A generalization of Milnor's formula

We describe a generalization of Milnor's formula for the Milnor number of an isolated hypersurface singularity to the case of a function $f$ whose restriction $f|(X,0)$ to an arbitrarily singular reduced complex analytic space $(X,0) \subset (\mathbb C^n,0)$ has an isolated singularity in the stratified sense. The corresponding analogue of the Milnor number, $\mu_f(\alpha;X,0)$, is the number of Morse critical points in a stratum $\mathscr S_\alpha$ of $(X,0)$ in a morsification of $f|(X,0)$. Our formula expresses $\mu_f(\alpha;X,0)$ as a homological index based on the derived geometry of the Nash modification of the closure of the stratum $\mathscr S_\alpha$. While most of the topological aspects in this setup were already understood, our considerations provide the corresponding analytic counterpart. We also describe how to compute the numbers $\mu_f(\alpha;X,0)$ by means of our formula in the case where the closure $\overline{ \mathscr S_\alpha} \subset X$ of the stratum in question is a hypersurface.


Summary of results
We start by a discussion of the Milnor number similar to the one found in [STV05]. The Milnor number µ f is one of the central invariants of a holomorphic function f : (C n , 0) → (C, 0) with isolated singularity. It has -among others -the following characterizations, cf. [Mil68,Chapter 7] and [AGZV85, Chapter 2]. 1) It is the number of Morse critical points in a morsification f η of f .
2) It is equal to the middle Betti number of the Milnor fiber 3) It is the degree of the map 1 | df | df : ∂B ε → S 2n−1 for some choice of a Hermitian metric on (C n , 0). 4) It is the length of the Milnor algebra where Jac(f ) = ∂f ∂x1 , . . . , ∂f ∂xn is the Jacobian ideal of f . In this note we consider the more general setup of an arbitrary reduced complex analytic space (X, 0) ⊂ (C n , 0) and a holomorphic function f : (C n , 0) → (C, 0), whose restriction f |(X, 0) to (X, 0) has an isolated singularity in the stratified sense. To this end, we will assume that (X, 0) is endowed with a complex analytic Whitney stratification S = {S α } α∈A with finitely many connected strata S α . There always exists a Milnor fibration for the restriction f |(X, 0) of any function f to (X, 0), regardless of whether or not f |(X, 0) has isolated singularity; see [L87], or [GM88]. Denote the corresponding Milnor fiber by where X is a suitable representative, B ε a ball of radius ε centered at the origin in C n , and ε δ > 0 sufficiently small.
We introduce invariants µ f (α; X, 0) of f |(X, 0) -see Definition 3.6 -which generalize the classical Milnor number simultaneously in all of these four characterizations. Let X α = S α be the closure of the stratum S α and d(α) its (complex) dimension. Then for every α ∈ A the number µ f (α; X, 0) is 1') the number of Morse critical points on the stratum S α in a morsification of f .
For the definition of morsifications in this context see Section 3.1. 2') the number of direct summands for α in the homology decomposition of the Milnor fiber M f |(X,0) , see Proposition 3.8. 3') the Euler obstruction Eu df (X α , 0) of the 1-form df on (X α , 0), see Definition 3.13 and Corollary 3.19. 4') the homological index i.e. as an Euler characteristic of a finite complex of coherent O X -modules, cf. Theorem 4.3 and Corollary 4.4.
Generalizations similar to those of 1), 2), and 3) have been made by J. Seade, M. Tibȃr and A. Verjovsky in [STV05]. The Euler obstruction of a 1-form was introduced by W. Ebeling and S. Gusein-Zade in [EGZ05]. Contrary to these previous topological considerations, we will describe the Euler obstructions Eu df (X α , 0) as an analytic invariant in Theorem 4.3. This allows us to also generalize the characterization 4) of the Milnor number to 4'). A description for how to compute the numbers µ f (α; X, 0) whenever S α is an algebraic hypersurface, f is also algebraic and both are defined over a finite extension field of Q, is described in Section 5.
