Abstract Abstract. We introduce graded, enriched characteristic cycles as a method for encoding Morse modules of strata with respect to a constructible complex of sheaves. Using this new device, we obtain results for arbitrary complex analytic functions on arbitrarily singular complex analytic spaces. §1. Introduction Let X be a complex analytic space, with arbitrary singularities, and let p ∈ X. As we are interested in local questions near p, we may assume that X is a (closed) complex analytic subspace of a connected, open subset U ⊆ C n+1 , and z := (z 0 , . . . , z n ) are coordinates on C n+1 . Letf : U → C be a complex analytic function, and let f :=f |X . Let S := {S α } be a complex analytic Whitney stratification of X, with connected strata. Fix a base ring R that is a regular, Noetherian ring with finite Krull dimension (e.g., Z, Q, or C). Let F §1. Introduction • be a bounded complex of sheaves of R-modules on X, which is constructible with respect to {S α }. §1. Introduction In this extremely general situation, we are interested in producing effectively calculable algebraic data which provides information about the topology and geometry of the hypersurface V (f ). In particular, we are interested in Thom's a f condition, and Milnor fibration data. Thus, we are led to study the nearby and vanishing cycles of F §1. Introduction • along f ; that is, respectively, ψ f F • and φ f F • . §1. Introduction Our method for producing algebraic data is motivated by our results on Lê cycles and Lê numbers ( [M3] , [M4] , [M5] ), which corresponds to the case of an affine hypersurface where F §1. Introduction • is the constant sheaf C • U . Much of our past work has centered around the problems of extended our work on Lê cycles to the general setting of this paper. In [M8], we associated cycles to arbitrary complexes of sheaves. In [M1] , [M2], we studied the special cases of ψ f F §1. Introduction • and φ f F • . In [M7] , we gave a somewhat satisfactory generalization; we defined Lê-Vogel cycles and numbers. §1. Introduction The Lê cycles of [M7] were produced by starting with the characteristic cycle of the complex F §1. Introduction • , and then using a Lê-Vogel process on an ideal, J, defining the image of df in T * U. It was crucial that we had a result which told us how the characteristic cycle of φ f F §1. Introduction • is obtained from the characteristic cycle of F §1. Introduction • via blowing-up J; we refer to this result as the vanishing index theorem. The fundamental weakness of the results of [M7] lies in the fact that the characteristic cycle of F §1. Introduction • contains only Euler characteristic data on the Morse modules to strata, and thus disposes of a great deal of cohomological data which describes the structure of F §1. Introduction • . In [M7] and [M9], we used the functorial properties of perverse cohomology to extract more refined data from characteristic cycles. In [M6] , we showed with some difficulty how the microlocal theory of Kashiwara and Schapira could be used to obtain better results in the special case of an isolated critical point. §1. Introduction In this paper, we show how our results from [M7] can easily be turned into stronger resultswithout using the devices of perverse cohomology or microlocal theory -simply by using cycles with module-coefficients; these are our enriched cycles. Since we wish for our enriched cycles to be able to encode the cohomology modules of normal data to strata (with coefficients in F §1. Introduction • ), we consider graded, enriched cycles and define gecc §1. Introduction • (F • ), the graded, enriched, characteristic cycle of F §1. Introduction • . The fact that gecc §1. Introduction explains why we may use gecc §1. Introduction • to obtain any of the results of [M7] that used the standard characteristic cycle and perverse cohomology. §1. Introduction The fundamental philosophy of this paper is: any result on perverse sheaves which is proved by using characteristic cycles and intersection theory can be generalized to arbitrary bounded, constructible complexes of sheaves by using graded, enriched characteristic cycles and a corresponding enriched intersection theory. Most of this paper consists of demonstrating applications of this general philosophy. §1. Introduction However, there are several important/interesting results in this paper which are not simply enriched forms of earlier results. The equivalent characterizations of isolating coordinates given in Theorem 5.10 yield a substantial improvement over [M7] and [M8] in the level of genericity required of our choice of coordinates when generalizing absolute polar varieties; this is very important since we are trying to produce a method for making effective computations.The result of Theorem 4.8 -necessary and sufficient conditions for Thom's a f condition -is