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"Curve-counting Invariants and Crepant Resolutions of Calabi-Yau Threefolds by David Christopher Steinberg  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July 2012 c David Christopher Steinberg 2012  \u000CAbstract The Donaldson-Thomas (DT) theory of a Calabi-Yau threefold X gives rise to subtle deformation invariants. They are considered to be the mathematical counterparts of BPS state counts in topological string theory compactified on X. Principles of physics (see [46], [50]) indicate that the string theory of a singular Calabi-Yau threefold and that of its crepant resolution ought to be equivalent, so one might expect that the DT theory of a singular Calabi-Yau threefold ought to be equivalent to that of its crepant resolution. There is some difficulty in defining DT when X is singular, but the authors of [14] have (in some generality) defined DT theory in the case where X is the coarse moduli space of an orbifold X . The crepant resolution conjecture of [14] gives a formula determining the DT invariants of the orbifold in terms of the DT invariants of the crepant resolution. In this dissertation, we begin a program to prove the crepant resolution conjecture using Hall algebra techniques inspired by those of Bridgeland [12].  ii  \u000CTable of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  Dedication  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  2 \u03C0-stable pairs  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8  3 Stability conditions . . . . . . . . . . . . . . . . . . . . . . . . .  20  4 The torsion pair and the stability condition . . . . . . . . .  29  5 The motivic Hall algebra . . . . . . . . . . . . . . . . . . . . .  37  6 Equations in the infinite-type Hall algebra and the fake proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  41  7 Equations in the Laurent Hall algebra and the true proof  51  8 Conclusion  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  62  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  63  iii  \u000CAcknowledgements I wish to thank my Ph. D. advisor, Jim Bryan, for his endless patience, enthusiasm, and support in the face of adversity. I would also like thank Arend Bayer for help with lemma 3.16, Brian Conrad for help with lemma 2.16, S\u00B4 andor Kov\u00B4 acs, Kai Behrend, John Calabrese, Kalle Karu, Andrew Morrison, and mathoverflow.com for their helpful conversations. Special thanks are due to Tom Bridgeland. On top of offering invaluable comments and suggestions on an early draft of this dissertation, his brilliantly written papers [11] and [12] were the inspiration for most everything that follows. Finally, I would like to thank my parents, whose continued support has made all the difference.  iv  \u000CDedicated to my friends.  v  \u000CChapter 1  Introduction Donaldson-Thomas (DT) theory of a Calabi-Yau threefold X gives rise to subtle deformation invariants. They are considered to be the mathematical counterparts of BPS state counts in topological string theory compactified on X. Principles of physics (see [46], [50]) indicate that the string theory of a singular Calabi-Yau threefold and that of its crepant resolution ought to be equivalent, so one might expect that the DT theory of a singular CalabiYau threefold ought to be equivalent to that of its crepant resolution. There is some difficulty in defining DT when X is singular, but the authors of [14] have (in some generality) defined DT theory in the case where X is the coarse moduli space of an orbifold X . The crepant resolution conjecture of [14] gives a formula determining the DT invariants of the orbifold in terms of the DT invariants of the crepant resolution. In this dissertation, we begin a program to prove the crepant resolution conjecture using Hall algebra techniques inspired by those of Bridgeland [12].  Donaldson-Thomas theory Let Y be a smooth projective Calabi-Yau threefold. Let K(Y ) be the numerical K-theory of Y , i.e. the quotient of the K-group of coh(Y ) by the kernel of the Chern character map to cohomology. The Hilbert scheme of Y , Hilb\u03B1 (Y ), parametrizes quotients OY \u2192 OZ , such that the class of OZ in K(Y ) is \u03B1. The group K(Y ) is filtered by the dimension of the support: F0 K(Y ) \u2282 F1 K(Y ) \u2282 F2 K(Y ) \u2282 F3 K(Y ) = K(Y ).  1  \u000CIn this dissertation, we will focus on curves, i.e., \u03B1 \u2208 F1 K(Y ), with ch(\u03B1) = \u223C Z is (0, 0, \u03B2, n), where \u03B2 \u2208 H 4 (Y, Z) is a curve class, and n \u2208 H 6 (Y, Z) = the holomorphic Euler characteristic. In [43], an obstruction theory for this moduli space is constructed, which produces (by [5]) a virtual fundamental cycle. Donaldson-Thomas invariants are defined by integrating over the zero-dimensional virtual fundamental class: DT \u03B1 (Y ) =  1. [Hilb\u03B1 (Y )]vir  Since the obstruction theory is symmetric, we may also express the invariants as the Euler characteristic of Hilb\u03B1 (Y ) weighted by Behrend\u2019s microlocal function [3]: n\u03C7(\u03BD \u22121 (n)),  DT \u03B1 (Y ) = n\u2208Z  where \u03BD : Hilb\u03B1 (Y ) \u2192 Z is Behrend\u2019s function. Following [35], we assemble the invariants into a partition function DT\u03B1 (Y )q \u03B1 .  DT(Y ) = \u03B1\u2208F1 K(Y )  Remark 1.1. In [29], Donaldson-Thomas invariants are greatly generalized, from the case of structure sheaves of curves to that of arbitrary sheaves. The price of admission to this generality is the formidable machinery of Joyce [24, 25, 26, 27, 28]. An even more ambitious program of generalization is being lead by Kontsevich and Soibelman [31].  The Donaldson-Thomas crepant resolution conjecture We follow [14] in our treatment of the crepant resolution conjecture. An orbifold CY3 is defined to be a smooth, quasi-projective, DeligneMumford stack X over C of dimension three having generically trivial sta-  2  \u000Cbilizers and trivial canonical bundle, KX \u223C = OX . The definition implies that the local model for X at a point p is [C3 /Gp ] where Gp \u2282 SL(3, C) is the (finite) group of automorphisms of p. The orbifold CY3s that appear in this dissertation will all be projective and satisfy the hard Lefschetz condition ([16, definition 1.1]), which in this case is equivalent ([15, lemma 24]) to the condition that all Gp are finite subgroups of SO(3) \u2282 SU (3) or SU (2) \u2282 SU (3). Let X denote the coarse space of X . A crepant resolution of X is a resolution of singularities \u03C0 : Y \u2192 X such that \u03C0 \u2217 KX \u223C = KY . Lemma 1 and proposition 1 of [48] prove that R \u2022 \u03C0\u2217 O Y = O X .  (1.2)  The results of [13] and [18] prove that one distinguished crepant resolution of X is Y = Hilb[Op ] (X ),  (1.3)  the Hilbert scheme parametrizing substacks in the class [Op ] \u2208 F0 K(X ). The hard Lefschetz condition implies that the resolution is semi-small (i.e., that the fibres of \u03C0 are zero- or one-dimensional), and that the singular locus of X is one-dimensional; see [8, 15]. Furthermore, [13] and [18] prove that there is a Fourier-Mukai isomorphism \u03A8 : Db (Y ) \u2192 Db (X ) defined by E \u2192 Rq\u2217 p\u2217 E where p : Z \u2192 Y, q : Z \u2192 X are the projections from the universal substack Z \u2282 X \u00D7 Y onto each factor.  3  \u000CThis isomorphism descends to an isomorphism of K-theory also denoted \u03A8 : K(Y ) \u2192 K(X ). It does not respect the filtration by dimension. However, the hard Lefschetz condition implies that the image of F0 K(X ) is contained in F1 K(Y ), under the inverse \u03A6 of \u03A8. We call the image Fexc K(Y ); its elements can be represented by formal differences of sheaves supported on the exceptional fibres of \u03C0 : Y \u2192 X. We define the multi-regular part of K-theory, Fmr (X ), to be the preimage of F1 K(Y ) under \u03A8. Its elements can be represented by formal differences of sheaves supported in dimension one where at the generic point of each curve in the support, the associated representation of the stabilizer group of that point is a multiple of the regular representation. The following filtrations are respected by \u03A8: Fexc K(Y ) \u2282 F1 K(Y ) \u2282 K(Y ) F0 K(X ) \u2282 FmR K(X ) \u2282 K(X ). Define the exceptional DT generating series of Y , the multi-regular generating series, and degree zero generating series of X to be: DT \u03B1 (Y )q \u03B1 ,  DTexc (Y ) = \u03B1\u2208Fexc K(Y )  DT \u03B1 (X )q \u03B1  DTmr (X ) = \u03B1\u2208Fmr K(X )  DT \u03B1 (X )q \u03B1  DT0 (X ) = \u03B1\u2208F0 K(X )  We state the crepant resolution conjecture of [14, conjecture 1]: Conjecture 1.4. Let X be an orbifold CY3 satisfying the hard Lefschetz condition. Let Y be the Calabi-Yau resolution of X given by equation 1.3. Then using \u03A8 to identify the variables, we have an equality DT(Y ) DTmr (X ) = . DT0 (X ) DTexc (Y ) This dissertation makes progress towards proving this conjecture. 4  \u000C\u03C0-stable pairs Objects of the Hilbert scheme may be viewed as two-term complexes, \u03B3  OY \u2192 G, where the cokernel of \u03B3 must be zero, and where G may be any sheaf admitting such a map \u03B3. The new invariants introduced in this dissertation, \u03C0-stable pairs, are a modification of this idea. They have been constructed with a view towards proving the crepant resolution conjecture, and as such, \u03C0  they depend on a crepant resolution Y \u2192 X as described in the previous section. The objects of our moduli space allow more variation in our cokernels, but less in the sheaf G. In particular, a two-term complex \u03B3  OY \u2192 G is a \u03C0-stable pair (c.f. definition 2.7) if: 1. R\u2022 \u03C0\u2217 coker(\u03B3) is a zero-dimensional sheaf on X, and 2. G admits only the zero map from any sheaf P with the property above, namely that R\u2022 \u03C0\u2217 P is a zero-dimensional sheaf. Remark 1.5. These pairs were inspired by, and are a generalization of, the stable pairs of Pandharipande and Thomas [39]. In fact, when X = Y and \u03C0 is the identity map, the above definition reduces to their definition of stable pairs. Below, we prove that there is a finite-type algebraic space, \u03C0-Hilb\u03B1 parametrizing these objects with [G] = \u03B1 \u2208 K(Y ). We may then define invariants n\u03C7(\u03BD \u22121 (n)),  \u03C0-PT\u03B1 (Y ) = n\u2208Z \u03B1  where \u03BD : \u03C0-Hilb \u2192 Z is Behrend\u2019s microlocal function. Note that if \u03C0 : Y \u2192 X is the identity, then \u03C0-PT\u03B1 (Y ) = PT\u03B1 (Y ), the usual PandharipandeThomas invariants of Y . As with Donaldson-Thomas theory, we collect the 5  \u000Cinvariants into a generating series, \u03C0-PT\u03B1 (Y )q \u03B1 .  \u03C0-PT(Y ) = \u03B1\u2208F1 K(Y )  Main result The following theorem rests the work of Bridgeland [12] and Joyce\u2013Song [29], and we therefore require our Calabi-Yau threefold Y to satisfy H 1 (Y, OY ) = 0. Theorem 1.6. Let X be a projective Calabi-Yau threefold that is the coarse space of an CY3 orbifold X that satisfies the hard Lefschetz condition. Let \u03C0 : Y \u2192 X be the resolution given by equation 1.3. Then the generating series for the \u03C0-stable pair invariants and the DT invariants are related by the equation \u03C0-PT(Y ) =  DT(Y ) . DTexc (Y )  The aim of this dissertation is to prove this theorem. We summarize the chapters below. In chapter 2, we describe a torsion pair (P\u03C0 , Q\u03C0 ) that is crucial to our definition of \u03C0-stable pairs. We explain the similarities between \u03C0-stable pairs and PT stable pairs and objects of the Hilbert scheme. The chapter ends by establishing results about the moduli space of \u03C0-stable pairs. In chapter 3, we recall the concept of a stability condition in the sense of Joyce. We then define the stability condition that we will use through out. The rest of the chapter is dedicated to proving that we may apply Joyce\u2019s powerful machinery. In chapter 4, we introduce the Harder-Narasimhan filtration for our stability condition, which will be our main tool to prove the relationship between the stability condition and the torsion pair from chapter 2. In chapter 5, we introduce the motivic Hall algebra. In chapter 6, we introduce the infinite-type Hall algebra as a purely pedagogical tool. It helps us to give the essence of the idea of many re6  \u000Csults, without having to concern ourselves with convergence issues, which are handled in the next chapter. In chapter 7, we introduce the Laurent Hall algebra, address the convergence issues alluded to in the previous chapter, and prove theorem 1.6. Remark 1.7. To prove the crepant resolution conjecture, we need to prove that \u03C0-PT(Y ) = PT(X ) and then use ([2]) Bayer\u2019s proof of the PT/DT correspondence on X , PT(X ) =  DTmR (X ) . DT0 (X )  The hope is that the Fourier-Mukai isomorphism \u03A8 takes \u03C0-stable pairs (as an object in Db (Y )) to a PT pair on X . In his recent article [17], John Calabrese proves a relationship between the DT invariants of a Calabi-Yau threefold and its flop. This problem is similar in many respects to the crepant resolution conjecture studied in this thesis, and Calabrese uses many similar techniques. He constructs a torsion pair and new counting invariants which he relates to invariants on the flop via equations in the Hall algebra and the integration map. While this is very similar to our approach in outline, the actual torsion pair and counting invariants that Calabrese considers (even when adapted to the orbifold setting) are quite different from ours. It would be very interesting to find the precise relationship between the two approaches.  