Example 1.1. The following will serve us as a guiding example throughout this article. Let X ⊂ C 3 be the Whitney umbrella given by the equation h = y 2 − xz 2 = This stratification is known to satisfy the Whitney conditions A and B.
Note that f does not have isolated singularity on C 3 . Its restriction f |X to X, however, has only isolated critical points at 0 = 0, 0, 0 and p 6,7 = 3 2 , ± 3 √ 2 , −3 . It will become clear later, why we label the last two of these points with indices 6 and 7. We will usually have to neglect these points, since we are interested in the local behaviour of f on the germ (X, 0) of X at the origin 0 ∈ C 3 . As we shall see in Example 3.7, we have µ f (0; X, 0) = 1, µ f (1; X, 0) = 1, µ f (2; X, 0) = 5 for the restriction f |(X, 0) at this point.

Acknowledgements
The author is indebted to J. Seade for discussions on the homological index, the law of conservation of number, and Euler obstructions. These conversations took place at the meeting ENSINO V at João Pessoa, Brazil in July 2019 and the author wishes to also thank the organizers A. Menegon Neto, J. Snoussi, M. da Silva Pereira, and M. F. Zanchetta Morgado for the opportunity to give a course on determinantal singularities together with M.A.S. Ruas. The author would also like to thank J. Schürmann for encouraging discussions during the conference "Nonisolated singularities and derived geometry" at Cuernavaca, Mexico the same year. It is the ideas from this period that cumulated in this article. The pictures for the examples were produced using the software "Surfer" by O. Labs et. al.

Background and motivation
Suppose the function f , the space (X, 0), and its stratification have been chosen as in 1') to 4') from Section 1. In [STV05] the Euler obstruction Eu f (X, 0) of the function f on (X, 0) plays the role of the µ f (α; X, 0) for the top-dimensional stratum -up to sign. The Euler obstruction of a function was introduced in [BMPS04] and it is defined as follows. Let ν :X → X be the Nash modification of (X, 0). Then there always exists a continuous alteration v of the gradient vector field grad f on (C n , 0) which is tangent to the strata of (X, 0), and a lift ν * v of v to the Nash bundleT onX. Over the link K = ∂B ε ∩ X of (X, 0) this lift is well defined as a non-zero section inT up to homotopy. Now Eu f (X, 0) is the obstruction to extending ν * v as a nowhere vanishing section to the interior ofX.
To understand how our approach came about to also include 4') in this discussion, we have to consider the article [STV05] in the context of a series of articles by various authors on different indices of vector fields and 1-forms on singular varieties. A thorough survey of the results from that time is [EGZ06].
One of these indices -the GSV index of a vector field -is particularly close to the idea of the Euler obstruction. The GSV index was first defined in [GMSV91, Definition 2.1 ii)] for the following setup: Let (X, 0) = (g −1 ({0}), 0) ⊂ (C n+1 , 0) be an isolated hypersurface singularity and v the germ of a vector field on C n+1 , 0 which has an isolated zero at the origin and is tangent to (X, 0). The GSV index Ind GSV (v, X, 0) of v on (X, 0) is the obstruction to extending the section v|K as a C ∞ -section of the tangent bundle from the link K = X ∩ ∂B ε to the interior of the Milnor fiber B ε ∩ g −1 ({δ}). Here we deliberately identify the link K with the boundary ∂B ε ∩ g −1 ({δ}) of the Milnor fiber and the section v|K with its image under this identification.
In [GM98], X. Gómez-Mont introduces the homological index of a vector field v on (X, 0) as above in order to compute the GSV index algebraically 1 . It is defined as Ind hom (v, X, 0) = χ(Ω • X,0 , v), i.e. the Euler characteristic of the complex where Ω p X,0 denotes the module of universally finite Kähler differentials on (X, 0) and v is the homomorphism given by contraction of a differential form with the vector field v. Later on in his article, X. Gomez-Mont generalizes the GSV index in the obvious way to the setting of an arbitrary complex space (X, 0) with an isolated singularity and a fixed smoothing X of (X, 0). In [GM98, Theorem 3.2] he proves that with k(X, X ) a constant depending only on (X, 0) and the chosen smoothing, i.e. independent of the vector field v. Finally, he shows in [GM98, Section 3.2] that whenever (X, 0) = (g −1 ({0}), 0) is an isolated hypersurface singularity with its canonical smoothing X = B ε ∩ g −1 ({δ}), ε δ > 0, then k(X, X ) = 0. From our point of view, the main novum in the approach by X. Gómez-Mont was the introduction of derived geometry in this setting and its comparison with topological invariants. To prove Equation (1), he procedes as follows.