important in Theorem 6.5 for similar reasons: we use 4.8 and 5.10 to produce a useful notion of "generic coordinates" in the relative situation. Proposition 6.11 tells us that, when the dimension of the critical locus is small, we may check whether our coordinates are generic enough "on-the-fly". Another interesting result which falls out of the enriched approach is a sort of converse to the theorem that: if F §1. Introduction • is perverse, then φ f F §1. Introduction • is perverse. In Corollary 3.7, we show that: if f is in the square of the maximal ideal of X at p and φ f F §1. Introduction • is perverse in a neighborhood of p, then F • is perverse in a neighborhood of p. §1. Introduction The remaining sections of this paper are organized as follows. §1. Introduction In Section 2, we define enriched cycles, describe a corresponding enriched intersection theory (for proper intersections in smooth manifolds), and define and give some basic properties of the graded, enriched characteristic cycle of a complex of sheaves. §1. Introduction Section 3 contains enriched versions of a number of previous results. Of particular importance is Theorem 3.4, an enriched form of our vanishing index theorem. The only new results in this section are the corollaries to 3.4. §1. Introduction In Section 4, we discuss partitions, stratifications, Thom's a f condition, and essentially transverse coordinates. For most of our results, we do not need the full power of a Whitney stratification, instead we use F §1. Introduction • -partitions. This weakening of the type of "stratification" used leads to stronger results. The genericity that we use on the coordinates in Section 6 will be that the coordinates are essentially transverse to a type of partition. §1. Introduction In Section 5, we look at extreme generalizations of polar varieties and polar multiplicities; we define the graded, enriched characteristic polar cycles and the characteristic polar modules. We define F §1. Introduction • -isolating coordinates and characterize them in terms of intersections with gecc • (F • ). This provides the fundamental relation between enriched intersections with gecc §1. Introduction • (F • ) and iterated vanishing and nearby cycles. §1. Introduction In Section 6, we apply the results of Section 5 to the special case of φ f F §1. Introduction • . By combining the work of Section 5 with an enriched version of the vanishing index theorem, we obtain our generalization of the Lê cycles and Lê numbers: the graded, enriched Lê-Vogel cycles and the Lê-Vogel modules. Our work on essentially transverse coordinates and the a f condition from Section 4 are used in this section to give sufficient conditions on the choice of coordinates. §1. Introduction In Section 7, we give a sample calculation of Lê-Vogel cycles. We compare and contrast this with other methods of analyzing the perverse cohomology. Section 8 contains some concluding remarks and questions. §2. Enriched Cycles We will now define enriched cycles, graded enriched cycles, the graded enriched characteristic cycle, and operations on them. The notions that we define are easy and/or obvious; hence, this section consists of a series of definitions with some remarks. Section 8 contains some concluding remarks and questions. §2. Enriched Cycles We continue with all of the notation from the Introduction. We also need to establish some more notation that we shall use throughout the remainder of this paper. Section 8 contains some concluding remarks and questions. §2. Enriched Cycles The cotangent bundle T * U η − → U is a trivial bundle; we will frequently use that T * U ∼ = U × C n+1 and write simply that η is the projection from U × C n+1 onto U. A linear choice of coordinates z for U determines a basis dz 0 , . . . , dz n for the cotangent spaces, and we use w := (w 0 , . . . , w n ) for the cotangent coordinates. Section 8 contains some concluding remarks and questions. §2. Enriched Cycles Using these coordinates, the image of df in T * U, im df , is given by Section 8 contains some concluding remarks and questions. §2. Enriched Cycles As T * U is (complex) conic, we may projectivize in the cotangent directions, and consider P(T * U) ∼ = U × P n . In this and the remaining sections of this paper, it will be important for us to consider all four projection maps: Section 8 contains some concluding remarks and questions. §2. Enriched Cycles The conormal bundle to a stratum S α is given by Section 8 contains some concluding remarks and questions. §2. Enriched Cycles We are frequently interested in the closure, Section 8 contains some concluding remarks and questions. §2. Enriched Cycles Finally, it is fundamental throughout this paper that the functors ψ f and φ f , shifted by −1, commute, up to natural isomorphism, with (middle perversity) perverse cohomology µ H * ; see [BBD] , 