or <cit.> for the main ideas behind them. By now, an implementation of the method we used is available to the public, as the magma intrinsic WeilPolynomialOfDegree2K3Surface.

  * Moreover, there are a few isolated examples
for (√(13)).
For instance, there is the following.





Let X
be the minimal desingularisation of the double cover
of ^2,
given by

    w^2 = (x - 4z)(5x - 9y - 8z)f_4(x,y,z)  ,

for f_4 := x^4 - 2x^3y - 5x^2y^2 - 26x^2z^2 + 6xy^3 + 104xyz^2 + 9y^4 - 130y^2z^2 + 52z^4.



  * Then X
is a
K3 surface
over 
of geometric Picard
rank 16.

  * There is strong evidence that
X() has
real multiplication
by (√(13)).


Proof of i).
The quartic
form f_4
is the norm of a linear form over the cyclic quar­tic number field

    K = (√(13 - 3 √(13)))

of
conductor 8 · 13 = 104.
Thus, the branch locus is the union of six lines in general position. This shows that
X_≥ 16,
while an upper bound
of 17
may, once again, be obtained using van Luijk's method. Let us note that
X
has bad reduction only at the primes
2,
3,
and 13.

In order to verify that
X_≠ 17,
a modification of the method described in <cit.> applies. Indeed, the
(/)-representation
H^2_((X_p)__p, _l(1))
splits into a direct summand of
dimension 16
corresponding to the obvious rank 16 part of
X_
and a
complement V_6.
Geometric Picard
rank 17
would cause a free

of rank one being contained 
in V_6
that is acted upon
by (/).
However, the characteristic polynomial of
_5
on V_6
turns out to be
(Z^2 + 1)(Z^4 - 2/5 Z^2 + 1).
Thus, there is no eigenvalue
±1,
which is a contradiction.


ii)
Evidence for real multiplication is as follows. For every good
prime p < 500,
the characteristic polynomial
χ^_p
splits over
(√(13))
or
χ^_p^2
is a square, as predicted by Proposition <ref>.B.a). Moreover, for every good prime
p ≡ 2,5,6,7,8,11 13
up
to 1000,
one has
#X(_p) ≡ 1  p,
cf. Proposition <ref>.B.b). In fact,
#X'(_p) = p^2+p+1
for X'
the double cover
of ^2
underlying X.



For the surface in Example <ref>, the Picard group is defined
over
K = (√(13 - 3 √(13))).




  §.§.§ Periods of
K3 surfaces
having real or complex multiplication




Let r ∈{1,…,20}
be an integer and
κ
a perfect pairing
on ^22.



  a) (Periods of the families generically having RM.) 


Let K
be a totally real number field of
degree d.
Then there is an at most countable
union M_K,κ,r⊆ Q_κ,r
of quadrics of dimension
22-r/d-2
such that the following holds.

-1
Let x ∈Ω_κ,r⊂ Q_κ,r
be the period point of a marked
K3 surface
(X, i),
for which
c_k ∈ H^2(X, )
(cf. <ref>.ii)) is algebraic,
for k = 22-r+1, …, 22,
and the Picard rank
of X
is exactly r.
Then T ⊂ H^2(X, )
is acted upon
by K
if and only if
x ∈ M_K,κ,r.

  b) (Periods of the families generically having CM.) 


Let K
be a CM field of
degree d.
Then there is an at most countable union
M_K,κ,r⊆ Q_κ,r
of projective subspaces of dimension
22-r/d-1
such that the following is true.

-1
Let x ∈Ω_κ,r⊂ Q_κ,r
be the period point of a marked
K3 surface
(X, i),
for which
c_k ∈ H_2(X, )
(cf. <ref>.ii)) is algebraic,
for k = 22-r+1, …, 22,
and the Picard rank
of X
is exactly r.
Then T ⊂ H^2(X, )
is acted upon
by K
if and only if
x ∈ M_K,κ,r.


Proof.
By Lemma <ref>, one has that
X = (i(x))^⊥⊇(c^1, …, c^22-r)^⊥.
However, the assumption
= r
ensures equality, such that the transcendental lattice
of X is
T = ( X ⊗_)^⊥ = (c^1, …, c^22-r).

a)
Being acted upon by the totally real
field K
means that
K operates

on T,
keeping the distinguished cohomology class
x_1 c^1 + ⋯ + x_22-r c^22-r
as a simultaneous eigenvector.

For this, take a primitive element
u ∈ K.
To fix the operation
of K,
one simply has to choose a self-adjoint endomorphism
U T → T,
whose minimal polynomial is the same as that
of u.
If this is possible then
d | (22-r).
Moreover, there are certainly only countably many choices. Thus, let us consider
U
as being fixed.

Then U_
has
d eigenvalues,
the numbers
σ_1(u), …, σ_d(u),
for
σ_i K →
the real embeddings, and the eigenspaces
E_1, …, E_d
are of dimension
22-r/d each.
Since U
is self-adjoint, the eigenspaces are perpendicular to each other. In particular, they are non-degenerate quadratic spaces.

To complete the proof, let us note that, for
T being
acted upon
by K,
the vector
x_1 c^1 + ⋯ + x_22-r c^22-r
needs to be contained in
E_i ⊗_,
for one of the eigenspaces.

b)
Take a primitive element
u ∈ K.
Similarly to the above, for
T
being acted upon by the CM field
K,
one has to choose an endomorphism
U T → T,
whose minimal polynomial is the same as that
of u.
Again, if this is possible then
d | (22-r),
and, again, there are only countably many choices, so that we may consider
U
as being fixed.

Then U_
has
d eigenvalues,
the numbers
σ_1(u), …, σ_d(u),
for
σ_i K →
the complex embeddings, and the eigenspaces
E_1, …, E_d
are of dimension
22-r/d each.
And, clearly, for
T
being acted upon
by K,
the vector
x_1 c^1 + ⋯ + x_22-r c^22-r
needs to be contained in one of the
eigenspaces E_i.
This provides us with projective subspaces of dimension
22-r/d-1,
as desired.

Moreover, U
must fulfil, together with the linear map
V,
associated with
u,
the self-adjointness relation
(U_(x), y) = (x, V_(y)).
In particular,
for x ∈ E_i,
this yields

    σ_i(u) (x, x) = (U_(x), x) = (x, V_(x)) = σ_i(u) (x, x)

and hence
(x, x) = 0,
as 
σ_i(u)
is certainly non-real, when
K is
a CM field and
u
a primitive element. This shows that the projective subspaces
(E_i),
associated with the
eigenspaces E_i,
are already contained in the
quadric Q_κ,r.



[Semicontinuity]

Let r ∈{1,…,20}
be an integer and
κ
a perfect pairing
on ^22.



  a) (RM) 
Let K
be a totally real number field of
degree d.
Assume that
22-r/d≥ 3.
Then there is an at most countable
union V_K,κ,r⊂ M_K,κ,r
of analytic subsets such that the following is true.

Let x ∈Ω_κ,r⊂ Q_κ,r
be the period point of a marked
K3 surface
(X, i),
for which
c_k ∈ H^2(X, )
(cf. <ref>.ii)) is algebraic,
for k = 22-r+1, …, 22.
Then X
has real multiplication
by K
and
X = r
if and only if
x ∈ M_K,κ,r∖ V_K,κ,r.

  b) (CM) 
Let K
be a CM field of
degree d.
Assume that
22-r/d≥ 2.
Then there is an at most countable
union V_K,κ,r⊂ M_K,κ,r
of analytic subsets such that the following is true.