On the one hand, the GSV index is constant under small perturbations of v. This is immediate from the definition, since small perturbations of v do not change the homotopy class of the non-zero section v|K. On the other hand, the homological index Ind hom (v, X, 0) satisfies the law of conservation of number, i.e. for suitable representatives andṽ sufficiently close to v one has Ind hom (v, X, 0) = p∈X Ind hom (ṽ, X, p). This is due to a technical but fundamental result based on derived geometry for the complex analytic setting from [GGM02] which states that, more generally, the Euler characteristic of a complex of coherent sheaves with finite dimensional cohomology satisfies the law of conservation of number. To conclude the proof of Equation (1), observe that at smooth points p ∈ X reg , the GSV index and the homological index coincide. Since the space of holomorphic vector fields on (X, 0) with isolated singularity at the origin is connected, the difference Ind GSV (v, X, X ) − Ind hom (v, X, 0) must be a constant k(X, X ) and in particular independent of the vector field v.
In this article, we will not be dealing with vector fields, but with holomorphic 1forms. In fact, the original definition of the Euler obstruction by R. MacPherson in [Mac74] was phrased in terms of radial 1-forms and only later the use of vector fields became popular following the work of J.P. Brasselet and M.H. Schwartz [BS81]. The use of 1-forms is more natural in the context of morsifications and it has several further advantages. For example, we can drop the tangency conditions to (X, 0) which we had to impose on any vector field v.
It is straightforward -and even easier -to also define the Euler obstruction Eu ω (X, 0) of a 1-form ω with isolated zero on (X, 0): Again, let ν :X → X be the Nash modification. Then there is a natural pullback ν * ω of ω to a section of the dual of the Nash bundle and this section does not vanish on ν −1 (∂B ε ∩X) whenever ω has an isolated zero on (X, 0) in the stratified sense. The Euler obstruction of such an ω on (X, 0) is the obstruction to extending ν * ω as a nowhere vanishing section to the interior ofX.
There is a natural notion of the homological index for a 1-form ω with isolated zero on any purely n-dimensional complex analytic space (X, 0) with isolated singularity. In [EGZS04], W. Ebeling, S.M. Gusein-Zade, and J. Seade define where Ω • X,0 , ω ∧ − is the complex / / 0 with differential given by the exterior multiplication with ω. Note that in case (X, 0) ∼ = (C n , 0) is smooth and ω = df is the differential of a function f with isolated singularity on (X, 0), the homological index coincides with the classical Milnor number. This is due to the fact that the complex (2) is the Koszul complex in the partial derivatives ∂f ∂xi of f which is known to be a free resolution of the Milnor algebra for an isolated hypersurface singularity.
When (X, 0) has isolated singularity, there is no immediate interpretation for the homological index of ω in terms of previously known invariants. However, it is relatively easy to see with the same reasoning as for indices of vector fields that the the difference is also a constant, independent of the 1-form ω: The Euler obstruction Eu ω (X, 0) is a homotopy invariant and Ind hom (ω, X, 0) satisfies the law of conservation of number. Suppose we have chosen a suitable representative X of (X, 0) and a sufficiently small ball B ε . Then for any a holomorphic 1-form ω on X which has only isolated zeroes on the smooth part X reg of X and which is sufficiently close to the original 1-form ω, we have This holds because, again, Eu ω (X, p) = Ind hom (ω , X, p) at smooth points p ∈ X reg . The general claim now follows from the fact that the set of those holomorphic 1-forms on X with only only isolated zeroes on X reg is open and connected.