10.3.13 of [K-S2] and 6.0.6 of [S] (but, be aware that the definition of the vanishing cycles used in [K-S2] is shifted by 1 from the standard definition that we use). In particular, the shifted nearby and vanishing cycles are perverse functors, i.e., they take perverse sheaves to perverse sheaves. For the nearby cycles, this was first proved by . The first proof of which we are aware that the shifted vanishing cycle functor is perverse appears in 1.7 of [Br] . For these reasons, we shall always include a shift of −1 when we use the nearby and vanishing cycle functors; we shall write ψ f [−1] and φ f [−1] for the functors ψ f and φ f composed with a shift by −1. Section 8 contains some concluding remarks and questions. §2. Enriched Cycles Definition 2.1. An enriched cycle, E, in X is a formal, locally finite sum V E V [V ] , where the V 's are irreducible analytic subsets of X and the E V 's are finitely-generated R-modules. We refer to the V 's as the components of E, and to E V as the V -component module of E. Two enriched cycles are considered the same provided that all of the component modules are isomorphic. The underlying set of E is |E| := ∪ E V =0 V . Section 8 contains some concluding remarks and questions. §2. Enriched Cycles If C = n V [V ] is an ordinary positive cycle in X, i.e., all of the n v are non-negative integers, then there is a corresponding enriched cycle [C] enr in which the V -component module is the free R-module of rank n V . If R is an integral domain, so that rank of an R-module is well-defined, then an enriched cycle E yields an ordinary cycle [E] ord := V (rk(E V )) [V ] . If q is a finitely-generated module and E is an enriched cycle, then we let qE := V (q ⊗E V ) [V ] ; thus, if R is an integral domain and E is an enriched cycle, [qE] ord = (rk(q)) [E] ord and if C is an ordinary positive cycle and n is a positive integer, then [nC] enr Section 8 contains some concluding remarks and questions. §2. Enriched Cycles The (direct) sum of two enriched cycles D and E is given by ( Section 8 contains some concluding remarks and questions. §2. Enriched Cycles There is a partial ordering on enriched cycles given by: D E if and only if there exists an enriched cycle P such that D + P = E. This relation is clearly reflexive and transitive; moreover, anti-symmetry follows from the fact that if M and N are Noetherian modules such that Section 8 contains some concluding remarks and questions. §2. Enriched Cycles If two irreducible analytic subsets V and W intersect properly in U, then the (ordinary) intersection cycle [V ] · [W ] is a well-defined positive cycle; we define the enriched intersection product of [V ] enr and [W ] enr by [V ] Section 8 contains some concluding remarks and questions. §2. Enriched Cycles enr . If D and E are enriched cycles, and every component of D properly intersects every component of E in U, then we say that D and E intersect properly in U and we extend the intersection product linearly, i.e., if D = V D V [V ] and Section 8 contains some concluding remarks and questions. §2. Enriched Cycles Suppose that g := (g 0 , . . . , g d ) is a (d+1)-tuple of analytic functions on U and the D = D V [V ] is an enriched cycle in U. Then, for each V , the blow-up of the ideal generated by the restriction of g to V , Bl g (V ) , yields an irreducible analytic subset of U × P d ; in addition, we obtain an exceptional divisor Ex g (V ) , which is an (ordinary) positive cycle in U × P d . We define the blow-up and exceptional divisor of D along g to be the enriched cycles in Section 8 contains some concluding remarks and questions. §2. Enriched Cycles enr , respectively. If we had started with an ordinary cycle D, we would analogously obtained ordinary cycles for Bl g (D) and Ex g (D). A graded, enriched cycle E • is simply an enriched cycle E i for i in some bounded set of integers. An single enriched cycle is considered as a graded enriched cycle by being placed totally in degree zero. The analytic set V is a component of E A graded, enriched cycle E • if and only if V is a component of E i for some i, and the underlying set of E A graded, enriched cycle E • is |E A graded, enriched cycle E • yields an ordinary cycle A graded, enriched cycle E If q is a finitely-generated module and E • is a graded enriched cycle, then we define the graded enriched cycle qE A graded, enriched cycle E • by (qE A graded, enriched cycle E . The (direct) sum of two graded enriched cycles D A graded, enriched cycle E • and E • is given by (D A graded, enriched cycle E properly intersects E j for all i and j, then we say that D A graded, enriched cycle E • and E • intersect properly and we define the intersection product by A graded, enriched cycle E Let τ : W → Y be a proper morphism between analytic spaces. If C = n V [V ] is an ordinary positive cycle in W , then the