Let x ∈Ω_κ,r⊂ Q_κ,r
be the period point of a marked
K3 surface
(X, i),
for which
c_k ∈ H^2(X, )
(cf. <ref>.ii)) is algebraic,
for k = 22-r+1, …, 22.
Then X
has complex multiplication
by K
and
X = r
if and only if
x ∈ M_K,κ,r∖ V_K,κ,r.


Proof.
We present the proof only for a), as that for b) is completely analogous.

There are two ways for the property stated
on X
to fail. Either
X > r
or
X = r,
but
_(T) ⫌ K.

Case 1:
X > r.

This means that, besides the linear combinations of
c_22-r+1,…,c_22,
a further cohomology class
c ∈ H^2(X, )
is algebraic. According to Lemma <ref>, that is equivalent to
x ∈ (i^* c)^⊥,
which clearly defines an analytic subset. As there are only countably many possibilities
for c,
this case indeed contributes
to V_K,κ,r
a countable union of analytic subsets.

Case 2:
X = r
and
_(T) ⫌ K.

If _(T) = K' ⫌ K
then
x ∈ M_K',κ,r,
which, according to Theorem <ref>.a), defines a countable union of analytic subsets
of M_K,κ,r.
As there are, up to isomorphism, only countably many number fields, the assertion follows.




This shows, in particular, the following. As long as the Picard rank remains unchanged, the endomorphism field
_(T)
cannot shrink under specialisation. This fact actually has been obtained before and in more generality <cit.>. The restrictive assumption on the Picard ranks is in fact unnecessary.




§ TRACING THE PREIMAGE OF A CURVE IN THE PERIOD SPACE



As a particular case of Theorem <ref>.a), we see that in a sufficiently general family of
K3 surfaces
of Picard
rank 16,
not containing an isotrivial subfamily, those surfaces that are acted upon by a real quadratic number field form families over
curves C ⊂ Y,
cf. <cit.>.




Let X
be an isolated example of a
K3 surface
that has real multiplication by a quadratic
field (√(d)).
Assume that
X
is given as the minimal desingularisation of a double cover of the form (<ref>). The strategy below describes how to find the

family of RM surfaces,
X
belongs to.



  * Run Algorithm <ref>
on X.
The resulting elements
c_α_1, …, c_α_6∈ H^2(X, )/P
yield a class of markings
on X,
as described in Theorem <ref>.

As step <ref>, Algorithm <ref> includes running Algorithm <ref>. In particular, open neighbourhoods
≅ U(a_0) ∋ a_0,
…,
≅ U(d_0) ∋ d_0
are chosen in such a way that, for every

    (a,b,c,d) ∈ U := U(a_0) ×⋯× U(d_0)  ,

no three of the resulting six lines
in ^2_
have a point in common. Thus, any
marking i
from the class above extends to the whole family
over U.
There is the associated restricted period map

    Π U ⟶^5(),    (a,b,c,d) ↦Π_X_(a,b,c,d), i_(a,b,c,d) ,

which is independent of the choice
of i,
cf. Theorem <ref>.c).






  * Calculate the period point
of X = X_(a_0,b_0,c_0,d_0),
based on Theorem <ref>, using numerical integration, and identify the three linear relations between the six periods that encode real multiplication.

These define, together with the quadric induced by the cup product pairing, a conic
C ⊂^5()
in the restricted period space.

  * 
Trace the curve
Π^-1(C) ⊂ U ≅^4
using a numerical continuation method <cit.>.

  * Use the singular-value decomposition in order to find algebraic relations between the coordinates of the points found. Control, by using Gröbner bases, that they indeed define an irreducible algebraic curve.


We provide a few details on how this strategy was implemented in Remarks <ref>, below.




  §.§.§ The result




Consider the family of double
covers X'_(a,b,c,d)
of ^2,
given by

    w^2 = (x + ay + bz)(x + cy + dz)f_4(x,y,z)  ,

for f_4 := x^4 - 2x^3y - 5x^2y^2 - 26x^2z^2 + 6xy^3 + 104xyz^2 + 9y^4 - 130y^2z^2 + 52z^4.



  a.i) 
Then the branch locus is the union of six lines, which are in general position for a generic choice
of (a,b,c,d) ∈^4.

  a.ii) 
Consider the closed subscheme
C ⊂^4,
given by the equations

    0    =    630 272a - 11 421bd^5 + 411 400bd^3 - 871 552bd - 272 976c^2d^2 + 315 136c^2 
           + 98 982cd^4 - 3 508 064cd^2 + 2 205 952c + 233 496d^4 - 6 409 856d^2 + 4 411 904 , 
    
    0    =    78 784bc - 243bd^4 + 37 040bd^2 + 110 528b - 5808c^2d + 2106cd^3 - 319 792cd 
           + 4968d^3 - 714 688d ,
    
    0    =    243bd^6 - 8960bd^4 + 29 952bd^2 - 26 624b + 5808c^2d^3 - 11 648c^2d - 2106cd^5 
           + 76 432cd^3 - 144 768cd - 4968d^5 + 140 608d^3 - 259 584d , 
    
    0    =    2c^3 + 28c^2 - 3cd^2 + 98c - 8d^2 + 104 .

Then C
is a geometrically irreducible, nonsingular curve of
genus 1.

  b) 
Moreover, there is strong evidence that, for generic
(a,b,c,d) ∈ C(),
the
K3 surface
X_(a,b,c,d)
obtained as the minimal desingularisation
of X'_(a,b,c,d)
is of Picard
rank 16
and has real multiplication
by (√(13)).


Proof of a).
i)
Putting (a,b,c,d) := (0,-4,-9,-8),
the surface from Example <ref> is obtained.

ii)
This is easily obtained by a calculation in any computer algebra system.

The curve
C
is the result of Strategy <ref>, taking the surface from Example <ref> as the starting point.


Evidence for b).
For every prime
p < 500
and every
(a_0,b_0,c_0,d_0) ∈ C(_p)
of the kind that
X_(a_0,b_0,c_0,d_0)
is a nonsingular surface
over _p,
the characteristic
polynomial χ^_p
either splits over
(√(13))
or
χ^_p^2
is a square, as predicted by Proposition <ref>.B.a). Moreover, for every prime
p ≡ 2,5,6,7,8,11 13
up
to 1000
and every
(a_0,b_0,c_0,d_0) ∈ C(_p),
one has that
#X_(a_0,b_0,c_0,d_0)(_p) ≡ 1  p,
cf. Proposition <ref>.B.b). To be more precise,
#X'_(a_0,b_0,c_0,d_0)(_p) = p^2+p+1.



The genus 1
curve C
has

points. Taking any of them as the origin, the Mordell–Weil group
of C
is isomorphic
to .


[Some details on the experiment]



  * When running Algorithm <ref>, we used 14 transcendental cohomology classes, which were represented by tori and obtained as explained in Section <ref>.

For numerical integration, the Gauß-Legendre method of degree
30
[i.e.order 60]
was applied, using floats of 30 digits.

In step <ref> of Algorithm <ref>, we found six singular values within a factor
of 100,
while the next one was smaller by nine orders of magnitude.

In the basis chosen, the cup product form found
on P^⊥
had coefficients only from
{±1, ±1/2, 0},
up to errors less than
10^-10.