There are other instances of very similar discussions. In [EGZS04, Proposition 4.1], for example, there is a comparison of the homological index and the radial index Ind rad (ω, X, 0) (cf. [EGZS04, Definition 2.1]) of a 1-form ω with isolated zero on an equidimensional complex analytic space (X, 0) with isolated singularity. Their difference is an invariant ν(X, 0) which coincides with the Milnor number of (X, 0) whenever (X, 0) is an isolated complete intersection singularity.
Coming back to the comparison of Eu ω (X, 0) with Ind hom (ω, X, 0) in Equation (3), the introduction of k (X, 0) as a new invariant of the germ (X, 0) seems to be rather unmotivated. Instead we propose a modification of the homological index in Section 4 which directly computes the Euler obstruction. This is Theorem 4.3. The new homological index will be based on the Nash modification ν :X → X of (X, 0) and the complex of sheaves Ω • , ν * dω ∧ − onX rather than Ω • X,0 , ω ∧ − . For the definition of this complex see Sections 3.3 and 4. The direct computation of Eu ω (X, 0) as an Euler characteristic of finite O n -modules comes at the price that one has to take the derived pushforward along ν of the complex of sheaves Ω • , ν * dω ∧ − . However, as a side effect of this, we may drop the assumption on (X, 0) to have only isolated singularity.

Generalizations of the Milnor number
We briefly recall the necessary definitions of singularity theory on stratified spaces, cf. [L87]. Let U ⊂ C n be an open domain, X ⊂ U a closed, reduced, equidimensional complex analytic set and f : U → C a holomorphic function. The existence of complex analytic Whitney stratifications was shown by H. Hironaka [Hir77]. In [TT81, Corollaire 6.1.8] Lê D. T. and B. Teissier constructed a canonical Whitney stratification for reduced, equidimensional complex analytic spaces, and in [Tei82] it was shown that this stratification is minimal. Whenever one of these strata consists of several components, we shall in the following consider each one of these components as a stratum of its own and -unless otherwise specified -use this stratification on any given reduced equidimensional complex analytic space X.
Definition 3.2. We say that f has an isolated singularity at (X, p), if there exists a neighborhood U of p such that all points x ∈ U ∩ X \ {p} are regular points of f in the stratified sense for the canonical Whitney stratification of X.
We give a brief definition of the Milnor fibration of f |(X, p) in this setting. Let B ε be the ball of radius ε around p in C n . By virtue of the Curve Selection Lemma, there exists ε 0 > 0 such that for every ε 0 ≥ ε > 0 the intersections ∂B ε ∩ X and ∂B ε ∩ X ∩ f −1 ({f (p)}) are transversal. Fix one such ε > 0. Then for sufficiently small ε δ > 0 the restriction of f is a proper C 0 -fiber bundle over the punctured disc D * δ ⊂ C of radius δ > 0 around f (p) -the Milnor fibration of f |(X, p). The fiber is unique up to homeomorphism and thus an invariant of f |(X, p).
Remark 3.3. Many authors prefer to work with a holomorphic function g : (X, p) → (C, g(p)) instead of an embedding ι : (X, p) → (C n , p) and a restriction f |(X, p) of a function f : (C n , p) → (C, f (p)). It is clear that for every g one can find ι and f such that f |(X, p) = g. Moreover, it can be shown that the canonical stratification of (X, p) does not depend on the embedding [L87]. Neither does the Milnor fiber M g = M f |(X,p) . If the reader intends to start from g defined on (X, 0), he/she is supposed to make the necessary translations throughout the rest of the article.
3.1. Morsifications. For functions on stratified spaces the most simple singularities are the stratified Morse critical points. They generalize the classical Morse critical points of a holomorphic function in the sense that every function f with an isolated singularity on (X, p) can be deformed to a function with finitely many stratified Morse critical points on X, cf. Corollary 3.17. Thus, they are the basic building blocks for the study of isolated singularities on stratified spaces.