7  \u000CChapter 2  \u03C0-stable pairs In this section, we define \u03C0-stable pairs, and prove some basic results.  Categorical constructions Let A be an abelian category. Here we recall the notion of torsion pairs. Definition 2.1. Let (P, Q) be a pair of full subcategories of A. We say (P, Q) is a torsion pair if the following conditions hold. \u2022 Hom(T, F ) = 0 for any T \u2208 P and F \u2208 Q. \u2022 Any object E \u2208 A fits into a unique exact sequence, 0 \u2192 T \u2192 E \u2192 F \u2192 0,  (2.2)  with T \u2208 P and F \u2208 Q. We borrow the following lemma from Toda [45]. Lemma 2.3. Suppose that A is a noetherian abelian category. (i) Let P \u2282 A be a full subcategory which is closed under extensions and quotients in A. Then for Q = {E \u2208 A : Hom(P, E) = 0}, the pair (P, Q) is a torsion pair on A. (ii) Let Q \u2282 A be a full subcategory which is closed under extensions and subobjects in A. Then for P = {E \u2208 A : Hom(E, Q) = 0}, the pair (P, Q) is a torsion pair on A. Proof: We only show (i), as the proof of (ii) is similar. Take E \u2208 A with E\u2208 / Q. Then there is T \u2208 P and a non-zero morphism T \u2192 E. Since P is 8  \u000Cclosed under quotients, we may assume that T \u2192 E is a monomorphism in A. Take an exact sequence in A, 0 \u2192 T \u2192 E \u2192 F \u2192 0.  (2.4)  By the noetherian property of A and the assumption that P is closed under extensions, we may assume that there is no T  T \u2282 E with T \u2208 T . Then  we have F \u2208 Q and (2.4) gives the desired sequence.  Example 2.5. Let P = {0-dimensional sheaves on Y }, and let Q = {E \u2208 coh(Y ) : Hom(P, E) = 0 for all P \u2208 P}. Lemma 2.3 easily proves that the pair (P, Q) is a torsion pair. Let C = coh\u22641 (Y ) denote the full subcategory of coherent sheaves on Y whose support is of dimension no more than one. We make the following definitions: P\u03C0 = {P \u2208 C|R\u2022 \u03C0\u2217 P is a zero-dimensional sheaf on X}, and Q\u03C0 = {F \u2208 C| for all P \u2208 P\u03C0 , Hom(P, F ) = 0} = P\u03C0\u22A5 . Lemma 2.6. The pair (P\u03C0 , Q\u03C0 ) is a torsion pair in C. Proof: By lemma 2.3, it suffices to prove that P\u03C0 is closed under extensions and quotients. Let P , P \u2208 P\u03C0 , and consider the short exact sequence 0 \u2192 P \u2192 P \u2192 P \u2192 0. We are to show that such a P must live in P\u03C0 . Consider now the long exact sequence, 9  \u000C0 \u2192 \u03C0\u2217 P \u2192 \u03C0\u2217 P\u03C0 \u2192 \u03C0\u2217 P \u2192 R1 \u03C0\u2217 P \u2192 R1 \u03C0\u2217 P\u03C0 \u2192 R1 \u03C0\u2217 P \u2192 0. Since P , P \u2208 P\u03C0 , we know that R1 \u03C0\u2217 P = 0 and R1 \u03C0\u2217 P = 0, so R1 \u03C0\u2217 P = 0. We also know that \u03C0\u2217 P and \u03C0\u2217 P are zero-dimensional sheaves, and so it is clear then that \u03C0\u2217 P must be so as well. This proves that P\u03C0 is closed under extensions. Let P \u2208 P\u03C0 , and consider the quotient P \u2192 B \u2192 0. Denote the kernel of this map by K. As before, we get a long exact sequence, 0 \u2192 \u03C0\u2217 K \u2192 \u03C0\u2217 P \u2192 \u03C0\u2217 B \u2192 R1 \u03C0\u2217 K \u2192 R1 \u03C0\u2217 P \u2192 R1 \u03C0\u2217 B \u2192 0. Since P \u2208 P\u03C0 , R1 \u03C0\u2217 P = 0, and so R1 \u03C0\u2217 B = 0. It remains to show that \u03C0\u2217 B is zero-dimensional. We know that \u03C0\u2217 K is zero dimensional, since it is a subsheaf of (the zero-dimensional sheaf) \u03C0\u2217 P . The support of R1 \u03C0\u2217 K is contained in the singular locus. Suppose dim supp(R1 \u03C0\u2217 K) = 1. Then, K must have been supported in dimension two, however this contradicts the fact that K \u2208 coh\u22641 (Y ). Hence R1 \u03C0\u2217 K is zero dimensional. Now, \u03C0\u2217 B is the extension of zerodimensional sheaves, so it too is zero-dimensional. This completes the proof that B \u2208 P\u03C0 , and that (P\u03C0 , Q\u03C0 ) is a torsion pair.  Definition 2.7. A map \u03B3 : OY \u2192 G is a \u03C0-stable pair if G \u2208 Q\u03C0 and coker(\u03B3) \u2208 P\u03C0 . Remark 2.8. Our notion of \u03C0-stable pair is a generalization of the stable pairs of Pandharipande and Thomas [39]. In the trivial case when X = Y and \u03C0 = the identity, we have that (P\u03C0 , Q\u03C0 ) = (P, Q) of example 2.5, and the \u03C0-stable pairs are exactly PT stable pairs. Definition 2.9. Two \u03C0-stable pairs \u03B31 : OY \u2192 G1 and \u03B32 : OY \u2192 G2 are isomorphic if there exists a isomorphism of sheaves \u03B8 : G1 \u2192 G2 making the 10  \u000Cfollowing diagram commute: \u03B31  G G1 gg gg \u03B32 ggg \u03B8 3 \u0015  OY g  G2  Remark 2.10. In [12], the tilt of A with respect to the torsion pair (P, Q) of example 2.5 is denoted A# , and lemma 2.3 of [12] proves that A# epimorphisms of the form OY \u2192 F are precisely stable pairs. The abelian category generated by OY and C has a tilt whose epimorphisms of the form OY \u2192 G are precisely \u03C0-stable pairs. This is analogous to the tilt used in [12]. However, since it is not strictly necessary for any of our arguments, we will not present a proof here. We associate to every \u03C0-stable pair \u03B3 : OY \u2192 G a short exact sequence 0 \u2192 OC \u2192 G \u2192 P \u2192 0, where P = coker(\u03B3) \u2208 P\u03C0 and OC = OY / ker(\u03B3). Proposition 2.11. Let G be a non-zero sheaf. If OY \u2192 G is a \u03C0-stable pair, then G is not supported exclusively on exceptional curves, and R1 \u03C0\u2217 G = 0. Proof: Let 0 \u2192 OC \u2192 G \u2192 P \u2192 0 be the associated short exact sequence. We will first show that R1 \u03C0\u2217 OC = 0. Consider the short exact sequence 0 \u2192 IC \u2192 OY \u2192 OC \u2192 0. Pushing forward yields 0 \u2192 \u03C0\u2217 IC \u2192 \u03C0\u2217 OY \u2192 \u03C0\u2217 OC \u2192 R1 \u03C0\u2217 IC \u2192 R1 \u03C0\u2217 OY \u2192 R1 \u03C0\u2217 OC \u2192 0 which is exact since the dimension of the fibres of Y \u2192 X is at most one. Thus the vanishing of R1 \u03C0\u2217 OY by equation 1.2 implies that of R1 \u03C0\u2217 OC . 11  \u000CNow consider the following long exact sequence. 0 \u2192 \u03C0\u2217 OC \u2192 \u03C0\u2217 G \u2192 \u03C0\u2217 P \u2192 R1 \u03C0\u2217 OC \u2192 R1 \u03C0\u2217 G \u2192 R1 \u03C0\u2217 P \u2192 0. From above, we know R1 \u03C0\u2217 OC = 0. As well, P \u2208 P\u03C0 implies R1 \u03C0\u2217 P = 0. Thus R1 \u03C0\u2217 G = 0. Now, if C consists of only exceptional curves, then \u03C0\u2217 OC is zero-dimensional. This implies that \u03C0\u2217 G is the extension of zero-dimensional sheaves, and therefore zero-dimensional. This means that G \u2208 P\u03C0 . By definition of \u03C0stable pair, G \u2208 Q\u03C0 . By definition of Q\u03C0 the only map from an object of P\u03C0 to an object of Q\u03C0 is the zero map, hence the identity of G is the zero map, and G is the zero object. Let us introduce some terminology and results taken from [12] (modified for our purposes, since we are only interested in sheaves supported in dimension no more than one). Let M denote the stack of objects of coh\u22641 (Y ). It is an algebraic stack, locally of finite type over C. Let M(O) denote the stack of framed sheaves, that is, the stack whose objects over a scheme S are pairs (E, \u03B3) where E is a S-flat coherent sheaf on S \u00D7 Y , of relative \u03B3  dimension no more than one, together with a map OS\u00D7Y \u2192 E. Given a morphism of schemes f : T \u2192 S, and an object (F, \u03B4) over T , a morphism in M(O) lying over f is an isomorphism \u03B8 : f \u2217 (E) \u2192 F such that the following diagram commutes f \u2217 (OS\u00D7Y ) \u0015  f \u2217 (\u03B3) \u2217 G f (E)  can  OT \u00D7Y  \u03B4  \u0015  (2.12)  \u03B8  G F.  The symbol \u201Ccan\u201D denotes the canonical isomorphism of pullbacks.  12  \u000CThere is a natural map q  M(O) \u2192 M  (2.13)  sending a sheaf with a section to the underlying sheaf. The following lemmas are 2.4 and 2.5 of [12]. Lemma 2.14. The stack M(O) is algebraic and the morphism q is representable and of finite type. Lemma 2.15. There is a stratification of M by locally-closed substacks Mr \u2282 M such that objects F of Mr (C) are coherent sheaves satisfying dimC H 0 (Y, F ) = r. Furthermore, the pullback of the morphism q to Mr is a Zariski fibration with fibre Cr . Both of these are proven in [12]. Lemma 2.16. The moduli space of \u03C0-stable pairs with a fixed Hilbert polynomial is an algebraic space of finite type. Proof: Let \u03C0-Hilb(\u03B2,n) denote the stack of \u03C0-stable pairs on Y whose sheaf G has Chern character (0, 0, \u03B2, n). Notice that \u03C0-Hilb(\u03B2,n) \u2282 M(O). Lemma 2.14 states that M(O) is an algebraic stack. Using the forth-coming description of P\u03C0 and Q\u03C0 in terms of stability, and using theorem 4.8 of [26], we know that P\u03C0 and Q\u03C0 are algebraic substacks of M. We use this fact to proven that \u03C0-Hilb is an algebraic substack of M.  13  \u000CConsider the following cartesian diagram, Q\u03C0 (O)  G M(O)  \u0015  \u0015  q  G M.  Q\u03C0  Since Q\u03C0 and M(O) are algebraic stacks, their fibre product, Q\u03C0 (O) is an algebraic stack, too. There is a map of stacks of stack c : Q\u03C0 (O) \u2192 M which takes \u03B3 : OY \u2192 G to the cokernel coker(\u03B3). Since \u03C0-Hilb fits into the following cartesian diagram, G P\u03C0  \u03C0-Hilb \u0015  Q\u03C0 (O)  c  \u0015 G M,  we conclude that it too is an algebraic stack. Now we prove \u03C0-Hilb(\u03B2,n) is of finite type. In lemma 3.16, we prove that the sheaves of Chern character (0, 0, \u03B2, n) underlying \u03C0-stable pairs form a bounded family. In particular, this means that there exists a coherent sheaf E such that every sheaf underlying a \u03C0-stable pair G of Chern character (0, 0, \u03B2, n) is a quotient of E. Now if we choose a polarization O(1) of Y , we have a finite type scheme Quot(E, P ) where P is the Hilbert polynomial P (t) = \u03B2 \u00B7 O(1)t + n. There is a morphism Quot(E, P ) \u2192 M that takes a quotient E \u2192 G and sends it to G. This map is of finite type. Consider the diagram: T f  \u0015  M(O)  f  G Quot(E, P ) q q  \u0015  GM  where T is the pullback of q by q. The image of the map q contains all sheaves underlying \u03C0-stable pairs with Hilbert polynomial P . This means that the image of T in M(O) contains all 2-term complexes [OY \u2192 G] where G is a sheaf underlying a \u03C0-stable pair with Hilbert polynomial P . 14  \u000CIn particular, the image of T contains all \u03C0-stable pairs. Now since q is of finite type, so must f be of finite type. The scheme Quot(E, P ) is of finite type over C, so then we may conclude that T is finite type over C. Thus the image of T under f is of finite type. The stack of all \u03C0-stable pairs is in this image, hence of finite type. It remains to prove that this stack is in fact an algebraic space. It suffices to prove that all closed points of the stack \u03C0-Hilb(\u03B2,n) have trivial automorphism. Let OY \u2192 G be a \u03C0-stable pair. We will show that it has only the trivial automorphism. Consider the associated short exact sequence, 0 \u2192 OC \u2192 G \u2192 P \u2192 0. An automorphism of this \u03C0-stable pair leads to a diagram of the form, 0  G OC \u03B3 G G h G P  0  g  id  \u0015  G OC  \u03B3  \u0015  \u0015  G0 g  G 0.  GG h GP  We will show that g is the identity map. Consider the following diagram, obtained by subtracting the identity from the diagram above, 0  G OC \u03B3 G G h G P zero  0  \u0015  G OC  g\u2212 id  \u0015 \u00D2 \u0015 GG h GP  G0  (2.17)  g\u2212id  G 0.  