  * -1
In step <ref> of Strategy <ref>, in the language of <cit.>, we applied a predictor–cor­rec­tor method. More precisely, we used the Euler predictor <cit.>, followed by Newton corrector steps. We did not care too much about rounding errors, as we worked with floats of high precision.
Neither did we implement a step length adaptation, but worked, as simply as possible, with a constant step length. For numerical integration, the Gauß-Legendre method of degree
100
[i.e.order 200]
was used.

Based on this, we determined a sample of 101 points
on Π^-1(C) ⊂^4,
each with a numerical precision of 80 digits. Due to the constant stepsize, these points are essentially equidistant. There are further particularities, caused by the limitations of our approach. First of all, all the points are real. Moreover, they have a limited distance from the starting point, i.e. the parameters of the surface from Example <ref>, because we are forced to stop sampling when approaching the first singular surface along the path.

  * Polynomials of degree
≤3
in four variables form a vector space of
dimension 35.
When looking for cubic relations between the 101 points found, we ended up with 25 singular values in the range from
1714
to 6.08 · 10^-41,
the other ten being less
than 10^-80.
Thus, the curve sought is contained in an intersection of ten cubics
in ^4.
The equations given form a Gröbner basis for the ideal generated by them.





§ EXPLICIT DESCRIPTION OF TRANSCENDENTAL COHOMOLOGY CLASSES—
PROOFS OF THE MAIN RESULTS




  §.§.§ Almost C^1-maps


In the application below, it turns out to be convenient to work with the following technical condition on maps between smooth manifolds.



A continuous map
φ S → X
between two smooth manifolds
is called almost C^1,
if there exists a auxiliary
C^1-map
ι S' → S
being a homeomorphism from another smooth manifold
S'
that satisfies the following two conditions.



  * The map
φ∘ι S' → X
is C^1.

  * For a suitable Lebesgue null set
N ⊂ S',
the restriction
φ |_S ∖ι(N) S ∖ι(N) → X
is C^1.






  * Every
C^1-map
between smooth manifolds is
almost C^1.

  * The map
φ→, x ↦√(x),
is
almost C^1.






  * One has that
ι(N)
is a Lebesgue null set, according to <cit.>. Thus, being almost
C^1
implies being 
C^1
outside of a null set.

  * In particular, if
φ S → X
is almost
C^1
and
η
a
C^1
differential form
on X
then the pull-back
φ^* η
is defined as a

on S,
outside of a Lebesgue null set.

  * Rather generally, for a differential form
ξ
that is defined
on S,
up to a Lebesgue null
set N,
we put
∫_S ξ = ∫_S ∖Nξ,
as soon as the integral to the right is existing. 



From now on, as in the sections above,
X
always denotes a complex
K3 surface.

-1

Let the cohomology class
c_φ∈ H^2(X, )
be given by a compact oriented
 S,
together with an almost
C^1-map φ S → X
(cf. Definition <ref>). Moreover, let
w ∈ H^2(X, )
be represented by a closed, smooth
 η.
Then the extended cup product pairing may be evaluated as the

integral

    (c_φ,w) = ∫_S φ^*η .


Proof.
First step.
The case that
φ
is C^∞.

One has
(c_φ,w) = ⟨ w ∪ c_φ, z_X⟩ = ⟨φ^* (w), z_S⟩,
according to the definition of the cup product pairing and the first claim of Lemma <ref>.b). Interpreting the term to the right in the de Rham cohomology theory, one indeed has
⟨φ^* (w), z_S⟩ = ∫_S φ^*η.

Second step.
The case that
φ
is C^1.

There is a
C^1
homotopy
H S × [0,1] → X
connecting
φ
with a
C^∞-map φ,
cf. the proof of <cit.>. As homotopic maps induce the same homomorphism on cohomology, one has
c_φ = c_φ
and, therefore,
(c_φ,w) = (c_φ,w) = ∫_S φ^*η,
according to the first step.

Moreover, the homotopy formula, cf. <cit.>, shows that

    φ^*η - φ^*η = j_0^* H^*η - j_1^* H^*η = d(LH^*η) + L(dH^*η)  ,

for L E^1(S × [0,1]) → E^1(S)
the operator of integration along the fibre. Here,
dH^*η = 0,
as the form
η
is closed. And hence
∫_S φ^*η - ∫_S φ^*η = ∫_S d(LH^*η) = 0,
due to Stokes' Theorem <cit.>.

Third step.
The general case that
φ
is
almost C^1.

Let ι S' → S
be an auxiliary map
for φ,
as in Definition <ref>.
Since ι
is a ho­me­o­morphism, one has
c_φ = φ_!(1) = φ_!(ι_!(1)) = (φ∘ι)_!(1) = c_φ∘ι.
Consequently,

    (c_φ,w) = (c_φ∘ι, w) = ∫_S' (φ∘ι)^* η ,

according to the step before. The integral to the right is the same as

    ∫_S' ∖ N(φ∘ι)^* η = ∫_S' ∖ Nι^* (φ^* η)  .

Note here that
φ
is C^1
on ι(S' ∖ N) = S ∖ι(N).
Since ι
is a

and bijective, the last integral coincides with
∫_S ∖ι(N)φ^* η = ∫_S φ^* η,
as required.





  §.§.§ The main construction–Technical details


Our idea for the proof of Theorem <ref> is to apply Proposition <ref>. This means that we have to provide a model of α_Γ,
respectively α_Γ,b,
allowing a lift
to X
that is not only continuous, but
almost C^1.



  
Step 1. 

tori from compact
 – Adding an imaginary direction


As before, we extend the
C^1-map
γ→^2 ⊂^2()
to a
C^1-map
in two variables by putting

    γ'^2 ⟶^2 ⊂^2(),      (t,u) ↦γ(t) +  ub  .


Then
lim_t→±∞γ'(t, u)
exists
in ^2(),
for every
u ∈.
Indeed, this is obvious in the case of a curve encircling a polygon, when simply
lim_t→±∞γ'(t, u) = γ(∞) +  ub.
On the other hand, for a deformed line, we have

    γ'(t,u) = γ_0(t) + (t + u)b  ,

γ_0
being of compact support. This shows that, independently
of u,
the limit is, on
^2(),
the point on the line at infinity, in the direction of the
vector b.
Thus, in either case,
γ' defines
a continuous
map γ'
from
^1() ×
to ^2().

Moreover,
lim_u→±∞γ'(t, u)
exists
in ^2()
and is independent
of t ∈^1().
In all cases, it is the point on the infinite line
of ^2()
in the direction of the vector
b ∼ b.
Therefore, γ'
actually provides a continuous map

    α' = ^1() ×^1() →^2()  .



Here, in the case of a deformed line, for the imaginary
direction b,
we take the real direction of the deformed line near infinity. In the case of a curve encircling a polygon, the imaginary direction
b ∈^2 ∖( (A_12, -A_11) ∪⋯∪(A_62, -A_61) )
is to be specified later. The actual choice turns out to be irrelevant, cf. Theorem <ref>.


[An unusual differential structure on the
]
We equip the topological space
^1() ×^1()
with the structure of a smooth
,
in a way that fits our purposes. On the open subset
^1() ×^1() ∖{(∞,∞)},
we take the natural
C^∞
differential structure, given as an open submanifold of the product. We extend this structure to the whole
of ^1() ×^1(),
by adding the chart

    φ_∞,∞ O    ⟶   ^2 
    
    ((t_0:t_1), (u_0:u_1))    ↦   {[                                           (0,0)                        if  (t_0,u_0) = (0,0)  ,; (t_0/t_1, u_0/u_1)/√((t_0/t_1)^2 + (u_0/u_1)^2)                                     otherwise , ].

for
O := ^1() ×^1() ∖ (^1() ×{0}∪{0}×^1()) ⊂^1() ×^1().
It is obvious that the chart
φ_∞,∞
is compatible with those on

    ^1() ×^1() ∖{(∞,∞)} ,

so that altogether they form a
C^∞-atlas
for ^1() ×^1().