proper push-forward τ * (C) = n V τ * ( [V ] ) is a well-defined ordinary cycle. If A graded, enriched cycle E is an enriched cycle in W , then we define the proper push-forward of E A graded, enriched cycle E • by τ to be the graded enriched cycle τ A graded, enriched cycle E The ordinary projection formula for divisors ( [F] , 2.3.c) immediately implies the following enriched version. Let E • be a graded enriched cycle in X. Let W := |E • |. Let τ : W → Y be a proper morphism, and let g : Y → C be an analytic function such that g • τ is not identically zero on any component of E A graded, enriched cycle E • . Then, g is not identically zero on any component of τ A graded, enriched cycle E The standard intersection result on conservation of number generalizes easily to: A graded, enriched cycle E conservation of module: A graded, enriched cycle E Let E be a purely k-dimensional enriched cycle in U and let f := (f 1 , . . . , f k ) ∈ (O U ) k be such that E and V (f ) intersect properly in the isolated point p. A graded, enriched cycle E where ǫ > 0 is sufficiently small, A graded, enriched cycle E is an open ball of radius ǫ centered at p, and |t 0 | ≪ ǫ. Definition 2.2. If D • is a graded, enriched cycle and g := (g 0 , . . . , g d ) is a (d + 1)-tuple of analytic functions on U, then we define the blow-up and exceptional divisor of D Definition 2.2. If D • along g to be the graded, enriched cycles in Definition 2.2. If D Later, we will be especially interested in the case where g is the tuple defining im df , • is a bounded complex of sheaves, which is constructible with respect to an analytic Whitney stratification {S α }, in which the strata are connected. Let Definition 2.2. If D is a pair consisting of a normal slice and complex link (see [G-M2] ), respectively, to the stratum S α , then, for each integer k, the isomorphism-type of the module Definition 2.2. If D Remark 2.4. There are no canonical choices for defining the the normal slices or complex links of strata (again, see [G-M2] Definition 2.2. If D ord , provided that the base ring is an integral domain. Definition 2.2. If D On a different note, recall that a complex F • is called pure (of shift 0) provided that gecc Definition 2.2. If D .2 and 9.5). If P Definition 2.2. If D • is a perverse sheaf on X, then P • is pure; the converse of this is also true (see 9.5.2 of [K-S1] or Remark 2.6). Definition 2.2. If D Finally, note that, for all k, gecc Definition 2.2. If D Proposition 2.5. For all k, there is an equality of enriched cycles given by Definition 2.2. If D Furthermore, there are equalities of sets given by supp Definition 2.2. If D Proof. The first equality follows immediately from the fact that perverse cohomology commutes with taking the shifted vanishing cycles; the argument is precisely the same as that of Proposition 2.1 in [M9] . If P • is a perverse sheaf, then supp(P • ) = η(| gecc 0 (P • )|); this is essentially the last part of Lemma 3.1 of [M1] , and the proof is the same. The second equality of the proposition follows by applying this to Definition 2.2. If D , and then using the first equality. As supp (F • Definition 2.2. If D , the last equality of the proposition follows from the second. Definition 2.2. If D Alternatively, the last two equalities could be concluded from the fact that, at a generic point in an irreducible component of supp (F • ) , the Morse modules simply yield stalk cohomology. Definition 2.2. If D Remark 2.6. The equality of supp Definition 2.2. If D , we recover the result of 9.5.2 of [K-S1] that pure complexes are perverse. Definition 2.2. If D We should also remark here that the equality of supp(F • ) and η(| gecc Definition 2.2. If D This is equivalent to saying that S α has a non-zero Morse module, with respect to F Definition 2.2. If D • , in some degree, i.e., provided that T * Sα U is a component of gecc Definition 2.2. If D Remark 2.8. In terms of the definitions in 2.7, the last equality of Proposition 2.5 can be restated as supp (F • . Enriched Forms of Previous Results In this section, we describe the enriched versions of a number of known results. We provide no proofs, since all one has to do is rewrite previous Morse-theoretic proofs, using enriched cycles in place of ordinary cycles. . Enriched Forms of Previous Results We continue with all of our notation from the previous two sections. . Enriched Forms of Previous Results U , then we define the relative graded enriched conormal cycle, . Enriched Forms of Previous Results The following result is an enriched version of Theorem 2.3 of [M1] ; it follows trivially by "enriching" the cycles in the Morse theoretic proof given in [M1] . Alternatively, one can use the device of perverse cohomology, as in [M9] , to obtain the result directly from the statement of Theorem 2.3 of [M1] . . Enriched Forms of Previous Results Theorem 