Since the left-most vertical arrow is zero, a diagram chase proves that the dotted morphism exists and commutes with the diagram. However, P \u2208 P\u03C0 and G \u2208 Q\u03C0 , so this map must be zero. As a consequence, g \u2212 id = 0. Homological algebra then implies that g \u2212 id must be the zero map, forcing g = id. This proves that closed points of the stack of \u03C0-stable pairs have no automorphisms, and that \u03C0-stable pairs form an algebraic space.  15  \u000CThe Behrend function identity We state and prove a variation of [12, theorem 3.1] of Bridgeland. Lemma 2.18. Let \u03B3 : OY \u2192 G be a \u03C0-stable pair. Then there is an equality of Behrend\u2019s microlocal functions \u03BDM(O) (\u03B3) = (\u22121)\u03C7(G) \u03BDM (G). Proof: The case when G is a stable pair is taken care of by theorem 3.1 of [12]. Thus, we may assume that the cokernel P of \u03B3 : OY \u2192 G has one-dimensional support. Let OC \u2282 G be the image of \u03B3. It is the structure sheaf of a subscheme C \u2282 Y of dimension 1. There is a line bundle L on Y such that H i (Y, G \u2297 L) = 0  (2.19)  for all i &gt; 0, and there is a divisor H \u2208 |L| such that H meets C at finitely many points, none of which are in the support of coker(\u03B3). This claim is verified in lemma 2.22. From here, the proof is identical to Bridgeland\u2019s, but we include a portion of it to illustrate his ideas. There is a short exact sequence s  0 \u2192 OY \u2192 L \u2192 OH (H) \u2192 0 where s is the section of L corresponding to the divisor H. Tensoring it with G, and using the above assumptions yields a diagram of sheaves OY f 0  ff \u03B3 ff\u03B4f ff \u0015 3 GG \u03B1 GF  (2.20) \u03B2  GK  G0  where F = G \u2297 L. The support of the sheaf K is zero-dimensional, and  16  \u000Cdisjoint from the support of coker(\u03B3). In particular, HomY (K, F ) = 0.  (2.21)  Consider two points of the stack M(O) corresponding to the maps \u03B3 : OY \u2192 G and \u03B4 : OY \u2192 F. The statement of lemma 2.18 holds for the map \u03B4 because lemma 2.15 together with equation 2.19 implies that q : M(O) \u2192 M is smooth of relative dimension \u03C7(F ) = H 0 (Y, F ) over an open neighbourhood of the point F \u2208 M(C). On the other hand, tensoring sheaves with L defines an automorphism of M, so the microlocal function of M at the points corresponding to G and F are equal. To prove the lemma, it suffices to show that (\u22121)\u03C7(G) \u00B7 \u03BDM(O) (\u03B3) = (\u22121)\u03C7(F ) \u00B7 \u03BDM(O) (\u03B4). Consider the stack W whose S-valued points are diagrams of S-flat sheaves on S \u00D7 Y of the form OS\u00D7Y  pp pp\u03B4S pp pp \u0015 \u03B1S 5 G GS G FS \u03B3S  0  \u03B2S  G KS  G 0.  There are two morphisms p : W \u2192 M(O),  q : W \u2192 M(O),  taking such a diagram to the maps \u03B3S and \u03B4S respectively. By passing to an open substack of W , we may assume that equation 2.21 holds for all C-valued points of W . It follows that p and q induce injective maps on 17  \u000Cstabilizer groups of C-valued points, and hence are representable. Recall that Behrend\u2019s microlocal function satisfies the property that when f : T \u2192 S is a smooth morphism of relative dimension d, there is an identity [3, proposition 1.5] \u03BDT = (\u22121)d f \u2217 (\u03BDS ). Using this identity, it will be enough to show that at the point w \u2208 W (C) corresponding to the diagram 2.20, the morphisms p and q are smooth of relative dimension \u03C7(K) and 0, respectively. For the proof of these facts, see the [12, pages 11\u201313].  Lemma 2.22. Given a \u03C0-stable pair \u03B3 : OY \u2192 G, we may choose a very ample divisor H on X such that its pull-back is equal to its proper transform (we denote both by H), it satisfies supp(coker \u03B3) \u2229 H = \u2205, H \u2229 supp G is 0-dimensional, and H 1 (Y, G(H)) = 0. Proof: First we collect a little notation. Let E be the exceptional locus of Y , let E be the image of E in X. Define a subset Z to be Z = {p \u2208 X : G|\u03C0\u22121 (p) is one dimensional} \u2282 E . Notice that Z is a finite collection of points. Since coker \u03B3 \u2208 P\u03C0 , we have that \u03C0(supp(coker \u03B3)) is zero dimensional. Moreover, \u03C0(supp G) is one dimensional. Thus we may choose an ample divisor H on X so that H \u2229 \u03C0(supp(coker \u03B3)) = \u2205 and H \u2229 \u03C0(supp G) is zero dimensional and does not contain any of the points in Z. It follows that H \u2229 supp(coker \u03B3) is empty, and H \u2229 supp(G) is zero dimensional. Moreover, by Serre vanishing, we may assume that H is  18  \u000Csufficiently ample on X so that H 1 (X, (\u03C0\u2217 G)(H)) = 0.  (2.23)  We now show that H 1 (Y, G(H)) = 0. By proposition 2.11, we know that R1 \u03C0\u2217 G = 0, since OY \u2192 G is a \u03C0-stable pair. The sequence 0 \u2192 G \u2192 G(H) \u2192 G(H)|He \u2192 0 gives . . . \u2192 R1 \u03C0\u2217 G \u2192 R1 \u03C0\u2217 G(H) \u2192 R1 \u03C0\u2217 G(H)|He \u2192 0. However, we know R1 \u03C0\u2217 G = 0 and G(H)|He is supported on points so R1 \u03C0\u2217 G(H)|He = 0, so R1 \u03C0\u2217 G(H) = 0. Now by the Leray spectral sequence, H 1 (Y, G(H)) = H 1 (X, \u03C0\u2217 (G(H)) = H 1 (X, \u03C0\u2217 (G \u2297 \u03C0 \u2217 OX (H))) = H 1 (X, (\u03C0\u2217 (G))(H)) = 0, where the last equality comes from equation 2.23.  19  \u000CChapter 3  Stability conditions In this section, we define a stability condition on C = coh\u22641 Y . We follow Joyce\u2019s treatment of stability conditions as found in section 4 of [26], though not in as great generality. Let N1 (Y ) denote the abelian group of cycles of dimension one modulo numerical equivalence. We begin by quoting lemmas 2.1 and 2.2 of [12]. Lemma 3.1. An element \u03B2 \u2208 N1 (Y ) has only finitely many decompositions of the form \u03B2 = \u03B21 + \u03B22 with \u03B2i effective. Lemma 3.2. The Chern character map induces an isomorphism ch = (ch2 , ch3 ) : F1 K(Y ) \u2192 N1 (Y ) \u2295 Z. Define \u2206 = {[E] \u2208 F1 K(Y ) : E \u2208 C} to be the positive or effective cone of F1 K(Y ). Definition 3.3. A stability condition on C is a triple (T, \u03C4, \u2264) where (T, \u2264) \u03C4  is a set T with a total ordering \u2264, and \u03C4 is a map \u2206 \u2192 T from the effective cone to T , satisfying the following condition: whenever \u03B1 + \u03B2 = \u03B3 in \u2206, then \u03C4 (\u03B1) &lt; \u03C4 (\u03B3) &lt; \u03C4 (\u03B2), or \u03C4 (\u03B2) &lt; \u03C4 (\u03B3) &lt; \u03C4 (\u03B1), or \u03C4 (\u03B1) = \u03C4 (\u03B3) = \u03C4 (\u03B2). 20  \u000CA triple (T, \u03C4, \u2264) is called a weak stability condition if it satisfies the weaker condition that whenever \u03B1 + \u03B2 = \u03B3 in \u2206, \u03C4 (\u03B1) \u2264 \u03C4 (\u03B3) \u2264 \u03C4 (\u03B2) or \u03C4 (\u03B2) \u2264 \u03C4 (\u03B3) \u2264 \u03C4 (\u03B1). Definition 3.4. A non-zero sheaf G is 1. \u03C4 -semistable if for all S \u2282 G, such that S \u223C = 0, we have that \u03C4 (S) \u2264 \u03C4 (G/S); 2. \u03C4 -stable if for all S \u2282 G, such that S \u223C = 0, we have that \u03C4 (S) &lt; \u03C4 (G/S); 3. \u03C4 -unstable if it is not \u03C4 -semistable. f  Lemma 3.5. Let F and G be \u03C4 -semistable sheaves, and let F \u2192 G be a map of sheaves. Then either \u03C4 (F ) \u2264 \u03C4 (G) or f = 0. Proof: Consider the inclusion map \u03B9 : im(f ) \u2192 G. Since G is semistable, we know either im(f ) = 0 or \u03C4 (im(f )) \u2264 \u03C4 (G). Consider now the corestriction of f , cor(f ) : F \u2192 im(f ). Since F is semistable, we know that either im(f ) = 0 or \u03C4 (F ) \u2264 \u03C4 (im(f )). This implies either im(f ) = 0 or \u03C4 (F ) \u2264 \u03C4 (G). Now we define a stability condition on C. Definition 3.6. Let E be the exceptional divisor of \u03C0. Choose an ample divisor H on X, let H denote the total transform of H in Y . Let L be a very ample line bundle on Y . Given a sheaf G in C, define the \u03C0-slope of G to be \u00B5\u03C0 (G) =  \u03C7(G) \u03C7(G) , \u2208 (\u2212\u221E, +\u221E] \u00D7 (\u2212\u221E, +\u221E], \u03B2\u00B7H \u03B2\u00B7L  where (\u2212\u221E, +\u221E] \u00D7 (\u2212\u221E, +\u221E] is ordered lexicographically, and \u03B2 = \u03B2G the homology class associated to the support of G. Theorem 3.7. The map \u00B5\u03C0 : \u2206 \u2192 (\u2212\u221E, +\u221E] \u00D7 (\u2212\u221E, +\u221E] 21  \u000Cdefines a weak permissible stability condition. Unpacking the definitions 4.1 and 4.7 of [26], we see that a weak permissible stability condition is one that satisfies three conditions: 1. (weak seesaw property) for any short exact sequence 0 \u2192 A \u2192 G \u2192 B \u2192 0, either \u00B5\u03C0 (A) \u2264 \u00B5\u03C0 (G) \u2264 \u00B5\u03C0 (B) or \u00B5\u03C0 (A) \u2265 \u00B5\u03C0 (G) \u2265 \u00B5\u03C0 (B), 2. \u00B5\u03C0 is dominated by a permissible stability condition, and 3. the family of all \u00B5\u03C0 -stable sheaves of a fixed Chern class is bounded. We will prove theorem 3.7 by proving the above three properties in lemma 3.8, corollary 3.10, and lemma 3.16. Lemma 3.8. The function \u00B5\u03C0 satisfies the seesaw property. Proof: Let 0 \u2192 A \u2192 G \u2192 B \u2192 0 be a short exact sequence of sheaves, and suppose \u00B5\u03C0 (A) \u2264 \u00B5\u03C0 (G). Less concisely, we are supposing \u03C7(A)  \u03C7(A) \u03C7(G) \u03C7(G) \u2264 , , \u03B2 \u00B7 L \u03B2A \u00B7 H A \u03B2G \u00B7 H \u03B2G \u00B7 L ,  from which we are to deduce that \u00B5\u03C0 (G) \u2264 \u00B5\u03C0 (B). Before we start a caseby-case analysis, notice that \u03C7(G) = \u03C7(A) + \u03C7(B) and \u03B2G = \u03B2A + \u03B2B . case 1:  \u03C7(A) e \u03B2 A \u00B7H  &lt;  \u03C7(G) e \u03B2 G \u00B7H  and no denominator is zero.  Then this follows from the observation a+c a+c c a &lt; \u21D2 &lt; , b b+d b+d d provided b, d &gt; 0. In particular, we assume \u03C7(A) \u03B2A \u00B7 H  \u2264  \u03C7(G) \u03B2G \u00B7 H  .  22  \u000CRewriting the second term yields \u03C7(A) \u03B2A \u00B7 H  \u2264  \u03C7(A) + \u03C7(B) (\u03B2A + \u03B2B ) \u00B7 H  .  The observation above then proves that \u03C7(A) + \u03C7(B) (\u03B2A + \u03B2B ) \u00B7 H  \u2264  \u03C7(B) \u03B2B \u00B7 H  ,  as desired. case 2:  \u03C7(A) e \u03B2 A \u00B7H  =  \u03C7(G) \u03C7(A) e , \u03B2A \u00B7L \u03B2 G \u00B7H  We are given that  \u2264  \u03C7(G) \u03B2G \u00B7L  \u03C7(A) e \u03B2 A \u00B7H  =  and no denominator is zero. \u03C7(G) e, \u03B2 G \u00B7H  so  \u03C7(A)(\u03B2G \u00B7 H) = (\u03B2A \u00B7 H)\u03C7(G). Writing everything in terms of A and B, \u03C7(A)((\u03B2A + \u03B2B ) \u00B7 H) = \u03B2A \u00B7 H(\u03C7(A) + \u03C7(B)), which implies \u03C7(A)(\u03B2B \u00B7 H) = \u03B2A \u00B7 H\u03C7B . Since we assume that all denominators are non-zero, we have \u03C7(A) \u03B2A \u00B7 H  =  \u03C7(B) \u03B2B \u00B7 H  .  So we must show that \u03C7(G) \u03C7(B) &lt; . \u03B2G \u00B7 L \u03B2B \u00B7 L This follows from the same observation made in case 1. case 3: \u03B2A \u00B7 H = 0. Then +\u221E = \u03C7(A)+\u03C7(B) e (\u03B2A +\u03B2B )\u00B7H  \u03C7(A) e \u03B2 A \u00B7H  \u2264  \u03C7(A)+\u03C7(B) e. (\u03B2A +\u03B2B )\u00B7H  This implies that  = +\u221E, so (\u03B2A + \u03B2B ) \u00B7 H = 0, and hence \u03B2B \u00B7 H = 0. This  23  \u000Creduces us to Gieseker stability on Y , which we know satisfies the seesaw property. case 4: \u03B2G \u00B7 H = 0. We know \u03B2 \u00B7 \u03C0 \u2217 H = \u03C0\u2217 (\u03B2 \u00B7 \u03C0 \u2217 H) = \u03C0\u2217 (\u03B2) \u00B7 H \u2265 0, hence \u03B2 \u00B7 H \u2265 0 for any effective curve class \u03B2, so we must have \u03B2A \u00B7 H = 0 and \u03B2B \u00B7 H = 0. This lands us back in the case of Gieseker stability on Y , and lemma 3.8 is proven.  Let us now recall from Joyce the following facts and definitions. A stability condition \u03C4 on C is permissible if: \u2022 C is \u03C4 -artinian, i.e. there exists no infinite descending chain \u00B7 \u00B7 \u00B7 A2 \u2282 A1 \u2282 A in C such that Ai = Ai+1 , and \u03C4 (Ai+1 ) \u2265 \u03C4 (Ai /Ai+1 ) for all i; and \u2022 the substack of \u03C4 -semistable objects of a fixed Chern character of the stack parametrizing objects of C is a constructible substack of M. Thankfully, the second condition amounts to showing that the family of \u00B5\u03C0 semistable sheaves of a fixed Chern character is bounded. See the proof of theorem 4.30 from [26] for details. Joyce proves that these properties are inherited by domination. Recall that a stability condition \u03C4\u02DC is said to dominate \u03C4 if for any A, B in C with \u03C4 (A) \u2264 \u03C4 (B) then \u03C4\u02DC(A) \u2264 \u03C4\u02DC(B). Define the following stability condition: \u03B4(G) = \u2212 dim supp G \u2208 Z. In [26], it is proven that the abelian category coh(Y ) is \u03B4-artinian. To prove that C = coh\u22641 (Y ) is \u00B5\u03C0 -artinian, we will show that \u00B5\u03C0 is dominated by \u03B4. Lemma 3.9. The stability condition \u00B5\u03C0 is dominated by the stability condition \u03B4.  