We denote the

obtained in this way
by '.
The identity
𝕀' →
is then a

and a homeomorphism. The restriction

    𝕀' ∖{(∞,∞)}→∖{(∞,∞)}

is a diffeomorphism.




The map

    α'⟶^2()  ,

as defined in (<ref>), is
almost C^1.
In the case of a curve encircling a polygon,
α'
is C^1.
Otherwise,
α' is
C^1
except at the
point (∞,∞).

Proof.
First case.
Γ
is a deformed line.

Then, on
^2 ⊂,
the map
α'
is given by the formula

    ((t_0:t_1), (u_0:u_1))    ↦   γ_0(t_1/t_0) + (t_1/t_0+u_1/u_0)b 
       =   ( 1 : γ_0,1(t_1/t_0) + (t_1/t_0+u_1/u_0)b_1 : γ_0,2(t_1/t_0) + (t_1/t_0+u_1/u_0)b_2 ) 
       =   ( t_0u_0 : t_0u_0 γ_0,1(t_1/t_0) + (t_1u_0 + t_0u_1)b_1 : 
           : t_0u_0 γ_0,2(t_1/t_0) + (t_1u_0 + t_0u_1)b_2 ) ,

which immediately extends to
∖{((0:1),(0:1))} = ∖{(∞,∞)}
and defines a
C^1-map
on this open subset.

Thus, it remains to show that
α'
is
almost C^1.
We choose
𝕀' →
as the auxiliary map, so that it suffices to verify that
α' ∘𝕀' →^2()
is a

near (∞,∞).
For this, it needs to be shown that
α' ∘φ_∞,∞^-1
is C^1
in a neighbourhood
of (0,0).
But φ_∞,∞^-1
sends
(x,y)
to
((x(x^2+y^2):1),(y(x^2+y^2):1)),
so that this map is given by

    (x,y)    ↦   ( xy(x^2+y^2)^2 : xy(x^2+y^2)^2 γ_0,1(1/x(x^2+y^2)) + (y + x)(x^2+y^2)b_1 : 
           : xy(x^2+y^2)^2 γ_0,2(1/x(x^2+y^2)) + (y + x)(x^2+y^2)b_2 ) 
       =   ( xy(x^2+y^2)^2 : (y + x)(x^2+y^2)b_1 : (y + x)(x^2+y^2)b_2 ) 
       =   ( xy(y- x) : b_1 : b_2 )  ,

at least in a neighbourhood of the origin. Note here that
γ_0 = (γ_0,1, γ_0,2)
has compact support. Finally, the function defined by
xy(y- x)
is clearly continuously differentiable
at (0,0).

Second case.
Γ
is a curve encircling a polygon.

Here, for
α',
one has a formula that is valid
on ^1() ×⊂,

    ((t_0:t_1), (u_0:u_1))    ↦   γ(t_0:t_1) + u_1/u_0b 
       =   ( 1 : γ_1(t_0:t_1) +u_1/u_0b_1 : γ_2(t_0:t_1) +u_1/u_0b_2 ) 
       =   ( u_0 : u_0 γ_1(t_0:t_1) + u_1 b_1 : u_0 γ_2(t_0:t_1) + u_1 b_2 )  .

From this, one sees that
α'
extends as a
C^1-map
to ^1() ×^1().
Finally, let us note that, by our construction, the identity map
𝕀→^1() ×^1()
is C^1.




According to its definition, given in (<ref>) and (<ref>), the torus
α'
is constant
on ^1() ×{∞}.
Contracting this subset, the
2-torus goes over into a
2-sphere
with two points identified. Thus, one might possibly work with spheroids instead of tori.

However, for the sphere, it is seemingly much harder to explicitly write down a differential structure that makes the resulting
map C^1,
at least in the case of a deformed line. And this is what we need in view of Proposition <ref>, in order to write the periods as integrals.




  
Step 2. 
Lifting to the double cover, i.e. to the
K3
surface X,
except at
_1, …, _n.


We write
_1, …, _n
for the points
(t_1,0), …, (t_n,0)
on .



Assume that
b
is generic in the sense of Assumption (<ref>) in the case that
Γ
is a curve encircling a polygon. I.e., that

    b ∉(A_12, -A_11) ∪…∪(A_62, -A_61)  .

Then the torus
α'
meets the branch locus
V(l_1 ⋯ l_6)
only at
_i,
for
i = 1, …, n.
I.e., only in the three to five double points.

-1
Proof.
First of all,
γ'(t,u) ∈ V(l_1 ⋯ l_6)
implies
that u = 0.
Indeed, (x,y) ∈ V(l_i)
yields
A_i1 x + A_i2 y = 0.
On the other hand,
γ'(t,u) = ub = (ub_1, ub_2),
so that 
γ'(t,u) ∈ V(l_i)
is possible only when
u(A_i1 b_1 + A_i2 b_2) = A_i1 ub_1 + A_i2 ub_2 = 0.

As b ∉(A_i2, -A_i1),
one
has A_i1 b_1 + A_i2 b_2 ≠ 0
and hence
u=0,
as required. Thus, the assertion is shown
for α' |_^2.

In the case of a curve encircling a polygon, the same argument applies to
t = ∞, too.

Moreover, all points of

that are not yet covered are mapped under
α'
to the point
of ^2()
on the line at infinity, in the direction of the
vector b ∼ b.
This can not be the point at infinity of any of the
lines V(l_i),
for
i = 1, …, 6.
In fact, one has
b ∉(A_12, -A_11) ∪…∪(A_62, -A_61)
in either case. For deformed lines, this is due to Assumption A.<ref>.







  * Let
α'→^2()
be a torus as above. In the case that
Γ
is a curve encircling a polygon, assume
b ∈^2 ∖( (A_12, -A_11) ∪…∪(A_62, -A_61) ).
Then the restriction
α' |_∖{_1,…,_n}
allows a continuous lift
α”
to the
K3 surface X.

  * In either case,
α”
is automatically
almost C^1.


Proof.
a)
First step.
Reduction to a statement on fundamental groups.

Put X_0 ⊂ X
to be the preimage
of ^2() ∖ V(l_1 ⋯ l_6)
under the natural map.
As X_0
avoids the branch locus and, in particular, the locus that is blown up, the restriction
π_0 X_0 →^2() ∖ V(l_1 ⋯ l_6)
of π
is a covering projection <cit.>.

The assertion is that
α_0 := α' |_∖{_1,…,_n}∖{_1, …, _n}→^2() ∖ V(l_1 ⋯ l_6)
allows a continuous
lift α”,
as indicated in the diagram below,


    X_0 @->[d]^π_0
    ∖{_1, …, _n}@->[r]^α_0   @.>[ru]^      α”   ^2() ∖ V(l_1 ⋯ l_6)  .


In order to establish this claim, according to <cit.>, we have to show that
(α_0)_#π_1(∖{_1, …, _n},·) ⊆ (π_0)_#π_1(X_0,·).