3.2. There is an equality of graded enriched cycles given by . Enriched Forms of Previous Results In particular, . Enriched Forms of Previous Results The next theorem is the main result of [M6] , stated in a form that uses our current terminology. One could also obtain this result by using enriched cycles throughout the proof in [L] . . Enriched Forms of Previous Results where im df is considered as a graded, enriched cycle. . Enriched Forms of Previous Results The next theorem follows from Proposition 2.2 and Theorem 3.1 of [M9], but we could have obtained the result directly by using enriched cycles throughout the proof of Theorem 2.10 of [M1] . . Enriched Forms of Previous Results Theorem 3.4. There is an equality of closed subsets of X given by . Enriched Forms of Previous Results and, for all k, an equality of graded, enriched cycles given by . Enriched Forms of Previous Results In particular, for all k, there is an equality of sets . Enriched Forms of Previous Results Theorem 3.3 can be recovered very quickly as a special case of Theorem 3.4. . Enriched Forms of Previous Results The following corollary is immediate from Theorem 3.4. . Enriched Forms of Previous Results Corollary 3.5. Suppose that f (p) = 0, and that M is an R-module. If the enriched cycle Proof. The isomorphism is immediate from the fundamental distinguished triangle relating the nearby and vanishing cycles. By 3.4, since If the enriched cycle must be concentrated in degree 0 over a neighborhood of p. This yields the first statement. The second statement uses the same argument. If the enriched cycle As our final result of this section, we need to state the enriched version of Theorem I.2.20 of [M7] . We will use this result in Section 6, where it will enable us to actually perform calculations. We state the result in the form in which we shall use it. Recall the definition of the graded, enriched exceptional divisor from 2.2. Then, there exists an open neighborhood of h can be calculated via the following inductive process, in which all of the intersections are proper: the sum of all those components (with their component modules) of the intersection which are not contained (as sets) in V (h), and k ∆ n h will precisely equal the sum of the components of the intersection which are contained in V (h). Proceeding inductively, one writes the intersection k Π j+1 h ⊙ V (h j ) = k Π j h + k ∆ j h , In defining gecc • (F • ), we used a Whitney stratification of X with respect to which F • was constructible. However, in practice, we do not explicitly need Whitney's condition b) or the condition of the frontier. Since weaker hypotheses on the "stratifications" will yield stronger, more useful results, we wish to describe the types of partitions of X that we will use, and prove a few basic results. In defining gecc We continue with all of the notation from the previous sections except that {S α } is no longer assumed to be a Whitney stratification. Throughout the remainder of this paper, if Y is an analytic set, then when we write dim p Y 0, we mean that either p is an isolated point of In defining gecc Definition 4.1. A (complex analytic) partition of X is a locally-finite collection S := {S α } α of disjoint, connected, complex analytic manifolds whose union is all of X and such that, for all α, S α and S α − S α are complex analytic subsets of U. We still refer to the elements of S as strata. In defining gecc The strata of a partition are partially-ordered by: S β S α if and only if S β ⊆ S α . In defining gecc A partition S of X is a stratification provided that the condition of the frontier holds, i.e., for all S α ∈ S, S α is a union of strata. This means that S β S α if and only if S β ∩ S α = ∅. In defining gecc A partition S of X is a Whitney a) partition provided that all pairs of strata satisfy Whitney's condition a); in conormal terms this means that, for all S β ∈ S, In defining gecc In particular, this implies that α T * Sα U is equal to α T * Sα U and, hence, is closed. If F • is a bounded, constructible complex of sheaves on X, then a partition S of X is an If F In an If F Note that this extends our definition in 2.7 to the case where S need not be a Whitney stratification. Also note that, if S is an F If F • -partition of X, then, as before, Proposition 2.5 implies that If F in particular, this implies that the maximal elements of {S α ∈ S | S α ⊆ supp(F • )} are all visible, and that their union is equal to supp(F • ). If F Certainly, a Whitney stratification of X, with connected strata, with respect to which F • is constructible, is an F If F • -partition of X. However, in our results, it is only the properties of F • -partitions that we actually use. If F Definition 4.2. Suppose that S is a partition of X. Let S α ∈ S, and let p ∈ X (but p is not necessarily in S α ). If F The germ at p of an analytic