24  \u000CProof: Let \u00B5\u03C0 (A) \u2264 \u00B5\u03C0 (B). We need to show that this implies that \u03B4(A) \u2264 \u03B4(B). Expanding, we have that \u03C7(B) \u03C7(B) \u03C7(A) \u2264 . , \u03B2 \u00B7 L \u03B2A \u00B7 H A \u03B2B \u00B7 H \u03B2B \u00B7 L \u03C7(A)  ,  We proceed with a case-by-case analysis. case 1: the denominators are non-zero. Since we have restricted our attention to sheaves supported in dimension \u2264 1, it follows that neither A nor B is 0-dimensional. Thus, they are both one dimensional, and \u03B4(A) = \u03B4(B). In particular, \u03B4(A) \u2264 \u03B4(B). case 2: \u03B2B \u00B7 H = 0 and \u03B2A \u00B7 H = 0. Then dim supp A \u2265 1 \u2265 dim supp B. So \u03B4(A) \u2264 \u03B4(B). case 3: Both \u03B2B \u00B7 H = 0 and \u03B2A \u00B7 H = 0. Then \u00B5\u03C0 (A) \u2264 \u00B5\u03C0 (B) amounts to regular Geiseker stability, which is dominated by \u03B4 as demonstrated by Joyce [26, \u00A74.4].  Corollary 3.10. The category C is \u00B5\u03C0 -artinian. Proof: As mentioned above, this follows from the previous lemma since it is inherited by domination. To finish the proof that \u00B5\u03C0 is a permissible weak stability condition, it remains only to prove the family of all \u00B5\u03C0 -semistable sheaves of a fixed Chern class is bounded. This is proven in the next section.  Boundedness Let us recall the basic results concerning boundedness (cf. [23]). Definition 3.11. A subcategory U of coh(Y ) is bounded if there exists a scheme S of finite type and a sheaf U on X \u00D7 S such that for every object \u223C U |X\u00D7{s } . Ui of U, there exists a closed point si \u2208 S such that Ui = i  25  \u000CNotice that this definition still makes sense if we have a set of isomorphism classes of sheaves instead of a category. Definition 3.12. Let Y be a scheme, let O(1) be an ample line bundle, and let m be an integer. A sheaf F on Y is m-regular if, for all i &gt; 0, H i (Y, F (m \u2212 i)) = 0. A proof for the following may be found in [30], as well as in [37]. Lemma 3.13. If F is m-regular, then the following statements are true: 1. F is m -regular for all m \u2265 m. 2. F (m) is globally generated. 3. For all n \u2265 0, the natural map H 0 (Y, F (m))\u2297H 0 (Y, O(n)) \u2192 H 0 (Y, F (n+ m)) is surjective. Definition 3.14. The Mumford-Castelnuovo regularity of a sheaf F is the number reg(F ) = inf{m \u2208 Z : F is m-regular }. Lemma 3.15. Let U be a category of sheaves on Y . The following statements are equivalent. 1. U is bounded. 2. The set of Hilbert polynomials of objects Ui of U is finite, and there is an integer N such that for all objects Ui of U, reg Ui &lt; N . 3. The set of Hilbert polynomials of objects Ui of U is finite, and there exists a sheaf F such that each object of U is isomorphic to a quotient of F . Proof of this lemma may be found in [20]. Now we prove the boundedness result we require. \u03B3  Lemma 3.16. The family of \u03C0-stable pairs OY \u2192 G with a fixed Chern class is bounded. 26  \u000CProof: To each such \u03C0-stable pair there is an associated a short exact sequence, 0 \u2192 OC \u2192 G \u2192 P \u2192 0, where OC is the image of the map \u03B3, and P is the cokernel. We will show that the family of possibilities for OC and the family of possible P s are both bounded families. Once this is established, it is clear that the family of sheaves underlying a \u03C0-stable pair is a bounded family of sheaves. First we will consider the family of possibilities for OC . To show that this family is bounded, we will show: 1. the Hilbert polynomials of this family take on only a finite number of values; and 2. there exists a single sheaf that surjects onto each member of this family. The second requirement is trivially satisfied, since each member of this family is the structure sheaf of a subscheme of Y , and hence, admits a surjective maps from OY . It remains to find upper and lower bounds for the coefficients of the Hilbert polynomial of a general element from this family. In contrast to P T -theory, the support of OC is not equal to the support of G, since P is not necessarily zero-dimensional. However, we still have \u03B2G = \u03B2C +\u03B2P , where all \u03B2 are effective. We know that there are only finitely many decompositions of \u03B2G into the sum of two effective curve classes. This forces an upper and lower bound on the linear coefficient of the Hilbert polynomial of OC . It remains to find upper and lower bounds for the Euler characteristic of OC (the constant coefficient of the Hilbert polynomial). The Leray spectral sequence proves that \u03C7(P ) = \u03C7(R\u2022 \u03C0\u2217 P ) \u2265 0, the inequality following from the fact that R\u2022 \u03C0\u2217 P is zero dimensional. Now, \u03C7(G) = \u03C7(P ) + \u03C7(OC ), and \u03C7(P ) \u2265 0 implies \u03C7(OC ) \u2264 \u03C7(G). This gives us an upper bound, since the Chern character, and hence, the Euler characteristic, of G is fixed. For the lower bound, let \u03B1G = (\u03B2G , nG ) be the Chern character of G, and let \u03B1C = (\u03B2C , nC ) be the Chern character of C. In general, if Hilb(\u03B2,n) is non-empty (say one of its points represents a curve J), then dim Hilb(\u03B2,n+k) \u2265 3k since we get a 3k-dimensional space 27  \u000Cof curves coming from the curve J with k \u201Cwandering points.\u201D This line of reasoning tells us that dim Hilb(\u03B2C ,nG ) \u2265 3(nG \u2212 nC ). Rearranging this yields nC \u2265 nG \u2212  1 dim Hilb(\u03B2C , nG ). 3  This gives us a lower bound for nC , which completes the proof that the corresponding family is bounded. Now to show that the family of cokernels is bounded, we will show 1. there is a common upper-bound to the index of regularity; and 2. the Hilbert polynomials of this family take on only a finite number of values Using the Leray spectral sequence again, we note that for all P \u2208 P\u03C0 , H 1 (Y, P ) = H 1 (X, \u03C0\u2217 P ) = 0, thus, all P \u2208 P\u03C0 are 1-regular. To show this family is bounded, it remains to find upper and lower bounds for the coefficients of the Hilbert polynomial of a general object. As above, there are only a finite number of options for the support curve of P . This yields upper and lower bounds on the linear coefficient of the Hilbert polynomial. We know that \u03C7(G) = \u03C7(P ) + \u03C7(OC ). Since \u03C7(OC ) is bounded, and \u03C7(G) is fixed, so too must \u03C7(P ) be bounded. This completes the proof that the sheaves underlying a \u03C0-stable pair of fixed K-class forms a bounded family of sheaves. This also completes the proof that \u00B5\u03C0 is a permissible weak stability condition.  28  \u000CChapter 4  The torsion pair and the stability condition In this section, we show that P\u03C0 may be conveniently expressed in terms of the stability condition, and similarly for Q\u03C0 . First we give a rapid introduction to the modern Harder-Narasimhan property, a generalization of the Harder-Narasimhan filtration of [22]. Definition 4.1. A weak stability condition (T, \u03C4, \u2264) on C is said to have the Harder-Narasimhan property if for every sheaf G, there exists a unique filtration of G 0 = HN\u03C4 (G)0 \u2282 HN\u03C4 (G)1 \u2282 \u00B7 \u00B7 \u00B7 \u2282 HN\u03C4 (G)N \u22121 \u2282 HN\u03C4 (G)N = G (where the inclusions are strict) such that the quotients Qi = HN\u03C4 (G)i /HN\u03C4 (G)i\u22121 are \u03C4 -semistable and \u03C4 (Qi ) &gt; \u03C4 (Qi+1 ) for all i &gt; 0. When it is clear from context, most of the notation will be suppressed, and we will denote the Harder-Narasimhan filtration of G with respect to \u03C4 by 0 \u2282 G1 \u2282 G2 \u2282 \u00B7 \u00B7 \u00B7 \u2282 GN \u22121 \u2282 GN = G. The Gi are called the filtered objects of the Harder-Narasimhan filtration, and the Qi are called the quotient objects. We borrow the following definition and theorem from Joyce [26]. In [24, \u00A79], Joyce proves that the category of coherent sheaves satisfies assumptions 29  \u000C3.7 of [26]. This is enough for us to conclude that the assumptions are also true of C the category of coherent sheaves supported in dimension one or less. Theorem 4.2 ([26], Theorem 4.4). Let (T, \u03C4, \u2264) be a weak stability condition on an abelian category A. If A is Noetherian and \u03C4 -artinian, then (T, \u03C4, \u2264) has the Harder-Narasimhan property. Corollary 4.3. The weak stability condition \u00B5\u03C0 on C has the HarderNarasimhan property. Proof: The category C is Noetherian because it is a subcategory of the category of coherent sheaves, which is Noetherian. Corollary 3.10 proves that C is \u00B5\u03C0 -artinian. When we refer to the Harder-Narasimhan filtration in what follows, we will always be referring to the filtration with respect to the stability condition \u00B5\u03C0 . We present some notation before we state and prove the main result of this section. Recall that our slope function \u00B5\u03C0 takes values in the lexicographically ordered set (\u2212\u221E, +\u221E]\u00D7(\u2212\u221E, +\u221E]. To avoid awkwardly writing the ordered pairs (+\u221E, +\u221E) and (+\u221E, 0) through-out, let us denote \u221E := (+\u221E, +\u221E), and  \u221E := (+\u221E, 0). 2  Given an interval I \u2282 (\u2212\u221E, +\u221E] \u00D7 (\u2212\u221E, +\u221E], we define SS(I) \u2282 C to be the full subcategory of zero objects together with those one-dimensional sheaves whose Harder-Narasimhan quotients have \u00B5\u03C0 -value in the interval I. If a, b \u2208 (\u2212\u221E, +\u221E] \u00D7 (\u2212\u221E, +\u221E] such that a &lt; b, then we denote the closed interval between a and b by a \u2264  \u2264 b, and similarly for open, half-open,  etc. intervals.  30  \u000CLemma 4.4. P\u03C0 = SS(  \u2265  \u221E ), 2  and  \u221E ). 2 Proof: First we will show that P\u03C0 \u2282 SS( \u2265 \u221E 2 ). Q\u03C0 = SS(  &lt;  Case 1: let P \u2208 P\u03C0 be semi-stable. We will show that P \u2208 SS( Since P is semi-stable, it suffices to show that \u00B5\u03C0 (P ) \u2265  \u221E 2 .  \u2265  \u221E 2 ).  Now P \u2208 P\u03C0  implies that \u03C7(P ) \u2265 0. By ampleness of L, we know \u03B2P \u00B7 L \u2265 0. Hence \u00B5\u03C0 (P ) \u2265 (+\u221E, 0). Case 2: Let P \u2208 P\u03C0 be general, let the following be its Harder-Narasimhan (HN) filtration, 0 = P0 \u2282 P1 \u2282 \u00B7 \u00B7 \u00B7 \u2282 PN \u22121 \u2282 PN = P, and let Qi =  Pi Pi\u22121  be the ith quotient; we must show that \u00B5\u03C0 (Qi ) \u2265  \u221E 2 .  Since  PN = P \u2208 P\u03C0 and P\u03C0 is closed under quotients, it follows that QN \u2208 P\u03C0 . By definition of the HN filtration, QN is semi-stable, hence \u00B5\u03C0 (QN ) \u2265 and QN \u2208 SS(  \u2265  \u221E 2 )  by the previous case. Another defining property is  that \u00B5\u03C0 (Q1 ) &gt; \u00B5\u03C0 (Q2 ) &gt; \u00B7 \u00B7 \u00B7 &gt; \u00B5\u03C0 (QN ). Hence, \u00B5\u03C0 (Qi ) \u2265 other words, P \u2208 SS(  \u2265  \u221E 2  for all i, in  \u221E 2 ).  Now we will show that SS( \u221E 2 )  \u221E 2  Case 1: Let G \u2208 SS(  \u2265  implies that \u00B5\u03C0 (G) \u2265  \u221E 2 .  \u221E 2 )  \u2265  \u221E 2 )  \u2282 P\u03C0 .  be semi-stable. In this case, G \u2208 SS(  \u2265  Since (P\u03C0 , Q\u03C0 ) is a torsion pair, for every sheaf  there exists a uniquely associated short exact sequence, 0\u2192A\u2192G\u2192B\u21920 where A \u2208 P\u03C0 and B \u2208 Q\u03C0 . By the above, we know that A \u2208 SS(  \u2265  \u221E 2 ).  If  A itself was semistable, then we could conclude immediately that \u00B5\u03C0 (A) \u2265 \u221E 2 .  Using this argument, and inducting on the length of the HN filtration  of A, we may prove that \u00B5\u03C0 (Qi ) \u2265 \u00B5\u03C0 (Ai ) \u2265 \u00B5\u03C0 Qi+1 , where Qi denotes the ith quotient object of the HN filtration of A, and Ai denotes the ith filtered object. This implies that \u00B5\u03C0 (A) \u2265  \u221E 2  in general. 31  \u000C\u221E 2  We know that \u00B5\u03C0 (G) \u2265  = (+\u221E, 0) so \u03C7(G) \u2265 0 by non-negativity  of denominators in \u00B5\u03C0 . Similarly, \u03C7(A) \u2265 0. By the see-saw property and the semi-stability of G, we know that \u00B5\u03C0 (A) \u2264 \u00B5\u03C0 (G) \u2264 \u00B5\u03C0 (B), hence \u00B5\u03C0 (B) \u2265  \u221E 2  and as before we have \u03C7(B) \u2265 0.  