For this, let us note that
π_1(∖{_1, …, _n},·),
the fundamental group of the
 minus
n points,
is a free group on
(n+1) generators.
A generating system is provided by the homotopy classes of
n small
loops
ν_1, …, ν_n S^1 →∖{_1, …, _n},
each of which encircles exactly one of the points
_1, …, _n,
together with the homotopy class of the closed path
μ_1
running through
^1() ×{∞}
and the homotopy class of the closed path
μ_2
running through
{∞}×^1().

Second step.
The small loops.

Let us first show that
(α_0)_# [ν_k] ∈ (π_0)_#π_1(X_0,·),
for k = 1, …, n.
This claim is clearly equivalent to the liftability of the loop
α_0 ∘ν_k S^1 →^2() ∖ V(l_1 ⋯ l_6)
to X_0.
As the particular covering projection
π_0 X_0 →^2() ∖ V(l_1 ⋯ l_6)
is given by
w^2 = l_1 ⋯ l_6,
this just means that the loop

    ν_k = (l_1 ⋯ l_6) ∘α_0 ∘ν_k S^1    ⟶∖{0} ,
       z     ↦  (l_1 ⋯ l_6) (α_0(ν_k(z)))

lifts under the covering projection
∖{0}→∖{0}, w ↦ w^2.
And the latter is well-known to be possible if and only if the winding number
of ν_k
is even.

We may assume that
ν_k S^1 →∖{_1, …, _n}
is given by
z ↦ (t_k + ε z, ε z),
for some real number
ε > 0.
Then the composition
α_0 ∘ν_k S^1 →^2() ∖ V(l_1 ⋯ l_6) is

    α_0 ∘ν_k (cosφ, sinφ)    ↦   γ(t_k + εcosφ) + εsinφ· b 
       =   γ(t_i_k,j_k + εcosφ) + εsinφ· b 
       =    x_i_k,j_k + εcosφ· b_i_k,j_k + εsinφ· b  .

Hence, for
a = 1,…,6,
the triple composition
l_a ∘α_0 ∘ν_k S^1 →∖{0}
is given by

    l_a ∘α_0 ∘ν_k (cosφ, sinφ)    ↦    l_a'(x_i_k,j_k + εcosφ· b_i_k,j_k + εsinφ· b) 
       =    l_a'(x_i_k,j_k) + l_a(εcosφ· b_i_k,j_k + εsinφ· b) 
       =    l_a'(x_i_k,j_k) + εcosφ·l_a(b_i_k,j_k) + εsinφ·l_a(b)  .

Here, for
a ≠ i_k, j_k,
one has that
x_i_k,j_k∉V(l_a),
i.e. that
l_a'(x_i_k,j_k) ≠ 0.
This shows that the winding number of
l_a ∘α_0 ∘ν_k
is 0,
at least as long as
ε
is sufficiently small.

On the other hand, for
a = i_k
or a = j_k,
clearly
x_i_k,j_k∈ V(l_a),
and hence
l_a'(x_i_k,j_k) = 0.
Moreover, the constants
l_a(b_i_k,j_k)
and
l_a(b)
are both nonzero. Indeed, for the first, this follows from Assumption A.<ref>, while, for the second, this is either Assumption A.<ref> or Assumption (<ref>). Therefore, the winding number of
l_a ∘α_0 ∘ν_k
must be
1
or (-1).

Altogether, the winding number of
ν_k = (l_1 ⋯ l_6) ∘α_0 ∘ν_k
is an element
of {-2,0,2}
and, in particular, even, as claimed.

Third step.
The large closed paths.

One has that
α_0 ∘μ_1
is a constant map. In the case of a deformed line,
α_0 ∘μ_2
is a constant map, too.

Therefore, it suffices to consider
μ_2
in the case of a curve encircling a polygon. One has to verify that
(α_0)_# [μ_2] ∈ (π_0)_#π_1(X_0,·),
which is equivalent to the liftability of the closed path
α_0 ∘μ_2^1() ≅ S^1 →^2() ∖ V(l_1 ⋯ l_6)
to X_0.
For the particular covering projection
π_0 X_0 →^2() ∖ V(l_1 ⋯ l_6),
in exactly the same way as above, this just means to show that the winding number of

    μ_2 = (l_1 ⋯ l_6) ∘α_0 ∘μ_2^1()    ⟶∖{0} ,
       (u_0:u_1)     ↦  (l_1 ⋯ l_6) (α_0(μ_2(u_0:u_1)))

is even.

Assume that
μ_2^1() →∖{_1, …, _n}
is given by
(u_0:u_1) ↦ (∞, (u_0:u_1)),
for ∞ = (0:1),
as before. Then
α_0 ∘μ_2^1() ≅ S^1 →^2() ∖ V(l_1 ⋯ l_6) is

    α_0 ∘μ_2 (u_0:u_1)    ↦   γ(∞) + u_1/u_0 b 
       =   ( γ_1(∞) + u_1/u_0 b_1 : γ_2(∞) + u_1/u_0 b_2 : 1 ) 
       =   ( γ_1(∞) · u_0 +  b_1 · u_1 : γ_2(∞) · u_0 +  b_2  · u_1 : u_0 )  .

Hence, for
a = 1,…,6,
the triple composition
l_a ∘α_0 ∘μ_2^1() →∖{0}
is given by

    l_a ∘α_0 ∘μ_2 (u_0:u_1) ↦ l_a(γ_1(∞), γ_2(∞), 1) · u_0 +  l_a(b_1, b_2, 0) · u_1  .

Here, both constants,
l_a(γ_1(∞), γ_2(∞), 1)
and l_a(b_1, b_2, 0),
are nonzero. Indeed, for the first, this follows from Assumption A.<ref>, while, for the second, this is just Assumption (<ref>). Thus, the winding number of
l_a ∘α_0 ∘μ_2
is either
1
or (-1).

Altogether, the winding number of
μ_2 = (l_1 ⋯ l_6) ∘α_0 ∘μ_2
is even, as required. Its absolute value is bounded
by 6.

b)
The final assertion is an immediate consequence of Lemma <ref>, together with the fact that
π_0 X_0 →^2() ∖ V(l_1 ⋯ l_6)
is a local diffeomorphism.





  
Step 3. 
Widening the holes of the domain
∖{_1,…,_n}


Let us first construct a few auxiliary functions.



  ∙ 
We put

    ψ [0,∞) ⟶ [0,∞)  ,    r ↦{[                         0              if  r ≤1/2 ,; 10(r-1/2)^2 - 12(r-1/2)^3          if 1/2≤ r ≤ 1  ,;                         r              if  1 ≤ r  . ].

As is easily checked, the
function ψ
is monotonically increasing
and C^1.

  ∙ 
Working in polar coordinates and
using ψ,
we define

    Ψ^2 ⟶^2  ,    (r,φ) ↦ (ψ(r),φ)  .

This is a
C^1-map
being the identity
outside U_1(0)
and contracting
U_1/2(0)
into the origin. Consequently, for
p ∈^2,

    Ψ_p^2 ⟶^2  ,    x ↦ p + Ψ(x-p)

is C^1,
equal to the identity
outside U_1(p),
and contracts
U_1/2(p)
to p.

  ∙ 
As Ψ_p
differs from
𝕀
only within a compact subset
of ^2,
it extends uniquely to a self-map

    Ψ_p→

that is
again C^1,
contracts
U_1/2(p)
to p
and is the identity outside
U_1(p).