submanifold N ⊆ U is essentially transverse to S α at p (in U) provided that there is an open neighborhood of p in which N transversely intersects S α − {p} inside U. If F We say that the germ N is essentially transverse to S at p provided that N is essentially transverse to S α at p, for all S α ∈ S. If F If N is essentially transverse to S at p, then, in a neighborhood of p, there is an induced partition of X ∩ N in which the strata are the point-stratum {p}, together with the connected-components of the intersection (S α − {p}) ∩ N . If F The coordinates z for U are essentially transverse to S α at p provided that, for all i with 0 i n, V (z 0 − p 0 , . . . , z i − p i ) is essentially transverse to S α at p. This is clearly equivalent to requiring that, for all i If F The coordinates z for U are essentially transverse to S at p provided that z is essentially transverse to each S α at p. This is equivalent to: V (z 0 − p 0 ) is essentially transverse to S at p and, for all i such that 1 i n, V (z i − p i ) is essentially transverse to the induced partition of If F In the proposition below, we should point out that the only difference between the statements in b) and c) is that, in b), V (z i − p i ) explicitly appears in the intersection, while it does not in c). If F Proposition 4.3. Suppose that S is a partition of X. Let S α ∈ S, and let p ∈ X. Then, the following are equivalent: If F a) The coordinates z are essentially transverse to S α at p; Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Proof. That a) and b) are equivalent is simply a translation of transversality into conormal terms. Certainly, c) implies b). That we obtain equivalent statements if we replace 0 i n in b) and c) by the condition that 0 i −1 + dim S α is also trivial. The surprising implication is that b) implies the seemingly stronger c). Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Assume that b) holds. Suppose that we had the germ of a complex analytic curve q(t) := (x(t), [ω(t)]) ∈ U × P n such that x(0) = p and so that, for small t = 0, q(t) is in Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that , the above paragraph shows that, as germs over p, Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that where Y −1 := ∅. Now, b) yields a contradiction. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Corollary 4.4. Suppose that S is a Whitney a) partition of X. Then the coordinates z are essentially transverse to S at p ∈ X if and only if, for all S α ∈ S, for all i such that 0 i n, Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that In particular, if z is essentially transverse to S at p, then z is essentially transverse to S at all points near p. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Proof. This follows from 4.3.c and the fact that the Whitney a) condition implies that Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that As our final result of this section, we wish to characterize Thom's a f condition in terms of exceptional divisors; this is essentially Proposition 4.3 from [M9] . Recall the definition of the relative conormal space from 3.1. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Definition 4.5. Let M and N be analytic submanifolds of X such that f has constant rank on N . Then, the pair (M, N ) satisfies Thom's a f condition at a point x ∈ N if and only if we have Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that In particular, if f is, in fact, constant on N , then the pair (M, N ) satisfies Thom's a f condition at a point x ∈ N if and only if we have the containment T * Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that If f is constant on both M and N , then the a f condition reduces to Whitney's condition a). The notion of Thom's a f condition that we use below (and above) is slightly more general than is sometimes the case; we do not require the rank of f to be constant on the bigger stratum. If we were to require the rank of f to be constant on the bigger stratum, then we would be forced to write the more cumbersome " (Y reg −Σ(f |Y reg ), N ) satisfies Thom's a f condition at x". Moreover, if C is a component of Y on which f is constant, then saying that (Y reg , N ) satisfies the a f condition implies that (C reg , N ) satisfies Whitney's condition a); the condition that (Y reg − Σ(f |Y reg ), N ) satisfies Thom's a f condition would ignore what happens on a component such as C. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Definition 4.7. An a f partition of X with respect to Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that is a union is strata (the V (f ) strata) and such that for every F Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that • -visible S α and every V (f ) stratum S β , the pair (S α , S β ) satisfies the a f condition. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that The following theorem looks like a significant improvement of Proposition 4.3 of [M9]; however, the proof is essentially the same. In Section 6, this theorem will enable us to link Thom's a f condition with the vanishing cycles along f . Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that iii) there is the containment of fibres above x given by π(E) Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Proof. We first make a few simple observations. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that • Suppose (Y reg , N ) satisfies Thom's a f condition at x. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Then certainly i) and ii) follow; for we can use as relative conormal covectors every conormal to Y reg and every covector of the form d yf , where y ∈ Y reg . In addition, we proved in Proposition 4.3 of [M9] that the a f condition implies that Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that This proves one direction of the theorem. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that • Now suppose that i), ii), and iii) hold. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that If C is a component of Y on which f is constant, then limiting relative conormals to C reg are the same as limiting (absolute) conormals to C reg . As we are assuming that Whitney's condition a) holds, it follows that (C reg , N ) satisfies that a f at x. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that For components C of Y on which f is non-constant, the proof of Proposition 4.3 of [M9] shows that (C reg , N ) satisfies that a f at x. §5. Characteristic Polar Complexes, Modules, and Isolating Coordinates Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that In this section, we will define one our primary objects of study -the characteristic polar modules -and relate them to polar varieties, graded, enriched characteristic cycles, and iterated vanishing and nearby cycles. This will require us to investigate how generic a linear choice of coordinates must be in order to produce nice results. It is important throughout this section, and throughout the remainder of this paper, that our notion of isolating coordinates is an effective notion of "generic"; that is, in many situations, one can determine fairly easily whether the coordinates are isolating. Moreover, if the coordinates are isolating at a given point, then they are isolating at all nearby points -there is no need to re-choose the coordinates at each point. These properties are important, since our goal is to produce effectively calculable data that one can associate to a singularity. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that We continue with all of our previous notation, and also introduce new notation. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that We let Y be a new complex analytic subspace of U and we let A Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that • be a bounded, constructible complex of R-modules on Y . Let R := {R β } be an A Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that • -partition of Y . Our reason for introducing these new objects is that, later, we will return to the setting of the previous sections by considering the special case where Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Recall that we use z = (z 0 , . . . , z n ) to denote coordinates on U. Below, when the context makes the domains clear, we shall not distinguish in the notation between the coordinate functions z i and their restrictions to various subspaces. As we will be projectivizing conormal varieties, we assume that codim U Y 1. Throughout the remainder of this paper, it will be convenient to adopt the standard convention that, when m = 0, Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that This section contains a number of technical results. However, the reader should note that the main points of this section are: Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that • In Definition 5.1, we define the j-th characteristic polar module in degree k of A Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that • with respect to the coordinates z at the point p to be Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that • The coordinates z are A • -isolating at p provided that the support of Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that is at most 0-dimensional at p for all j. Despite the fact that this condition looks fairly unmanageable, in Theorem 5.10, we prove that this condition has a nice interpretation in terms of intersections with gecc Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Moreover, under the assumption that the coordinates are A • -isolating, general results on nearby cycles, vanishing cycles, and perverse cohomology allow us to conclude Theorem 5.18: a result which yields chain complexes, containing the k γ Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that , whose cohomology is isomorphic to the stalk cohomology of the perverse cohomology of A Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that • in each degree. This result is important because the modules appearing in these chain complexes can be calculated; see the following paragraph. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that • In Definition 5.20, under the assumption that the coordinates are A Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that • -isolating, we define the enriched j-th characteristic polar cycle in degree k of A Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that • with respect to the coordinates z to be Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that and we show in Theorem 5.23 that Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Definition 5.1. For all j such that 0 j n, for all a ∈ C j+1 , we define the j-th characteristic polar complex of A Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that • with respect to z at a to be Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that For all x ∈ Y , we define the j-th characteristic polar module in degree k of A • with respect to Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that is, naturally, defined to be the closure of the set {x Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that to be the closure of the set Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that We denote this support by supp( Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that ). By boundedness, there are a finite number of k such that Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that ). Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that If the base ring R is a domain, we define the j-th polar Euler number of A • with respect to z at x to be ord γ Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that We make the following trivial observation. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Proposition 5.2. Letz denote the "rotated" coordinate system (z 1 , z 2 , . . . , z n , z 0 ). Fix x 0 . For all j such that 1 j n, for all x ∈ V (z 0 − x 0 ), Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that and, for all k, k γ j−1 Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Proof. This is immediate from the definitions. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Only slightly less trivial is the following. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Proof. This follows immediately by inductively applying the fact that Euler characteristics are additive over long exact sequences to the long exact sequences on stalk cohomology, at x, which come from the distinguished triangles Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that While it is clear that Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that the reverse inclusion need not hold. In order to determine a more precise relationship between supp(Φ Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that ), we need a lemma and a proposition. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Lemma 5.4. Let h : Y → C be an analytic function, and leth : Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Then, for all k, there is an equality of fibres over p given by Proof. By the definition of Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that We need to deal with possible strata R β such that p ∈ R β − R β . Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that By 3.4, the set η | gecc Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Therefore, if , we may take x i ∈ R β such that x i → p and Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that As d ph ∈ T * R β U p , we conclude that the limits behave "nicely", and thus Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Definition 5.5 For all m such that 0 m n, we define Θ Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that and we define Γ . . , z m ) |R β together with some possible "degenerate" critical points on smaller strata. By Whitney's condition a), these degenerate points will be critical points of (z 0 , . . . , z m ) restricted to smaller strata. Hence, we have the containments Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that If the coordinates z are sufficiently generic at p, and dim R γ m, then crit(z 0 , . . . , z m ) |R γ is precisely the m-dimensional absolute polar variety of R γ at p. Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Proposition 5.7. For all m such that 0 m n, there is an equality of sets given by Moreover, we obtain equivalent statements if we replace the condition that 0 i n in b) and c) by the condition that Proof. We proceed by induction on m. Moreover, we obtain equivalent statements if we
{'paper_id': '17768641', 'title': 'Singularities and Enriched Cycles', 's2_url': 'https://api.semanticscholar.org/CorpusID:17768641', 'source': 'S2ORC'}