Notice that G must be supported on a fibre of \u03C0 because \u00B5\u03C0 (G) \u2208 {+\u221E} \u00D7 (0, +\u221E]. Hence B is also supported on a fibre. We claim that this forces H 0 (B) = 0. For suppose there was a non-zero map OY \u2192 B. This would yield non-trivial 0 \u2192 OC \u2192 B where C is the support of the map OY \u2192 B. From the proof of proposition 2.11, we know that R1 \u03C0\u2217 OC = 0, which implies that OC \u2208 P\u03C0 which contradicts the definition of Q\u03C0 . Hence H 0 (B) = 0. However, \u03C7(B) \u2265 0 so dim H 1 (B) \u2264 0. This implies that H 1 (B) = 0, which implies that R1 \u03C0\u2217 B = 0 (by the theorem of cohomology and basechange) and hence B \u2208 P\u03C0 . Since B \u2208 Q\u03C0 we conclude that B = 0 and G = A \u2208 P\u03C0 . Case 2: Let G \u2208 SS(  \u2265  \u221E 2 )  be general. We need to show that G \u2208 P\u03C0 .  Let 0 = G0 \u2282 G1 \u2282 \u00B7 \u00B7 \u00B7 \u2282 GN \u22121 \u2282 GN = G Gi Gi\u22121  be the HN filtration of G, and let Qi =  denote the corresponding \u221E 2 ; notice that G1 = Q1 , \u221E 2 . The previous case then  semistable quotients. By assumption, \u00B5\u03C0 (Qi ) \u2265 so we have that G1 is semistable and \u00B5\u03C0 (G1 ) \u2265  proves that for all i, Qi \u2208 P\u03C0 , so in particular Q1 = G1 \u2208 P\u03C0 . Now consider the following short exact sequence 0 \u2192 G1 \u2192 G2 \u2192 Q2 \u2192 0 We know G1 \u2208 P\u03C0 and Q2 \u2208 P\u03C0 hence G2 \u2208 P\u03C0 . Proceeding inductively, we conclude G \u2208 P\u03C0 . This completes the proof that P\u03C0 = SS( Now we prove that Q\u03C0 = SS( \u221E 2 )  &lt;  \u221E 2 ).  \u2265  \u221E 2 ).  First, we will show that SS(  &lt;  \u2282 Q\u03C0 . Case 1: Let G \u2208 SS(  &lt;  \u221E 2 )  be semi-stable. There is a unique short  32  \u000Cexact sequence associated to G 0\u2192A\u2192G\u2192B\u21920 such that A \u2208 P\u03C0 and B \u2208 Q\u03C0 . We will show that A = 0. Since G is semi-stable, either A = 0 or \u00B5\u03C0 (A) \u2264 \u00B5\u03C0 (G). Suppose that \u00B5\u03C0 (A) \u2264 \u00B5\u03C0 (G). Notice \u00B5\u03C0 (G) &lt; P\u03C0 = SS(  \u2265  \u221E 2 )  \u221E 2  so \u00B5\u03C0 (A) &lt;  \u221E 2  in this case. However, the proof that  tells us that if A \u2208 P\u03C0 then either A = 0 or \u00B5\u03C0 (A) \u2265  \u221E 2 .  Thus, A = 0. Case 2: Let G \u2208 SS(  &lt;  \u221E 2 )  be general. As above, we have that each of  the quotients associated to the HN filtration is an element of Q\u03C0 . Since Q\u03C0 is closed under extension, we then get that each Gi is in Q\u03C0 hence G \u2208 Q\u03C0 . Next, we will show that Q\u03C0 \u2282 SS(  &lt;  \u221E 2 ).  Case 1: Let B \u2208 Q\u03C0 be semi-stable. Assume B = 0. We need to show that \u00B5\u03C0 (B) &lt;  \u221E 2 .  This is equivalent to showing that either B is not  supported on a fibre, or that \u03C7(B) &lt; 0. Assume that B is supported on a fibre. As before, in this case H 0 (B) = 0. Notice that if H 1 (B) = 0 then B \u2208 P\u03C0 and so B = 0. Since we have assumed B = 0, this forces us to assume H 1 (B) = 0. Hence \u03C7(B) &lt; 0 and \u00B5\u03C0 (B) &lt;  \u221E 2 .  Case 2: Let B \u2208 Q\u03C0 be general. Let 0 = B0 \u2282 B1 \u2282 \u00B7 \u00B7 \u00B7 \u2282 BN \u22121 \u2282 BN = B be the HN filtration of B, and let Qi be the associated quotients. By definition of the HN filtration, B1 = Q1 is semi-stable. Since Q\u03C0 is closed under taking sub-objects, we know that B1 \u2208 Q\u03C0 . By the previous case, we know that \u00B5\u03C0 (B1 ) &lt;  \u221E 2 .  However, by definition of the HN filtration, we know  that \u00B5\u03C0 (Q1 ) &gt; \u00B5\u03C0 (Q2 ) &gt; \u00B7 \u00B7 \u00B7 &gt; \u00B5\u03C0 (QN ). Hence, \u00B5\u03C0 (Qi ) &lt; B \u2208 SS(  &lt;  \u221E 2  for all i, i.e.,  \u221E 2 ).  Before moving on, let us collect a functorial property of the HarderNarasimhan filtration (see [42]). It will tell us something about how it is  33  \u000Cpreserved by morphisms. Let G be a coherent OY -module, let Gi \u2282 G denote the Harder-Narasimhan subsheaves of G. For \u03B1 \u2208 (\u2212\u221E, +\u221E] \u00D7 (\u2212\u221E, +\u221E], we define  G  (\u03B1)  \uF8F1 \uF8F4 \uF8F2 Gk = G \uF8F4 \uF8F3 0  G if \u00B5\u03C0 (QG k ) \u2265 \u03B1 &gt; \u00B5\u03C0 (Qk\u22121 )  if \u00B5\u03C0 (QG i ) \u2265 \u03B1 for all i if \u00B5\u03C0 (QG i ) &lt; \u03B1 for all i.  Lemma 4.5. If f : A \u2192 G is a map of sheaves on Y , then f (A(\u03B1) ) \u2282 G(\u03B1) . Proof: First, we prove by induction that if f (Ak ) \u2282 G and with this property, then \u00B5\u03C0  (QG )  \u2265  is minimal  \u00B5\u03C0 (QA k ).  For k = 0, there is nothing to prove. For k = 1, suppose we have f (A1 ) \u2282 G where  is minimal with respect to this property. Then we get  a non-zero map A1 \u2192 QG . By lemma 3.5, we have that \u00B5\u03C0 (A1 ) \u2264 \u00B5\u03C0 (QG ). For general k, suppose we have f (Ak ) \u2282 G ,  minimal. Consider the  following short exact sequences i  0 \u2212\u2212\u2212\u2212\u2192 Ak\u22121 \u2212\u2212\u2212\u2212\u2192 Ak \u2212\u2212\u2212\u2212\u2192 QA \u2212\u2212\u2212\u2192 0 k \u2212 \uF8E6 \uF8E6 f 0 \u2212\u2212\u2212\u2212\u2192 G  q  \u22121  \u2212\u2212\u2212\u2212\u2192 G \u2212\u2212\u2212\u2212\u2192 QG \u2212\u2212\u2212\u2212\u2192 0.  If the composition q \u25E6 f \u25E6 i is zero, then we can define a non-zero map QA k  G \u2192 QG which implies that \u00B5\u03C0 (QA k ) \u2264 \u00B5\u03C0 (Q ), as desired.  If the composition q \u25E6 f \u25E6 i is non-zero, then we know that f (Ak\u22121 ) \u2282 G and f (Ak\u22121 ) \u2282 G ; by induction, we conclude that \u00B5\u03C0 (QG ) \u2265 \u00B5\u03C0 (QA k\u22121 ) &gt; \u00B5\u03C0 (QA k ), as desired. So: f (Ak ) \u2282 G , where \u00B5\u03C0  (QG )  \u2265  \u00B5\u03C0 (QA k ).  is minimal with this property, implies that  We will finish proving the lemma using this fact.  Fix an \u03B1 \u2208 (\u2212\u221E, +\u221E] \u00D7 (\u2212\u221E, +\u221E]. By definition, A(\u03B1) = Ak for A all indices k such that (QA k ) \u2265 \u03B1 &gt; \u00B5\u03C0 (Qk\u22121 ).  Choose the \u00B5\u03C0  (QG )  Fix such an index k.  minimal such that f (Ak ) \u2282 G . Notice that we now have  (\u03B1) by definition. \u2265 \u00B5\u03C0 (QA k ) \u2265 \u03B1, and thus G \u2282 G  34  \u000CThe following lemma will be useful later. Lemma 4.6. SS(a \u2264  \u2264 b) is closed under extension.  Proof: Let A and B \u2208 SS(a \u2264  \u2264 b). Consider j  i  0 \u2192 A \u2192 G \u2192 B \u2192 0. ). We know that A = A(a) , so i(A(a) ) =  First we show that G \u2208 SS(a \u2264  i(A) \u2282 G(a) . If we choose k minimal such that \u00B5\u03C0 (QG k ) &lt; a, then we also know that G(a) = Gk\u22121 . Notice that the possibilities for either it is the zero sheaf, or it is an element of SS( G i(A)  The inclusion i(A) \u2282 Gk\u22121 implies that B =  G Gk\u22121  are limited:  &lt; a). maps onto  G Gk\u22121 ;  denote  the quotient map by \u03C1. Now consider the chain of inclusions and identities G Gk\u22121  \u03C1(B (a) ) \u2282  (a)  \u2282  G = \u03C1(B); Gk\u22121  by assumption, \u03C1(B (a) ) = \u03C1(B), so all of the inclusions are actually equalities. In particular, G Gk\u22121 which means either  G Gk\u22121  (a)  =  G , Gk\u22121  is zero, or it is an element of SS(a \u2264  the above, we see that the only possibility is G=  G(a) ,  i.e. G \u2208 SS(a \u2264  G Gk\u22121  ). By  = 0 which means that  ).  Now we prove that G \u2208 SS(  \u2264 b). Let  0 = G0 \u2282 G1 \u2282 \u00B7 \u00B7 \u00B7 \u2282 GN \u22121 \u2282 GN = G denote the Harder-Narasimhan filtration of G, and let Qi be the associated quotients. Since G1 = Q1 and \u00B5\u03C0 (Q1 ) &gt; \u00B5\u03C0 (Qi ) for all i &gt; 1, it suffices to prove that \u00B5\u03C0 (Q1 ) \u2264 b. Consider the short-exact sequence 0 \u2192 A \u2229 G1 \u2192 G1 \u2192 j(G1 ) \u2192 0. Since the quotients associated to the Harder-Narasimhan filtration are semi35  \u000Cstable, and since G1 = Q1 , we know that G1 is semi-stable. This implies that \u00B5\u03C0 (G1 ) \u2264 \u00B5\u03C0 (j(G1 )), hence it suffices to prove that \u00B5\u03C0 (j(G1 )) \u2264 b. Let B be any subsheaf of B; we will show that \u00B5\u03C0 (B ) \u2264 b. We know that for any c &gt; b, B  (c)  \u2282 B (c) = 0,  which implies that for all c &gt; b, B  (c)  = 0. Hence B \u2208 SS(  \u2264 b) and  \u00B5\u03C0 (B ) \u2264 b. This allows us to conclude that \u00B5\u03C0 (G1 ) \u2264 b.  36  \u000CChapter 5  The motivic Hall algebra Here we provide a quick summary of the constructions and results of Bridgeland\u2019s papers [11], [12] (which came into existence as a gentle introduction to part of Joyce\u2019s theory of motivic Hall algebras [24], [25], [26], [27] ). Let S be a stack, locally of finite type over C and with affine stabilizers. Definition 5.1. The relative Grothendieck group K(St /S) of stacks over S is the Q-vector space spanned by symbols m  T \u2192S (where T is a finite-type stack and m is a morphism), subject to the following relations. a) T \u2192 S = U \u2192 S + F \u2192 S where U is an open substack of T and F is the corresponding closed complement. s\u25E6f  s\u25E6g  g  f  b) T1 \u2192 S = T2 \u2192 S , if T1 \u2192 B and T2 \u2192 B are Zariski fibrations1 s  over B with identical fibres, and B \u2192 S is a morphism of stacks. a  b  c) T \u2192 S = T \u2192 S if there exists a commutative diagram c GT RR \u00D9 \u00D9 RRa b\u00D9\u00D9 R% \u00D4\u00D9\u00D9  TR  S  c  such that the associated map on C-points T (C) \u2192 T (C) is an equivalence of categories. 1  a Zariski-local product space  37  \u000CThe vector space K(St /S) is a K(St / Spec C)-module, whose action we now describe. Let A \u2192 Spec C = A \u2208 K(St / Spec C), and let m  T \u2192 S \u2208 K(St /S). Then we define f  m  A \u00B7 T \u2192 S = A \u00D7Spec C T \u2212\u2192 S , where f is the composition of the projection A \u00D7Spec C T \u2192 T and the map T \u2192 S. We are most interested in the case where S = C, the stack of coherent sheaves on Y supported in dimension one or less. We denote this stack by M, and we denote K(St /M) by H(C). The vector space H(C) is the motivic Hall algebra; let us justify the name by endowing it with the structure of an algebra. First, we define M(2) to be the stack of short-exact sequence of sheaves on M. Now, given A \u2192 M and B \u2192 M we define the c\u25E6g  convolution product A \u2192 M \u2217 B \u2192 M to be Z \u2192 M where Z \u2192 M is defined by the following Cartesian diagram: Z \uF8E6 \uF8E6  g  \u2212\u2212\u2212\u2212\u2192  c  M(2) \u2212\u2212\u2212\u2212\u2192 M \uF8E6 \uF8E6(l,r)  A \u00D7 B \u2212\u2212\u2212\u2212\u2192 M \u00D7 M. The morphisms l, r are the \u201Cleft hand\u201D and \u201Cright hand\u201D morphisms, which project a short exact sequence to its left-most (resp. right-most) non-zero entry. The morphism c is the \u201Ccentre\u201D morphism. Intuitively, given families of sheaves A \u2192 M and B \u2192 M their product in the Hall algebra is the family Z \u2192 M parametrizing extensions of objects of B by objects of A. The motivic Hall algebra is useful tool. It holds enough information to allow us to retrieve Euler characteristics, yet is flexible enough to produce decompositions of elements in terms of extensions. We will describe an \u201Cintegration\u201D map on H(C) taking values in the ring of polynomials. Equations among elements of H(C) will be integrated to yield equations of polynomials. This entire framework will then be souped-up to incorporate Laurent series, and our theorem will be the result of applying the souped-up integration  38  \u000Cmap to equations in the souped-up Hall algebra. In [11], Bridgeland introduces regular elements. Let K(Var/C) denote the relative Grothendieck group of varieties over C (cf. definition 5.1). Let L denote the element [A1 \u2192 C], the Tate motive. Consider the maps of commutative rings, K(Var/C) \u2192 K(Var/C)[L\u22121 ] \u2192 K(St/C), and recall from [11] that H(C) is algebra over K(St/C). Define a K(Var/C)[L\u22121 ]module Hreg (C) \u2282 H(C) f  to be the span of classes of maps [V \u2192 M] with V a variety. We call an element of H(C) regular if it lies in this submodule. The following result is theorem 5.1 of [11]. Theorem 5.2. The submodule of regular elements is closed under the convolution product: Hreg (C) \u2217 Hreg (C) \u2282 Hreg (C), and is therefore a K(Var/C)[L\u22121 ]-algebra. Moreover the quotient Hsc (C) = Hreg (C)/(L \u2212 1) Hreg (C), is a commutative K(Var/C)-algebra. Bridgeland equips Hsc (C) with a Poisson bracket, defined by {f, g} =  f \u2217g\u2212g\u2217f . L\u22121  The integration map I is defined on Hsc (C). Now we work toward the polynomial ring in which it takes values. Recall that K(Y ) is the numerical K-theory of Y . Recall \u2206 \u2282 F1 K(Y ) is the effective cone of F1 K(Y ), that is, the collection of elements of the form [F ] where F is a one-dimensional sheaf. Define a ring C[\u0393] to be the vector  39  \u000Cspace spanned by symbols x\u03B1 for \u03B1 \u2208 \u2206 and defining the multiplication by x\u03B1 \u00B7 x\u03B2 = x\u03B1+\u03B2 . We equip C[\u2206] with the trivial Poisson bracket. We are now ready for the following theorem. Theorem 5.3 (5.1 of [12]). There exists a Poisson algebra homomorphism I : Hsc (C) \u2192 C[\u2206] such that f  I( Z \u2192 M\u03B1 ) = \u03C7(Z, f \u2217 (\u03BD))x\u03B1 , where \u03BD : M \u2192 Z is Behrend\u2019s microlocal function of M, and M\u03B1 denotes the component of M with fixed Chern character \u03B1.  