[Widening the holes]
For
α”∖{_1,…,_n}→ X,
put

    α”' := α”∘ (Ψ__1∘⋯∘Ψ__n) |_∖ (U_1/2(_1)∪…∪U_1/2(_n))∖( U_1/2(_1)∪…∪U_1/2(_n)) → X   .


Note that the order of the maps
Ψ__j
is irrelevant here, due to Assumption A.<ref>.




The almost
C^1-map

    α”'∖( U_1/2(_1)∪…∪U_1/2(_n)) → X

allows an extension to the manifold with boundary
∖( U_1/2(_1) ∪…∪ U_1/2(_n) )
that is again
almost C^1.
It is, in fact, a

on ∖{(∞,∞)}.

Proof.
By the constructions above, on an annulus
around _k
of inner radius
1/2
and outer
radius 1,
the projection
π∘α”'∖( U_1/2(_1)∪…∪U_1/2(_n)) →^2()
is given by

    (t_k + r cosφ, r sinφ) ↦ x_i_k,j_k + ψ(r) cosφ· b_i_k,j_k + ψ(r) sinφ· b ∈^2 ⊂^2()  .

The exceptional curve of the blowup
_x_i_k,j_k(^2())
of ^2()
in x_i_k,j_k
is covered by two open subschemes, both of which being affine
.
The projections down to
^2()
are given by
(x',y') ↦ x_i_k,j_k + (x',x'y')
and
(x',y') ↦ x_i_k,j_k + (x'y',y').
Let us work in the first chart, the other one being analogous.

Then, on the annulus above, the lift of
π∘α”'
to _x_i_k,j_k(^2())
is given by

    (t_k + r cosφ, r sinφ)   ↦( ψ(r) cosφ· (b_i_k,j_k)_1 +ψ(r) sinφ· b_1, ψ(r) cosφ· (b_i_k,j_k)_2 + ψ(r) sinφ· b_2/ψ(r) cosφ· (b_i_k,j_k)_1 + ψ(r) sinφ· b_1) 
       = ( ψ(r) · (cosφ· (b_i_k,j_k)_1 + sinφ· b_1), cosφ· (b_i_k,j_k)_2 + sinφ· b_2/cosφ· (b_i_k,j_k)_1 + sinφ· b_1)  .

Here, of course, the denominator in the second coordinate might vanish for certain values
of φ.
For these values, however, the numerator is nonzero and therefore other chart applies. Indeed, by Assumptions A.<ref>, A.<ref>, as well as (<ref>), the vectors
b_i_k,j_k
and b
are both real and different from the zero vector.

For U ⊂{φ∈ [0,2π) |cosφ· (b_i_k,j_k)_1 + sinφ· b_1 ≠ 0}
an open interval, the map (<ref>) clearly extends as a

from
(1/2, 1) × U
to [1/2, 1) × U.
In total, this shows that

    π∘α”' |_U_1(_k) ∖U_1/2(_k) U_1(_k) ∖U_1/2(_k)⟶^2()

allows an extension to
U_1(_k) ∖ U_1/2(_k)
that lifts as a

to _x_i_k,j_k(^2()).

We still have to eliminate the
projection π,
i.e. to lift to the
K3 surface X.
For this, recall that the double cover
π' X →_x_12, …, x_56(^2())
is a local diffeomorphism outside the branch locus. Moreover, the branch locus
on _x_i_k,j_k(^2())
consists of the union of six disjoint curves. Locally near
x_i_k,j_k,
it is the union of the two curves, given by
l_i_k(1,y') = 0
and l_j_k(1,y') = 0.
Thus, to complete the proof, it suffices to show that the image
of U_1/2(_k)
does not meet either of the two curves.

For this, let us note first that, by our assumptions,
l_i_k(1,y') = 0
or
l_j_k(1,y') = 0
is possible only when
y'
is real. Moreover, in the generic case that the vectors
b_i_k,j_k
and b
are linearly independent,

    cosφ· (b_i_k,j_k)_2 + sinφ· b_2/cosφ· (b_i_k,j_k)_1 + sinφ· b_1∈

implies that
cosφ = 0
or sinφ = 0.
I.e., that

    y' = (b_i_k,j_k)_2/(b_i_k,j_k)_1  or   y' = b_2/b_1 .

But l_i_k(1,b_2/b_1) = 0
or
l_j_k(1,b_2/b_1) = 0
would mean nothing but
b ∈(A_i_k,2, -A_i_k,1)
or
b ∈(A_j_k,2, -A_j_k,1),
which is excluded by Assumption A.<ref> or (<ref>), respectively. And

    l_i_k( 1,(b_i_k,j_k)_2/(b_i_k,j_k)_1) = 0   or  l_j_k( 1,(b_i_k,j_k)_2/(b_i_k,j_k)_1) = 0

would yield that
b_i_k,j_k∈(A_i_k,2, -A_i_k,1)
or
b_i_k,j_k∈(A_j_k,2, -A_j_k,1),
in contradiction with Assumption A.<ref>.

Finally, in the exceptional case that the vectors
b_i_k,j_k
and b
are linearly dependent, one has that
cosφ· (b_i_k,j_k)_2 + sinφ· b_2/cosφ· (b_i_k,j_k)_1 + sinφ· b_1

is
constantly b_2/b_1.
But both
l_i_k(1,b_2/b_1) = 0
and
l_j_k(1,b_2/b_1) = 0
are contradictory to our assumptions, as above.





  
Step 4. 
Extending to the whole
of .




  * The almost
C^1-map
α”'∖( U_1/2(_1)∪…∪U_1/2(_n)) → X allows
an extension
α→ X
to the whole
of 
that is again
almost C^1.

  * The map
α
may be chosen in such a way that, for
each k,
the image
α( U_1/2(_k))
is completely contained in an exceptional curve.


Proof.
We start with the extension
α””∖( U_1/2(_1) ∪…∪ U_1/2(_n) ) → X
that is provided by Proposition <ref>. Then,

    α””(∂U_1/2(_k)) ⊂ E_i_k,j_k ,

for each
k ∈{1, …, n},
according to our construction.
Moreover, for every point
x ∈∂U_1/2(_k)
and every tangent vector
v ∈ T_x
that is perpendicular to the boundary and points to the interior of
∖( U_1/2(_1) ∪…∪ U_1/2(_n) ),
one has

    Tα””(v) = 0⃗∈ T_α””(x) X  .


Moreover, as
E_i_k,j_k≅^1()
is simply-connected, the modification of <cit.> for the

yields a
C^1-homotopy
H∂U_1/2(_k)× [0,1/2] → E_i_k,j_k
connecting

    H_∂U_1/2(_k)×{1/2} = α”” |_∂U_1/2(_k)

with a constant map. Applying a suitable monotonously increasing, bijective
C^1-self-map
of [0,1/2],
if necessary, one may assume that

    TH(v) = 0⃗∈ T_H(x,1/2) E_i_k,j_k  and  TH(x,v) = 0⃗∈ T_H(x,0) E_i_k,j_k

for every tangent vector
v ∈ T_(x,1/2)(∂U_1/2(_k)× [0,1/2])
or
v ∈ T_(x,0)(∂U_1/2(_k)× [0,1/2]),
respectively, that is perpendicular to the boundary and points to the inside.

Putting, finally,

    α(tx) := H(x,t)

for
x ∈∂U_1/2(_k)
and t ∈ [0,1/2],
and gluing the maps together, one finds an extension
α
of α”'
with all the properties required.





  
Step 5. 
Projecting down
to X'.