40  \u000CChapter 6  Equations in the infinite-type Hall algebra and the fake proof For the sake of exposition only, we follow [12] and [17] by introducing an infinite-type version of the Hall algebra. This has the benefit of allowing non-finite-type stacks, but the devastating draw-back of not admitting an integration map. We use it because it will allow us to temporarily work without having to think about convergence of power series. Also, many of the arguments will be used again later. We end this chapter with a fake proof of our main result. It is our hope that this fake proof helps the reader to navigate the true one in the following chapter. The infinite-type Hall algebra is defined by considering symbols as in definition 5.1, but with T assumed only to be locally of finite type over C, and use relations as before, except that we do not use relation (a). (Admitting relation (a) in this case would make every infinite-type Hall algebra trivial). We denote it by H\u221E (C). Given a substack N \u2282 M, we let i  1N = [N \u2192 M] denote the inclusion i : N \u2192 M. Pulling back the morphism (2.13) to q  N \u2282 M gives a stack denoted N (O) with a morphism N (O) \u2192 N , and hence an element q  1O N = [N (O) \u2192 M] \u2208 H\u221E (C).  41  \u000CFor example, P\u03C0 and Q\u03C0 are full subcategories of C, and define substacks of M (see lemma 2.16), which we abusively denote with the same letters, P\u03C0 , Q\u03C0 \u2282 M. These substacks define elements of the infinite-type Hall algebra, 1P\u03C0 , 1Q\u03C0 \u2208 H\u221E (C). Other examples include H = [Hilb(Y ) \u2192 M], Hexc = [Hilbexc \u2192 M], and H\u03C0 = [\u03C0-Hilb(Y ) \u2192 M] \u2208 H\u221E (C), where Hilbexc denotes the Hilbert scheme of curves supported on fibres of \u03C0, and the map to M is given by taking OY \u2192 G to G. Note that all Hilbert schemes are restricted to the components parametrizing sheaves G of dimension one. Lemma 6.1. 1C = 1P\u03C0 \u2217 1Q\u03C0 This lemma reflects the fact that (P\u03C0 , Q\u03C0 ) is a torsion pair. Proof: Form the following Cartesian diagram: Z \uF8E6 \uF8E6  f  \u2212\u2212\u2212\u2212\u2192  M(2) \uF8E6 \uF8E6(a  b  \u2212\u2212\u2212\u2212\u2192 M  1 ,a2 )  (6.2)  i  P\u03C0 \u00D7 Q\u03C0 \u2212\u2212\u2212\u2212\u2192 M \u00D7 M. By lemma A.1 [11], the groupoid of T -valued points of Z can be described as follows. The objects are short exact sequences of T -flat sheaves on T \u00D7 Y of the form 0\u2192A\u2192G\u2192B\u21920 such that A and B define families of sheaves on Y lying in the subcategories P\u03C0 and Q\u03C0 respectively. The morphisms are isomorphisms of short exact 42  \u000Csequences. The composition, g =b\u25E6f :Z \u2192M sends a short exact sequence to the object G. Since P\u03C0 and Q\u03C0 are subcategories of C, it follows that the composition factors through C. This morphism induces an equivalence on C-valued points because of the torsion pair property: every object G of C fits into a unique short exact sequence of the form (6.2). Thus, the identity follows from the relations in the infinite Hall algebra. We will need a framed version of the previous lemma. Lemma 6.3. O O 1O C = 1P\u03C0 \u2217 1Q\u03C0 .  Proof: Form the following Cartesian diagram: U \u0015  P\u03C0 (O) \u00D7 Q\u03C0 (O)  p  q\u00D7id  GV \u0015  G P\u03C0 \u00D7 Q\u03C0 (O)  j  G M(2) \u0015  b  GM  (a1 ,a2 )  G M \u00D7 M.  O Then 1O P\u03C0 \u2217 1Q\u03C0 is represented by the composite map b \u25E6 j \u25E6 p : U \u2192 M.  Note that, by lemma 2.5 of [12], the map P\u03C0 (O) \u2192 P\u03C0 is a Zariski fibration with fibre H 0 (P ) over a point P \u2208 P\u03C0 . By pullback, the same is true of the morphism p. The groupoid of T -valued points of V can be described as follows. The objects are short exact sequences of T -flat sheaves on T \u00D7 Y of the form 0\u2192P \u2192G\u2192B\u21920 such that P and B define flat families of objects in P\u03C0 and Q\u03C0 respectively, together with a map OT \u00D7Y \u2192 B. We represent the objects of V as diagrams  43  \u000Cof the form: OT \u00D7Y \uF8E6 \uF8E6 0 \u2212\u2212\u2212\u2212\u2192 P \u2212\u2212\u2212\u2212\u2192 G \u2212\u2212\u2212\u2212\u2192  B  \u2212\u2212\u2212\u2212\u2192 0.  Consider the stack Z from lemma 6.1 with its map Z \u2192 M. Form the diagram h  W \u2212\u2212\u2212\u2212\u2192 C(O) \uF8E6 \uF8E6 \uF8E6 \uF8E6 q g  Z \u2212\u2212\u2212\u2212\u2192  C.  Since g induces an equivalence on C-valued points, so does h, so that the element 1O C can be represented by the map q \u25E6 h. We represent the objects of W as diagrams of the form: OT \u00D7Y \uF8E6 \uF8E6 \u03B4  0 \u2212\u2212\u2212\u2212\u2192 P \u2212\u2212\u2212\u2212\u2192  G  \u03B2  \u2212\u2212\u2212\u2212\u2192 B \u2212\u2212\u2212\u2212\u2192 0.  Setting \u03B3 = \u03B2 \u25E6\u03B4 defines a map of stacks W \u2192 V , which is a Zariski fibration with fibre an affine model of the vector space H 0 (P ) over a sheaf P . Now, since H 1 (P ) = 0, U \u2192 V is a Zariski fibration with fibre H 0 (P ), hence they represent the same element of the Hall algebra, namely Q\u03C0 .  Lemma 6.4. 1O C = H \u2217 1C This is lemma 4.3 of [12]. Intuitively, this amounts to the fact that every \u03B3  map O \u2192 G factors uniquely into a surjection O \u2192 im(\u03B3) and an inclusion im(\u03B3) \u2192 G. The following lemma is a restriction of the previous to the substack P\u03C0 . Lemma 6.5. 1O P\u03C0 = Hexc \u2217 1P\u03C0 44  \u000CProof: Form the Cartesian diagram: Z \uF8E6 \uF8E6  M(2) \uF8E6 \uF8E6(a  \u2212\u2212\u2212\u2212\u2192  b  \u2212\u2212\u2212\u2212\u2192 M  1 ,a2 )  i  Hexc \u00D7 P\u03C0 \u2212\u2212\u2212\u2212\u2192 M \u00D7 M. The groupoid of T -valued points of Z may be described as follows. The objects are short exact sequences of T -flat sheaves of T \u00D7 Y 0\u2192A\u2192G\u2192B\u21920 such that B \u2208 P\u03C0 , and A is supported on exceptional fibres, together with an epimorphism OY \u2192 A. We can represent these objects as diagrams of the form OT \u00D7Y \u03B3  0  \u0015  GA  GG \u03B2 GB  \u03B1  G 0.  Since OY \u2192 A is an epimorphism, we know that A is of the form OC for some one-dimensional subscheme C of Y . By the proof of proposition 2.11, we know that R\u03C0\u2217 OC = 0. Since A = OC has exceptional support, it follows that \u03C0\u2217 A is a zero-dimensional sheaf, hence A \u2208 P\u03C0 . Since B \u2208 P\u03C0 by design, and since P\u03C0 is closed under taking extensions, we conclude that in any such short exact sequence, G \u2208 P\u03C0 . There is a map h : Z \u2192 P\u03C0 (O) sending the above diagram to the composite map \u03B4 = \u03B1 \u25E6 \u03B3 : OT \u00D7Y \u2192 G. This morphism h fits into a commuting diagram of stacks ZT  G P\u03C0 (O) &#127;&#127; &#127;&#127;q &#127; &#127;&#127; &#127;  h  TT TT b\u25E6f TT &amp;  M  45  \u000CWe argue that the map h then induces an equivalence on C-valued points. \u03B4  Suppose OT \u00D7Y \u2192 G is an arbitrary map of sheaves, with G defining a family of sheaves in P\u03C0 . Then we get the following diagram: OT \u00D7Y \u0015  \u03B3  G im(\u03B4)  0  GG  G coker(\u03B4)  G 0.  Since G \u2208 P\u03C0 we know that the one-dimensional component of its support is exceptional, hence im(\u03B4) is also exceptional, so that OT \u00D7Y \u2192 im(\u03B4) defines a family of objects in Hilbexc . As well, we know that P\u03C0 is closed under taking quotients, so coker(\u03B4) is in P\u03C0 . This completes the proof. Morally, the next lemma is similar to lemma 6.4 since H\u03C0 may be thought of as the surjections OY \u2192 G in a tilt of the abelian category generated by O and C. We provide a direct proof since we have not constructed this tilt. Lemma 6.6. \u03C0 1O Q\u03C0 = H \u2217 1Q\u03C0 .  Proof: Form the following Cartesian diagram: Z \uF8E6 \uF8E6  f  \u2212\u2212\u2212\u2212\u2192  M(2) \uF8E6 \uF8E6(a  b  \u2212\u2212\u2212\u2212\u2192 M  1 ,a2 )  i  H\u03C0 \u00D7 Q\u03C0 \u2212\u2212\u2212\u2212\u2192 M \u00D7 M The groupoid of T -valued points of Z is described as follows. The objects are short exact sequences of T -flat sheaves on T \u00D7 Y 0\u2192A\u2192G\u2192B\u21920 with the property that G \u2208 Q\u03C0 , together with a map OT \u00D7Y \u2192 A that pulls back to a \u03C0-stable pair OY \u2192 At for every t \u2208 T . We can represent these objects as diagrams of the form:  46  \u000COT \u00D7Y  (6.7)  \u03B3  \u0015  GA  0  GG \u03B2 GB  \u03B1  G 0.  Since A and B are objects of Q\u03C0 , and Q\u03C0 is closed under extensions, we conclude G \u2208 Q\u03C0 . Thus, there is a map h : Z \u2192 Q\u03C0 (O) sending the above diagram to the composite map \u03B4 = \u03B1 \u25E6 \u03B3 : OT \u00D7Y \u2192 G. This map h fits into a commuting diagram of stacks G Q\u03C0 (O) &#127;&#127; &#127;&#127;q &#127; &#127;&#127; &#127;  h  ZT  TT TT b\u25E6f TT &amp;  M  (6.8)  The map h then induces an equivalence on C-valued points because of the \u03C3  following argument. Let OY \u2192 G be an arbitrary map, with G \u2208 Q\u03C0 . We need to produce a diagram, OY \u03B3  0  \u0015  GA  \u03B1  GG \u03B2 GB  G 0,  with OY \u2192 A a \u03C0-stable pair, B \u2208 Q\u03C0 , and \u03B1 \u25E6 \u03B3 = \u03C3. Consider the cokernel K of \u03C3. Since (P\u03C0 , Q\u03C0 ) is a torsion pair, we know that K fits into a short exact sequence 0\u2192P \u2192K\u2192Q\u21920 c  where P \u2208 P\u03C0 and Q \u2208 Q\u03C0 . Let G \u2192 K be the canonical map from G to d  the cokernel K of \u03C3, and let G \u2192 Q be the composition of G \u2192 K and  47  \u000CK \u2192 Q. Define A to be the kernel of d. Consider the following diagram. 0  G OY  =  G OY  G0  \u03C3  0  \u0015  \u0015  GA  GG  \u0015  \u0015  d  \u0015 GQ =  c  0  GP  GK  \u0015  0  \u0015  0  G0  \u0015  GQ  G0  \u0015  0.  We know that Q \u2208 Q\u03C0 , and a diagram chase proves that the dotted vertical morphisms exist and that P is the cokernel of OY \u2192 A. The sheaf A is a subsheaf of G \u2208 Q\u03C0 , and Q\u03C0 is closed under taking subsheaves, so A \u2208 Q\u03C0 . This proves that OY \u2192 A is a \u03C0-stable pair. This proves that the map h is surjective on C-valued points. It remains to show that h is injective on C-valued points. Now let \u03B3 : OY \u2192 G be an arbitrary map with G \u2208 Q\u03C0 . Since Q\u03C0 is closed under subobjects, the image of \u03B3 is an object of Q\u03C0 , im(\u03B3) \u2208 Q\u03C0 . Now we can use the description of Q\u03C0 in terms of stability to show that G im(\u03B3)  is also in Q\u03C0 . This proves that the map h is injective on C-valued  points. This completes the proof.  We end this section by giving a fake proof of theorem 1.6 that depends on a fake integration map. In fact, no such integration map is known to exist, but if there was one, the proof our theorem would be simpler. As it stands, we have a chapter dedicated to convergence issues to get around the fact that no such integration map exists on the infinite type Hall algebra. It is our hope that this fake proof will make the true one easier to follow. Fake proof: By lemma 6.6, we have H\u03C0 \u2217 1Q\u03C0 = 1O Q\u03C0 .  (6.9)  48  \u000CLemma 4.3 of [12] proves H \u2217 1C = 1O C. Using lemma 6.1, we may rewrite H \u2217 1C : H \u2217 1C = H \u2217 1P\u03C0 \u2217 1Q\u03C0 . Lemma 6.3 allows us to write O O 1O C = 1P\u03C0 \u2217 1Q\u03C0 .  Putting these together yields O H \u2217 1P\u03C0 \u2217 1Q\u03C0 = 1O P\u03C0 \u2217 1Q\u03C0 .  