We obtain the torus
α_Γ→ X'
or
α_Γ,b→ X',
respectively, as the composition
∘α.


Modifying Definition <ref> in an obvious way, one defines the concept of an almost C^∞-map.
Starting with a
C^∞-parametrisation of the

Γ,
it is certainly possible to work out the constructions just presented entirely within the setup of
almost C^∞-maps.
We prefer the use of
almost C^1-maps,
as this approach keeps Steps 3 and 4 slightly simpler.




  §.§.§ Periods as improper integrals



According to our construction, we may suppose that the lift
α→ X
of the torus
α_Γ
or
α_Γ,b
considered is an almost
C^1 map.
Therefore, Proposition <ref> applies and shows that
(c_α, [ω]) = ∫_α^* ω.

Moreover, the exceptional curves
E_i_1,j_1, …, E_i_n,j_n
are holomorphic curves
on X.
Hence, as
ω
is a
,
one has that the restrictions
ω |_E_i_k,j_k
are equal to the null form. In particular,
∫_E_i_k,j_kω = 0.
Pulling back, one finds that
∫_U_1/2(_k)α^* ω = 0.
Since, in addition,
∂ U_1/2(_k)
is a Lebesgue null set, this yields

    (c_α, [ω]) = ∫_∖ (U_1/2(_1)∪…∪U_1/2(_n))α^* ω = ∫_∖ (U_1/2(_1)∪…∪U_1/2(_n))α”' ^* ω = ∫_∖{_1,…,_n}α”^* ω = ∫_∖{_1,…,_n}α”^* ω'  .


Furthermore, noticing that
^2 ∖{_1,…,_n}
differs from
∖{_1,…,_n}
by a Lebesgue null set, we conclude that

    (c_α, [ω]) = ∫_^2 ∖{_1,…,_n}(α” |_^2 ∖{_1,…,_n} )^* ω' = ∫_^2 (α” |_^2 ∖{_1,…,_n} )^* ω'  .


Finally, a direct calculation of the differential

(α” |_^2 ∖{_1,…,_n} )^* ω'
in terms of the coordinates
t
and u,
making use of formulae (<ref>) and (<ref>), establishes the claim.




  §.§.§ Independence of the choice
of b
in the case of a curve encircling a polygon



First step.
Preparations.

By Corollary <ref>, it suffices to show that
α_Γ,b→ X'
is homotopic either to
α_Γ,b
or to
α_Γ,b→ X'.
Moreover, in doing so, it is clearly sufficient to assume that
b and b
lie in neighbouring sectors of the set
^2 ∖( (A_12, -A_11) ∪…∪(A_62, -A_61) )
of admissible vectors.

We let
s_0 ∈ (0,1)
be the unique real number such that
b_0 := b + s_0(b - b)
is not an admissible vector, cf. Figure <ref>. Furthermore, we denote by
i_0 ∈{1, …, 6}
the index, for which
b_0 ∈(A_i_0,2, -A_i_0,1).



Then, at first, the map

    H^2 × [0,1]    ⟶^2 ⊂^2()  , 
    
    (t,u,s)     ↦ γ(t) +  u[b + s(b - b)]  ,

clearly allows an extension to a continuous map

    H× [0,1] ⟶^2()  ,

which is a homotopy connecting
α'_b
with α'_b.

Moreover, one readily sees that
H^-1(V(l_1 ⋯ l_6)) = B, for
B ⊂× [0,1]
the subset

    B := ({_1}× [0,1]) ∪…∪({_n}× [0,1]) ∪⋃_k=1,…,nl_i_0(γ(t_k)) = 0({t_k}×^1() ×{s_0})  .

I.e., the map
H
hits the branch locus
V(l_1 ⋯ l_6)
exactly for the arguments
(t,u,s) ∈ B.

There are now two cases to be distinguished. Either,
V(l_i_0)
is one of the edges of the polygon encircled
by Γ.
Then
l_i_0(γ(t_k)) = 0
happens exactly two times, say
for k_1
and k_2.
In this case,
γ(t_k_1) = x_i_k_1,j_k_1
and γ(t_k_2) = x_i_k_2,j_k_2
are the two vertices of the
edge V(l_i_0).
Which means that
i_0
is one of the indices
i_k_1
and j_k_1,
as well as one of the indices
i_k_2
and j_k_2.
We take the ordering such that
t_k_1 < t_k_2.
Moreover, we assume, without loss of generality, that the parametrisation
γ
is of the kind that
γ((t_k_1, t_k_2))
is the direct path
on Γ
from
x_i_k_1,j_k_1
to x_i_k_2,j_k_2,
not meeting any other branch point, cf. Figure <ref> below.

Or, V(l_i_0)
is not an edge of the polygon.
Then l_i_0(γ(t_k)) = 0
does not happen, for
any k.



Second step.
Lifting to the double
cover X'.

As usual, the continuity of a lift
of H
to X',


    X_0  ⊂ X'@->[d]^π_0
    
    (× [0,1]) ∖ B @->[rrr]^H|_(× [0,1]) ∖ B@.>[rrru]^H'         ^2() ∖ V(l_1 ⋯ l_6)  ,

is governed by the image of the fundamental group
π_1((× [0,1]) ∖ B, ·)
under
(H|_(× [0,1]) ∖ B)_#.

By <cit.>, there would be a continuous lift if and only if

    (H|_(× [0,1]) ∖ B)_#π_1((× [0,1]) ∖ B, ·) ⊆ (π_0)_#π_1(X_0, ·)  .


So let us consider the fundamental group
π_1((× [0,1]) ∖ B, ·).
It is generated by the homotopy classes
[ν_1], …, [ν_n]
of small loops encircling
{_1}× [0,1], …, {_n}× [0,1],
respectively, together with the homotopy class
[μ_1]
of a closed path running through
^1() ×{∞}×{0},
and the homotopy class
[μ_2]
of the closed path running through
{∞}×^1() ×{0}.
In the more interesting case that
V(l_i_0)
is one of the edges of the polygon encircled
by Γ,
two further generators need to be taken into consideration, namely the homotopy classes
[λ_k_1]
and
[λ_k_2]
of small loops
λ_k_1, λ_k_2 S^1 → (× [0,1]) ∖ B encircling

    {t_k_1}×^1() ×{s_0}  and  {t_k_2}×^1() ×{s_0} ,

respectively.

One indeed has
(H|_(× [0,1]) ∖ B)_# ([ν_k]) ⊆ (π_0)_#π_1(X_0, ·),
for k = 1,…,n,
as well as
(H|_(× [0,1]) ∖ B)_# ([μ_1]) ⊆ (π_0)_#π_1(X_0, ·)
and
(H|_(× [0,1]) ∖ B)_# ([μ_2]) ⊆ (π_0)_#π_1(X_0, ·).
The calculations showing this are essentially the same as those performed in the second and third steps of the proof of Proposition <ref>.a). Thus, there is no need to repeat them here.

As a conclusion, one finds that
H
allows a continuous
lift H'× [0,1] → X'
in the case when
V(l_i_0)
is not an edge of the polygon.

In particular, in this case,
α_Γ,b
is homotopic either to
α_Γ,b
or to
α_Γ,b,
as claimed.

Third step.
A discontinuous lift.