Applying lemma 6.5 H \u2217 1P\u03C0 \u2217 1Q\u03C0 = Hexc \u2217 1P\u03C0 \u2217 1O Q\u03C0 . Using equation 6.9, we get H \u2217 1P\u03C0 \u2217 1Q\u03C0 = Hexc \u2217 1P\u03C0 \u2217 H\u03C0 \u2217 1Q\u03C0 . Now, for reasons we will explain in the next section, 1P\u03C0 and 1Q\u03C0 are invertible in the Hall algebra. We may therefore cancel the copies of 1Q\u03C0 and isolate H. H = Hexc \u2217 1P\u03C0 \u2217 H\u03C0 \u2217 1\u22121 P\u03C0 . The elements H, Hexc , H\u03C0 all lie in the subalgebra Hreg (C) since they are represented by schemes. As we will see in the next section, conjugation by 1P\u03C0 induces a Poisson homomorphism of Hreg (C) of the form: identity + terms expressed in the Poisson bracket. Since the Poisson bracket of the polynomial ring is trivial, these terms vanish when we apply the fake integration map, and we are left with I(H) = I(Hexc ) \u00B7 I(H\u03C0 ). 49  \u000CUp to signs arising from lemma 2.18, the \u201Cpolynomials\u201D I(H), I(Hexc ), and I(H\u03C0 ) are the generating series of DT(Y ), DTexc (Y ), and \u03C0-PT(Y ), respectively, and we see that the above equation is the formula claimed in theorem 1.6. or  I(H) = I(H\u03C0 ) I(Hexc )  which is exactly the statement of our theorem. The true proof will follow precisely these steps, fully justified, and with the appropriate convergence arguments. The next chapter describes the Laurent Hall algebra, which does have an integration map.  50  \u000CChapter 7  Equations in the Laurent Hall algebra and the true proof Laurent subsets In this section, we will formally modify the algebra H(C) and its integration map so that the modified integration map on the modified algebra takes valued in power series. This section is a summary of sections 5.2 and 5.3 of [12]. Definition 7.1. A subset S \u2282 \u2206 is Laurent if for all \u03B2 \u2208 N1 (Y ), the collection of elements of the form (\u03B2, n) \u2208 S is such that n is bounded below. Let \u03A6 denote the set of all Laurent subsets. It has the following properties: 1. if S, T \u2208 \u03A6 then so it S + T = {\u03B1 + \u03B2 : \u03B1 \u2208 S, \u03B2 \u2208 T } 2. if S, T \u2208 \u03A6, and \u03B1 \u2208 \u2206 then there are only finitely many ways to write \u03B1 = \u03B2 + \u03B3 such that \u03B2 \u2208 S and \u03B3 \u2208 T . Given a ring A graded by \u2206, A =  \u03B3\u2208\u2206 A\u03B3 ,  we can use the Laurent  subset to define a new algebra, which we will denote A\u03A6 . Elements of A\u03A6 are of the form a=  a\u03B3 \u03B3\u2208S  51  \u000Cwhere S \u2208 \u03A6, and a\u03B3 \u2208 A\u03B3 \u2282 A. Given an element a \u2208 A\u03A6 as above, we define \u03C1\u03B3 (a) = a\u03B3 \u2208 A. (Here, our notation differs from [12], since we are using the symbol \u03C0 for the map \u03C0 : Y \u2192 X.) The projection operator \u03C1 allows us to define a product \u2217 on A\u03A6 by: \u03C1\u03B3 (a \u2217 b) =  \u03C1\u03B1 (a) \u2217 \u03C1\u03B2 (b). \u03B3=\u03B1+\u03B2  A\u03A6 admits a natural topology that may be identified by declaring a sequence (aj )j\u2208N \u2282 A\u03A6 to be convergent if for any (\u03B2, n) \u2208 \u2206, there exists an integer K such that for all m &lt; n i, j &gt; K \u21D2 \u03C1(\u03B2,m) (ai ) = \u03C1(\u03B2,m) (aj ). Lemma 7.2. If A is a C-algebra and a \u2208 A\u03A6 satisfies \u03C10 (a) = 0 then any series cj aj j\u22651  with coefficients cj \u2208 C is convergent in the topological ring A\u03A6 . See Lemma 5.3 of [12] for a proof. Given two \u2206-graded algebras A and B, and a morphism f : A \u2192 B that preserves the \u2206-grading, we get an induced continuous map f\u03A6 : A\u03A6 \u2192 B\u03A6 by defining \u03C1\u03B3 (f\u03A6 (a)) = f (\u03C1\u03B3 (a)). Applying this process to the map of \u2206-graded algebras I : Hsc (C) \u2192 C[\u2206] yields a continuous map I\u03A6 : Hsc (C)\u03A6 \u2192 C[\u2206]\u03A6 . We call Hsc (C)\u03A6 the Laurent Hall algebra; it, too, is equipped with a Poisson bracket. Definition 7.3. A morphism of stacks f : W \u2192 C is \u03A6-finite if a) W\u03B1 = f \u22121 (C\u03B1 ) is of finite type for all \u03B1 \u2208 \u2206, and b) there is a Laurent subset S \u2282 \u2206 such that W\u03B1 is empty unless \u03B1 \u2208 S. 52  \u000CA \u03A6-finite morphism of stacks f : W \u2192 C defines an element of Hsc (C)\u03A6 by the formal sum f  [W\u03B1 \u2192 C]. \u03B1\u2208S  In [12], it is shown that [Hilb \u2192 M] is \u03A6-finite, and hence defines an element in Hsc (C)\u03A6 . Lemma 7.4. The maps Hilb\u03C0 \u2192 C,  Hilb \u2192 C,  Hilbexc \u2192 C,  are \u03A6-finite. The corresponding elements H\u03C0 , H and Hexc of Hsc (C)\u03A6 satisfy I\u03A6 (H\u03C0 ) =  (\u22121)n \u03C0-PT(\u03B2, n)x\u03B2 q n = \u03C0-PT(Y )(x, \u2212q), (\u03B2,n)\u2208\u2206  where we have written x\u03B2 = q (\u03B2,0) and q = q (0,1) . Similarly, (\u22121)n DT(\u03B2, n)x\u03B2 q n = DT(Y )(x, \u2212q)  I\u03A6 (H) = (\u03B2,n)\u2208\u2206  (\u22121)n DT (\u03B2, n)x\u03B2 q n = DTexc (Y )(x, \u2212q).  I\u03A6 (Hexc ) = (\u03B2,n)\u2208\u2206,\u03C0\u2217 \u03B2=0  Proof: Lemma 2.16 proves that \u03C0-Hilb is locally of finite type, and is of finite type once the Chern character is fixed. As well, the set of elements \u03B1 \u2208 \u2206 for which \u03C0-Hilb\u03B1 is non-empty is Laurent. To prove this, it suffices to show that for any curve class \u03B2, there exists an integer N such that for any n &lt; N , the moduli space \u03C0-Hilb(\u03B2,n) is empty. Fix a curve class \u03B2, and consider all \u03C0-stablepairs OY \u2192 G in that class. There is the associated short exact sequence, 0 \u2192 OC \u2192 G \u2192 P \u2192 0 and since there are only finitely many decompositions \u03B2 = \u03B21 + \u03B22 of \u03B2 into a sum of effective curve classes, we lose no generality in fixing the curve class of OC and P . Now the structure sheaf OC lives in a Hilbert scheme, and the set of elements (\u03B21 , n1 ) \u2208 \u2206 for which Hilb(\u03B21 ,n1 ) is non-empty is Laurent, so 53  \u000Cthere is a \u201Cminimal\u201D Euler characteristic of OC , which we denote by N1 . As for P , the Leray spectral sequence shows that \u03C7(P ) \u2265 0 (see lemma 3.16). This proves that we may take N = N1 , and for any n &lt; N , \u03C0-Hilb(\u03B2,n) is empty. The formulae then follow from lemma 2.18 and Behrend\u2019s description of DT invariants as a weighted Euler characteristic. Hilbexc is a subscheme of Hilb, so the desired properties follow from [12, lemma 5.5].  Lemma 7.5. Let I \u2282 (\u2212\u221E, +\u221E] \u00D7 (\u2212\u221E, +\u221E] be an interval bounded from below. Then 1SS(I) \u2192 C is \u03A6-finite. Proof: Since this is holds for Gieseker stability ([23, theorem 3.3.7]), it suffices to prove that for any b = (b1 , b2 ) \u2208 (\u2212\u221E, +\u221E] \u00D7 (\u2212\u221E, +\u221E] there exists a number M such that the family of all G with \u00B5\u03C0 (G) \u2265 b satisfies \u00B5(G) \u2265 M . Here, \u00B5 stands for Gieseker slope stability, namely \u00B5(G) =  \u03C7(G) . \u03B2\u00B7L  Case 1: \u03C7(G) &gt; 0 Here, we have 0&lt;  \u03C7(G) = \u00B5(G), \u03B2\u00B7L  so we may take M = 0 in this case. Case 2: \u03C7(G) &lt; 0 Now, since \u03B2 \u00B7 H \u2264 \u03B2 \u00B7 L and \u03C7(G) &lt; 0, we have b1 \u2264  \u03C7(G) \u03B2\u00B7H  \u2264  \u03C7(G) . \u03B2\u00B7L  In this case, we may take M = b1 . 54  \u000CCase 3: \u03C7(G) = 0 In this case, \u00B5\u03C0 (G) is either 0 or +\u221E, so we may take M = 0 in this case. Case-by-case analysis reveals that we may use M = min{0, b1 }.  Equations in the Laurent Hall algebra In this section, following [12] we establish equations in Hsc (C)\u03A6 , and ultimately prove theorem 1.6. Lemma 7.6. Let \u00B5 \u2208 (\u2212\u221E, +\u221E] \u00D7 (\u2212\u221E, +\u221E] such that \u00B5 &lt;  \u221E 2 .  Then the  following equality holds in Hsc (C)\u03A6 : 1SS(\u00B5\u2264  \u2264\u221E)  = 1P\u03C0 \u2217 1SS(\u00B5\u2264  &lt;\u221E ). 2  Proof: Form the following Cartesian diagram: f  Z  G M(2)  c  GM  (l,r)  \u0015  P\u03C0 \u00D7 SS(\u00B5 \u2264  &lt;  \u221E 2 )  \u0015  GM\u00D7M  T -valued points of Z are short exact sequences 0 \u2192 A \u2192 G \u2192 B \u2192 0 of T -flat sheaves on T \u00D7 Y such that A defines a family of objects in P\u03C0 and B a family in SS(\u00B5 \u2264  &lt;  \u221E 2 ).  By lemma 4.4, we know  \u221E 2  \u2264 \u00B5\u03C0 (A) \u2264  \u221E. Now by [12, lemma6.2], we know that G defines a family of objects in SS(\u00B5 \u2264  \u2264 \u221E).  Now let G \u2208 SS(\u00B5 \u2264  \u2264 \u221E). If G \u2208 P\u03C0 or SS(\u00B5 \u2264  &lt;  \u221E 2 )  then we are  done, since then G will be an extension where one term is zero (recall that all SS(a &lt;  &lt; b) include the zero objects). Otherwise, let 0 = G0 \u2282 G1 \u2282 . . . \u2282 GN \u22121 \u2282 GN = G  55  \u000Cbe the Harder-Narasimhan filtration of G, let Qi be the associated quotients. Then there exists an index j \u2208 N, 1 &lt; j &lt; N such that for all i &lt; j, \u00B5\u03C0 (Qi ) \u2265 \u221E 2  and \u00B5\u03C0 (Qj ) &lt;  \u221E 2 .  Finally, by the uniqueness of the Harder-Narasimhan  filtration, we have that Gj \u2208 P\u03C0 and G/Gj \u2208 SS(\u00B5 \u2264  &lt;  \u221E 2 ).  Remark 7.7. The proof of this lemma is strikingly similar to the proof of Lemma 6.1. This is no coincidence. The above is actually just a minor refinement of Lemma 6.1 which says that we may cut off the tail end of Q\u03C0 and have the corresponding result still hold. As we go on, we will be less explicit about the proofs of lemmas when the argument has been already made in the infinite-type case. Lemma 7.8. Let \u00B5 \u2208 (\u2212\u221E, +\u221E] \u00D7 (\u2212\u221E, +\u221E]. Then, as \u00B5 \u2192 \u2212\u221E, we have H \u2217 1SS(\u00B5\u2264  \u2264\u221E)  \u2212 1O SS(\u00B5\u2264  \u2264\u221E)  \u2192 0.  Proof: Fix (\u03B2, n) \u2208 \u2206, and consider (\u03B2, m) for m &lt; n. There are only finitely many decompositions (\u03B2, n) = (\u03B21 , n1 ) + (\u03B22 , n2 ) such that both \u03C1(\u03B21 ,n1 ) (H) and \u03C1(\u03B22 ,n2 ) (1SS(\u00B5\u2264  \u2264\u221E) )  are non-zero. This follows from the  fact that there are only finitely many decompositions \u03B2 = \u03B21 + \u03B22 with both \u03B2i effective. Now for each fixed \u03B2, there exist finitely many n such that both \u03C1(\u03B21 ,n1 ) (H) and \u03C1(\u03B22 ,n2 ) (1SS(\u00B5\u2264  \u2264\u221E) )  are non-zero).  By the boundedness of the Hilbert scheme, we may assume that \u00B5 is small enough so that for any of the decompositions, \u03B2 = \u03B21 + \u03B22 , all points OY \u2192 A of Hilb(\u03B21 ,n1 ) satisfy A \u2208 SS(\u00B5 \u2264  \u2264 \u221E). Consider a diagram of  sheaves, OY  (7.9)  \u03B3  0  \u0015  GA  \u03B1  GG \u03B2 GB  G0  with OY \u2192 A in Hilb(\u03B21 ,n1 ) and ch([G]) = (\u03B2, n). Now G \u2208 SS(\u00B5 \u2264  \u2264 \u221E) if and only if B \u2208 SS(\u00B5 \u2264  \u2264 \u221E). Since  56  \u000CBridgeland proves [12, prop 6.5] \u03C1(\u03B2,n) (H \u2217 1SS(\u00B5\u2264  \u2264\u221E) )  = \u03C1(\u03B2,n) (1O SS(\u00B5\u2264  \u2264\u221E) ),  the claim is proven.  Lemma 7.10. Let \u00B5 \u2208 (\u2212\u221E, +\u221E] \u00D7 (\u2212\u221E, +\u221E). Then, as \u00B5 \u2192 \u2212\u221E, we have H\u03C0 \u2217 1SS(\u00B5\u2264  &lt;\u221E ) 2  \u2212 1O SS(\u00B5\u2264  ) &lt;\u221E 2  \u2192 0.  Proof: Fix (\u03B2, n) \u2208 \u2206, and consider (\u03B2, m) for m &lt; n. There are only finitely many decompositions (\u03B2, n) = (\u03B21 , n1 ) + (\u03B22 , n2 ) such that both \u03C1(\u03B21 ,n1 ) (H\u03C0 ) and \u03C1(\u03B22 ,n2 ) (1SS(\u00B5\u2264  )) &lt;\u221E 2  are non-zero.  By the boundedness of the moduli space of \u03C0-stable pairs, we may assume that \u00B5 is small enough so that for any decompositions, \u03B2 = \u03B21 +\u03B22 , all points OY \u2192 A of \u03C0-Hilb(\u03B21 ,n1 ) satisfy A \u2208 SS(\u00B5 \u2264  &lt;  \u221E 2 ).  Consider a diagram  of sheaves, OY  (7.11)  \u03B3  0  \u0015  GA  \u03B1  GG \u03B2 GB  G0  with OY \u2192 A in \u03C0-Hilb(\u03B21 , n1 ) and [G] = (\u03B2, n). Using that SS(I) \u2217 SS(I) \u2282 SS(I), we see that B \u2208 SS(\u00B5 \u2264  &lt;  \u221E 2 )  if and only if G \u2208 SS(\u00B5 \u2264  &lt;  \u221E 2 ).  Now since A \u2208 Q\u03C0 and B \u2208 Q\u03C0 , we have that G \u2208 Q\u03C0 . Composing the map OY \u2192 A with the map A \u2192 G, yields a map OY \u2192 G; this represents an object of 1O Q\u03C0 . The proof of lemma 6.6, that \u03C0 1O Q\u03C0 = H \u2217 1Q\u03C0 ,  can be easily adapted to now prove that \u03C1(\u03B2,n) (H\u03C0 \u2217 1SS(\u00B5\u2264  &lt;\u221E )) 2  = \u03C1(\u03B2,n) (1O SS(\u00B5\u2264  ) ), &lt;\u221E 2  57  \u000Cusing lemma 4.6. This completes the proof that H\u03C0 \u2217 1SS(\u00B5\u2264  ) &lt;\u221E 2  \u2212 1O SS(\u00B5\u2264  ) &lt;\u221E 2  \u21920  as \u00B5 \u2192 \u2212\u221E.  