On the other hand,
(H|_(× [0,1]) ∖ B)_# ([λ_k]) ⊆ (π_0)_#π_1(X_0, ·)
is false for both indices,
k = k_1
and k_2.
In order to see this, let us restrict considerations
to [λ_k_1],
the arguments for the other one being exactly the same. Similarly to the above, we have to show that the winding number of
λ_k_1 := (l_1 ⋯ l_6) ∘ H|_(× [0,1]) ∖ B∘λ_k_1 S^1 →∖{0}
is odd.

Choose a real number
u_0 ≠ 0
and let
λ_k_1 S^1 → (× [0,1]) ∖ B
be given by

    (cosφ, sinφ) ↦ (t_k_1 + εcosφ, u_0, s_0 + εsinφ)  ,

for some real number
ε > 0.
Then
H|_(× [0,1]) ∖ B∘λ_k_1 S^1 →^2() ∖ V(l_1 ⋯ l_6) is

    H|_(× [0,1]) ∖ B∘λ_k_1 (cosφ, sinφ)    ↦   γ(t_k + εcosφ) +  u_0 [b + (s_0 + εsinφ)(b - b)] 
       =   γ(t_k + εcosφ) +  u_0[b_0 + εsinφ(b - b)] 
       =    x_i_k_1, j_k_1+ u_0 · b_0 +εcosφ· b_i_k_1, j_k_1+εsinφ· u_0 (b - b)  .

Hence, for
a = 1,…,6,
the triple composition
l_a ∘ H|_(× [0,1]) ∖ B∘λ_k_1 S^1 →∖{0}
is given by

    l_a ∘ H|_(× [0,1]) ∖ B∘λ_k_1    (cosφ, sinφ) ↦
       l_a(x_i_k_1, j_k_1 +  u_0 · b_0 + εcosφ· b_i_k_1, j_k_1 + εsinφ· u_0(b - b))
    
    =   l_a'(x_i_k_1, j_k_1) + u_0 ·l_a(b_0) + εcosφ·l_a(b_i_k_1, j_k_1) + εsinφ· u_0 l_a(b - b)  .

Here, for
a ≠ i_0,
the real number
l_a(b_0)
is nonzero.
As l_a'(x_i_k_1, j_k_1) ∈,
this means that the winding number of
l_a ∘ H|_(× [0,1]) ∖ B∘λ_k_1
is 0,
at least as long as
ε
is sufficiently small.

On the other hand,
l_i_0'(x_i_k_1, j_k_1) = 0
and
l_i_0(b_0) = 0.
Moreover, the constants
l_i_0(b_i_k_1, j_k_1)
and
l_i_0(b - b))
are both nonzero. This is due to Assumption A.<ref> for the first and obvious for the second. Therefore, the winding number of
l_i_0∘ H|_(× [0,1]) ∖ B∘λ_k_1
must be
1
or (-1).
Altogether, the winding number of the triple composition
λ_k_1 = (l_1 ⋯ l_6) ∘ H|_(× [0,1]) ∖ B∘λ_k_1
is
1
or (-1)
and, in particular, odd, as claimed.

As a conclusion, one finds that
H× [0,1] →^2()
does not allow any continuous lift
to X'.
There is, however, a discontinuous lift

    H'× [0,1] ⟶ X'

that is continuous
on (× [0,1]) ∖( (t_k_1, t_k_2) ×^1() ×{s_0}).
The discontinuity is of the kind that,
for (t,u) ∈
such that
t ∈ (t_k_1, t_k_2),
the limit
α_Γ,b_0^+ (t,u) := lim_s→ s_0+0 H'(t,u,s)
differs from
α_Γ,b_0^- (t,u) := lim_s→ s_0-0 H'(t,u,s)
by the
involution ζ.

In particular,
H'
provides us with two homotopies, one connecting
α_Γ,b
with
α_Γ,b_0^-
and another, connecting
α_Γ,b
or
α_Γ,b
with α_Γ,b_0^+.[3]

Fourth step.
Straightening the
 Γ.

It is thus our remaining task to show that
α_Γ,b_0^+
and
α_Γ,b_0^-→ X'
are homotopic to each other. For this, consider the following,

    D^1() ×× [0,1]    ⟶^2 ⊂^2()  , 
    
    (t,u,s)     ↦ {[ s ( t_k_2-t/t_k_2-t_k_1γ(t_k_1) + t-t_k_1/t_k_2-t_k_1γ(t_k_1) );                                         + (1-s) γ(t) +  u · b_0                                       if  t ∈ (t_k_1, t_k_2)  ,;                                                 γ(t) +  u · b_0                                                     otherwise . ].

The map
D
clearly allows an extension to a continuous map
D× [0,1] →^2(),
which is a homotopy between the maps
α'_Γ
and α'_Γ',
for
Γ'
a straightening
of Γ,
as indicated in the figure below.




Moreover, one has
D^-1(V(l_1 ⋯ l_6)) = B,
for

    B := ⋃_k=1,…,nk ≠ k_1,k_2( {_k}× [0,1] )  ∪⋃_k = k_1,k_2( {t_k}×^1() × [0,1] ) ∪( (t_k_1, t_k_2) ×^1() ×{1})  .




Indeed, the inclusion
“⊇”
is obvious. In order to verify the reverse
inclusion “⊆”,
we first observe that the convex hull of the union of the line segment
[γ(t_k_1), γ(t_k_2)]
with the direct path
γ((t_k_1, t_k_2))
does not contain any point
of V(l_1 ⋯ l_6)
in its interior.

In fact, a
line V(l_a)
through such a point would necessarily intersect
γ((t_k_1, t_k_2))
somewhere, in contradiction to the assumption that this is a direct path from one vertex of the encircled polygon to the next, cf. Figure <ref>, above.

Now let
(t,u,s) ∈× [0,1]
be any point such that
D(t,u,s) ∈ V(l_1 ⋯ l_6).
I.e., such that
l_a(D(t,u,s)) = 0
for
some a ∈{1,…,6}.

In the particular case
that t ∈ (t_k_1, t_k_2),
the reasoning just given shows that

    0 =  l_a(D(t,u,s)) = l_a( D(t,u,s))

is possible only
for s=1,
as claimed.

On the other hand, for
t ∉(t_k_1, t_k_2),
equation (<ref>) means that
0 = l_a(γ(t)),
which, according to the general assumptions on the
 Γ,
together with Notation <ref>.ii), may happen only
for t = t_1, …, t_n.
Moreover, in this case,

    0 =  l_a(D(t,u,s)) = l_a( D(t,u,s)) = u ·l_a(b_0)  ,

which is true only for
u = 0
or a = i_0.
But l_i_0(γ(t)) = 0
happens only for
t = t_k_1
and t = t_k_2,
which completes the argument for the 
inclusion “⊆”.

The domain
(× [0,1]) ∖ B
turns out to be disconnected into two components. We construct a continuous lift
D'× [0,1] → X'
of D
as follows.
For t ∉(t_k_1, t_k_2),
we
put D'(t,u,s) := α_Γ,b_0^+(t,u).
For (t,u,s) ∈ B,
there is a unique lift
of D(t,u,s),
anyway. Finally,
( (t_k_1, t_k_2) ×^1() × [0,1] ) ∖ B
is simply connected. Thus, there are two continuous lifts, interchanged by the
involution ζ,
and we may take the one that agrees
with α_Γ,b_0^+
on ×{0}≅.
Then D'
is automatically continuous on the whole
of × [0,1],
and a homotopy connecting
α_Γ,b_0^+
with α_Γ',b_0.

It is obvious that a completely analogous construction yields a homotopy connecting
α_Γ,b_0^-
with α_Γ',b_0.
The proof is hence complete.


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