Proposition 7.12. We have the following equality in the Laurent Hall algebra, Hsc (C)\u03A6 : H \u2217 1P\u03C0 = Hexc \u2217 1P\u03C0 \u2217 H\u03C0 . Proof: Using 1SS(\u00B5\u2264 1O SS(\u00B5\u2264  &lt;\u221E ), 2  \u2264\u221E)  = 1P\u03C0 \u2217 1SS(\u00B5\u2264  ) &lt;\u221E 2  and 1O SS(\u00B5\u2264  \u2264\u221E)  = 1O P1 \u2217 \u03C0  we can rewrite H \u2217 1SS(\u00B5\u2264  \u2264\u221E)  \u2212 1O SS(\u00B5\u2264  \u2264\u221E)  \u21920  as H \u2217 1P\u03C0 \u2217 1SS(\u00B5\u2264  ) &lt;\u221E 2  O \u2212 1O P\u03C0 \u2217 1SS(\u00B5\u2264  ) &lt;\u221E 2  \u2192 0,  as \u00B5 \u2192 \u2212\u221E. Multiplying H\u03C0 \u2217 1SS(\u00B5\u2264  O ) \u2212 1SS(\u00B5\u2264 &lt; \u221E ) &lt;\u221E 2 2  \u2192 0 on the left by 1O P\u03C0 , and  rewriting using 1O P\u03C0 = Hexc \u2217 1P\u03C0 yields Hexc \u2217 1P\u03C0 \u2217 H\u03C0 \u2217 1SS(\u00B5\u2264  &lt;\u221E ) 2  O \u2212 1O P\u03C0 \u2217 1SS(\u00B5\u2264  &lt;\u221E ) 2  \u21920  as \u00B5 \u2192 \u2212\u221E. Hence Hexc \u2217 1P\u03C0 \u2217 H\u03C0 \u2217 1SS(\u00B5\u2264 as \u00B5 \u2192 \u2212\u221E. Since 1SS(\u00B5\u2264  &lt;\u221E ) 2  &lt;\u221E ) 2  \u2212 H \u2217 1P\u03C0 \u2217 1SS(\u00B5\u2264  &lt;\u221E ) 2  \u21920  is invertible, we can cancel it from both  sides: H \u2217 1P\u03C0 = Hexc \u2217 1P\u03C0 \u2217 H\u03C0 .  58  \u000CThe proof of theorem 1.6 We first collect results. The next proposition is theorem 3.11 of [29], and is a very deep result whose proof depends on all the full power of the formalism of [24, 25, 26, 27, 28]. Proposition 7.13. For each slope \u00B5 \u2208 ((\u2212\u221E, \u2212\u221E), (+\u221E, +\u221E)], we can write 1SS(\u00B5) = exp( \u00B5 ) \u2208 H(C)\u03A6 with \u03BD\u00B5 = [C\u2217 ] \u00B7  \u00B5  \u2208 Hreg (C)\u03A6 a regular element.  Proof: The proof is identical to that of [12, theorem 6.3]. Bridgeland uses Joyce\u2019s machinery, which applies in our case just as it does in his. The following corollary corresponds to Bridgeland\u2019s 6.4 [12]. Corollary 7.14. For any \u00B5 \u2208 ((\u2212\u221E, \u2212\u221E), (+\u221E, +\u221E)], the element 1SS(\u00B5) \u2208 H(C)\u03A6 is invertible, and the automorphism Ad1SS(\u00B5) : H(C)\u03A6 \u2192 H(C)\u03A6 preserves the subring of regular elements. The induced Poisson automorphism of Hsc (C) is given by Ad1SS(\u00B5) = exp{\u03B7\u00B5 , \u2212}. Proof: The proof of this is identical to that of corollary 6.4 of [12]. Now we can prove theorem 1.6. We have H \u2217 1P\u03C0 = Hexc \u2217 1P\u03C0 \u2217 H\u03C0 . Rearranging yields H = Hexc \u2217 1P\u03C0 \u2217 H\u03C0 \u2217 (1P\u03C0 )\u22121 . By lemma 4.4, we can write 1P\u03C0 = SS( \u221E 2 \u2264  \u2264 \u221E), and by lemma 6.2 of 59  \u000C[12], we can write SS(  \u221E \u2264 2  \u2264 \u221E) =  1SS(\u00B5) . \u221E \u2264\u00B5\u2264\u221E 2  In [12, lemma 6.2], it is explained that given an interval J \u2282 (\u2212\u221E, +\u221E] \u00D7 (\u2212\u221E, +\u221E] that is bounded below, and an increasing sequence of finite subsets V1 \u2282 V2 \u2282 . . . \u2282 J the sequence 1SS(Vj ) in converges to 1SS(J) , where 1SS(Vj ) is defined to be 1SS(v) , v\u2208Vj  where the product is taken in descending order of slope. So, letting J denote the interval of slopes between  \u221E 2  and \u221E, including  \u221E 2  and excluding \u221E, we  can write H= Hexc \u2217  lim  finite V \u2282J  1SS(\u00B5N ) \u2217 . . . \u2217 1SS(\u00B51 ) \u2217 H\u03C0 \u2217 (1SS(\u00B51 ) )\u22121 \u2217 \u00B7 \u00B7 \u00B7 \u2217 (1SS(\u00B5N ) )\u22121 ,  where \u00B5i enumerate all the elements of V . Using proposition and corollary , we can rewrite H= Hexc \u2217  lim  finite V \u2282J  exp({\u03B7\u00B5N , exp{\u03B7\u00B5N \u22121 , . . . exp{\u03B7\u00B51 , \u2212} . . .}(H\u03C0 ).  Now hitting this equation with the integration map yields I\u03A6 (H) = I\u03A6 (Hexc ) \u00B7 I\u03A6  lim  finite V \u2282J  exp({\u03B7\u00B5N , exp{\u03B7\u00B5N \u22121 , . . . exp{\u03B7\u00B51 , \u2212} . . .}(H\u03C0 ) .  The integration map commutes with limits since it is continuous, thus I\u03A6 (H) =  60  \u000CI\u03A6 (Hexc ) \u00B7  lim  finite V \u2282J  I\u03A6 exp({\u03B7\u00B5N , exp({\u03B7\u00B5N \u22121 , . . . exp({\u03B7\u00B51 , \u2212}) . . .})(H\u03C0 ) .  Now, the Poisson bracket is a commutator which is trivial in the ring of Laurent series, so it vanishes after applying the integration map, and we are left with I\u03A6 (H) = I\u03A6 (Hexc ) \u00B7 I\u03A6 (H\u03C0 ). Applying lemma 7.4, we get DT(Y )(x, \u2212q) = DTexc (Y )(x, \u2212q) \u00B7 \u03C0-PT(Y )(x, \u2212q) and substituting q for \u2212q yields DT(Y ) = DTexc (Y ) \u00B7 \u03C0-PT(Y ), which is what we set out to prove.  61  \u000CChapter 8  Conclusion Summary of results In this dissertation, we introduced the notion of a \u03C0-stable pair, which may be thought of as a stable pair whose cokernel is allowed to be one-dimensional \u03C0  provided it is contained in the exceptional divisor of Y \u2192 X and has vanishing higher derived push-forwards.. Following ideas from [12], we described \u03C0-stable pairs in terms of a torsion pair (P\u03C0 , Q\u03C0 ). Then, we constructed a stability condition, and established a relationship between the stability condition and the torsion pair. Finally, after reviewing the construction of the various Hall algebras, we established equations of elements in the Laurent Hall algebra which integrate to give us the desired equality of generating series.  Future research This dissertation is progress toward proving the DT crepant resolution conjecture. The complete proof might take the following form. Let us consider the case of global quotients for simplicity, i.e., stacks of the form X = M/G. In this case, the derived category of X , Db (X ), is isomorphic to the Gb (M ). By the celequivariant derived category of the master space M , DG  ebrated results of Bridgeland, King, and Reid [13], we know that there is an equivalence \u03A8 between the derived category of Y and the G-equivariant derived category of M . The next step will be to show that this equivalence (or one closely related) takes \u03C0-stable pairs on Y to stable pairs on X .  62  \u000CBibliography [1] A. Bayer:  Polynomial Bridgeland stability conditions and the large  volume limit, arXiv: math.AG/0712.1083. [2] A. Bayer: (In preparation) [3] K. Behrend: Donaldson-Thomas type invariants via microlocal geometry, Ann. of Math. (2) 170 (2009), no. 3, 13071338, arXiv: math.AG/0507523. [4] K. Behrend, and J. Bryan: Super-rigid Donaldson-Thomas invariants, Mathematical Research Letters, 14(4):559\u2013571, 2007, arXiv: math.AG/0601203. [5] K. Behrend, and B. Fantechi: The intrinsic normal cone, Invent. Math. 128 (1997), 45-88. [6] K. Behrend, and B. Fantechi: Symmetric obstruction theories and Hilbert schemes of points on threefolds, Algebra &amp; Number Theory 2 (2008), no. 3, 313\u2013345, arXiv: math.AG/0512556. [7] A. Be\u02D8\u0131linson, J. Bernstein, P. Deligne: Faisceaux perverse, Analysis and topology on singular spaces, I (Luminy, 1981), 5171, Ast\u00B4eisque, 100, Soc. Math. France, Paris, 1982. [8] S. Boissiere, and A. Sarti; Contraction of excess fibres between the McKay correspondences in dimensions two and three, Ann. Inst. Fourier, 57(6):1839\u20131861,2007 arXiv: math.AG/0504360. [9] T. Bridgeland: Flops and derived categories, Invent. Math. 147, 613632 (2002). math.AG/0009053. 63  \u000C[10] T. Bridgeland: Stability conditions on triangulated categories, Ann. of Math, Vol 166, 317-345, 2007. [11] T. Bridgeland: An introduction to motivic Hall algebras, preprint arXiv: math.AG/1002.4372v1. [12] T. Bridgeland: Hall algebras and curve-counting invariants, J. Amer. Math. Soc. 24 (2011), no. 4, 969998. [13] T. Bridgeland, A. King, and M. Reid: The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535554 (electronic). [14] J. Bryan, C. Cadman, and B. Young:  The orbifold topological  vertex, Advances in Mathematics, 229 (1), pg. 531-595, arXiv: math.AG/1008.4205v1. [15] J. Bryan and A. Gholampour: The quantum McKay correspondence for polyhedral singularities, Inventiones Mathematicae, 178(3):655\u2013681, 2009, arXiv: math.AG/0803.3766. [16] J. Bryan; T. Graber: The crepant resolution conjecture, Algebraic geometrySeattle 2005. Part 1, 2342, Proc. Sympos. Pure Math., 80, Part 1, Amer. Math. Soc., Providence, RI, 2009 [17] J. Calabrese: Donaldson-Thomas invariants and flops, preprint arXiv: math.AG/1111.1670v1. [18] J. C. Chen; H. H. Tseng: A note on derived McKay correspondence, Math. Res. Lett. 15 (2008), no. 3, 435445. [19] P. Griffiths and J. Harris: Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley &amp; Sons], New York, 1978. [20] A. Grothendieck: Techniques de construction et theoremes d\u2019existence en geometrie algebrique IV: Les schemas de Hilbert, Seminaire Bourbaki 1960/61, no 221. 64  \u000C[21] D. Happel, I. Reiten, S. Smal\u00F8: Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+ 88 pp. [22] G. Harder, and M.S. Narasimhan: On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1975), 215\u2013248. [23] D. Huybrechts, and M. Lehn: The geometry of moduli spaces of sheaves, Aspects of Mathematics, Viewveg 1997. [24] D. Joyce: Configurations in abelian categories. I. Basic properties and moduli stacks, Advances in Mathematics 203 (2006), 194-255. [25] D. Joyce: Configurations in abelian categories. II. Ringel-Hall algebras, Adv. Math. 210 (2007), no. 2, 635706. [26] D. Joyce: Configurations in abelian categories. III. Stability conditions and identities, Adv. Math. 215 (2007), no. 1, 153219. [27] D. Joyce: Configurations in abelian categories. IV. Invariants and changing stability conditions, Advances in Mathematics 217 (2008), 125204. [28] D. Joyce: Motivic invariants of Artin stacks and \u2018stack functions\u2019, Q. J. Math. 58 (2007), no. 3, 345392. [29] D. Joyce and Y. Song: A theory of generalized Donaldson-Thomas invariants, Memoirs of the AMS, 2011, arXiv: math.AG/0810.5645. [30] S. Kleiman: Les Theoremes de finitude pour le foncteur de Picard, SGA de Bois Maris 1966/67, exp XIII. [31] M. Kontsevich, and Y. Soibelman:  Stability structures, motivic  Donaldson-Thomas invariants and cluster transformations, preprint arXiv: math.AG/0811.2435v1. [32] J. Le Potier: Syst`emes coh\u00B4erents et structures de niveau, Ast\u00B4erisque 214, Soci\u00B4et\u00B4e Math\u00B4ematique de France, 1993. 65  \u000C[33] M. Levine, and R. Pandharipande: Algebraic cobordism revisited, arXiv: math.AG/0605196. [34] J. Li: Zero dimensional Donaldson-Thomas invariants of threefolds, Geom. Topol. 10 (2006) 2117\u20132171. [35] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande: Gromov-Witten theory and Donaldson-Thomas theory, I. Compos. Math. 142, no. 5, 1263\u20131285. [36] B. Mitchell, The full imbedding theorem, Amer. J. Math., 86 (1964) 619637. [37] D. Mumford: Lectures on curves on an algebraic surface, Annals Math. Studies 59, Univ Press, Princeton 1966. [38] C. Okonek, M. Schneider, and H. Spindler: Vector bundles on complex projective spaces, Birkhauser, 1980. [39] R. Pandharipande, and R. P. Thomas: Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009), no. 2, 407\u2013447. [40] R. Pandharipande, and R. P. Thomas: The 3-fold vertex via stable pairs, arXiv: math.AG/0709.3823. [41] R. Pandharipande, and R. P. Thomas: Stable pairs and BPS invariants, arXiv: math.AG/0711.3899. [42] M. Reineke: Torsion pairs induced from Harder-Narasimhan filtration, www.math.uni-bielefeld.de/ sek/select/fahrtorsion.pdf, November, 2007 [43] R. P. Thomas: A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations, Journal of Differential Geometry 54, no. 2, 367\u2013438. [44] Y. Toda: Limit stable objects on Calabi-Yau 3-folds, Duke Math. J. 149 (2009), no. 1, 157208. 66  \u000C[45] Y. Toda: Curve counting theories via stable objects II. DT/ncDT/flop formula, arxiv: math.AG/0909.5129. [46] C. Vafa: String vacua and orbifoldized LG models Modern Phys. Lett. A, 4(12):1169\u20131185, 1989. [47] M. van den Bergh: Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423\u2013455. [48] E. Viewhweg: Rational singularities of higher dimensional schemes, Proc. Amer. Math. Soc. 63 (1977), no. 1, 68. [49] C. A. Weibel:  An introduction to homological algebra, Cambridge  Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994. xiv+450 pp. [50] E. Zaslow: Topological orbifold models and quantum cohomology rings, Comm. Math. Phys., 156(2):301\u2013331, 1993.  67  "@en
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