Consider a point p ∈ U and the germ f : such that f = f 0 . It is clear that whenever p ∈ X, any unfolding of f induces an unfolding F |(X, p) of f |(X, p).
Definition 3.5. Let (X, p) ⊂ (C n , p) be a reduced complex analytic space and f : (C n , p) → (C, f (p)) a holomorphic function with isolated singularity on (X, p). An unfolding F of f induces a morsification of f |(X, p), if there exists an open neighborhoods V ⊂ C n of p and an open disc T ⊂ C around the origin such that f t |X has only Morse critical points in X ∩ V for all 0 = t ∈ T .
For the existence of morsifications and the density of Morse functions in the stratified setting see for example [GM88]. We will usually take f t (x) = f (x)−t·l(x) for a generic linear form l ∈ Hom(C n , C), cf. Corollary 3.17.
We may choose the open neighborhood V in Definition 3.5 to be an open Milnor ball B ε for f |(X, p). Then for t = η = 0 sufficiently small, all Morse critical points of f η on X ∩ B ε arise from the original singularity of f 0 at 0 ∈ X and we can count the number of Morse critical points of f η on each stratum S α in X ∩ B ε .
Definition 3.6. We define the numbers µ f (α; X, 0) of f |(X, p) to be the number of Morse critical points on the stratum S α in a morsification of f |(X, p).
These numbers clearly depend on the choice of the stratification. However, it follows from [STV05, Proposition 2.3], that they do not depend on the choice of the morsification F |(X, p) of f |(X, p). This fact will also be a consequence of Theorem 4.3.
Example 3.7. We continue with Example 1.1. As a morsification of f |(X, 0) we may choose Clearly, µ f (0; X, 0) = 1, because S 0 is a one-point stratum and any such point is a critical point of a function f in the stratified sense. On S 1 the function f η has exactly one Morse critical point for η = 0. This can be verified by classical methods: Note that X 1 = S 1 is smooth and the restriction of f to X 1 is an ordinary A 1 singularity. The given morsification is moving this critical point -depicted in purple in Figure 2 -from x = 0 to x = −t/2 so that for t = 0 it really lies in the stratum S 1 .
In order to compute µ f (2; X, 0) let be the global curve of critical points of f t on the regular part X reg = S 2 of the whole affine variety X ⊂ C 3 . Using a computer algebra system, one can verify that Γ has seven branches. Five of these branches pass through the origin 0 ∈ C 3 , i.e. they arise from the critical point of f on (X, 0). Note that Γ 4,5 (t) does not have real coordinates for t ∈ R \ {0}, so we will not be able to illustrate these branches in real pictures. Nevertheless, the behaviour of Γ 4,5 (t) is symmetric to what happens with the real branches Γ 2,3 (t). Each one of these branches corresponds to a Morse critical point of f t on S 2 ⊂ X and we drew them as green dots in the picture on the right of Figure 2. Thus we have The remaining two branches are swept out from the points p 6 and p 7 and do not contribute to the number µ f (2; X, 0) of f |(X, 0) at the origin. They correspond to the blue dots in Figure 2.
3.2. Homology decomposition for the Milnor fiber. The Milnor fiber M f |(X,0) of a holomorphic function f on a complex analytic space (X, 0) ⊂ (C n , 0) is by construction a topologically stable object: By virtue of Thom's Isotopy Lemma, small perturbations of the defining equation For the previous example this is illustrated in the first two pictures of Figure 3.
It is a straightforward exercise to transfer the classical theory of morsifications (see e.g. [AGZV85]) to this setting and use stratified Morse theory [GM88, Part II] to deduce the following homology decomposition for the Milnor fiber: Proposition 3.8. Let (X, 0) ⊂ (C n , 0) be a complex analytic space, S = {S α } α∈A a complex analytic Whitney stratification of X with connected strata, L(X, S α ) the complex link of X along the stratum S α , C(L(X, S α )) the real cone over it, f : (C n , 0) → (C, 0) a holomorphic function with an isolated singularity on (X, 0) in the stratified sense, and M f |(X,0) its Milnor fiber on X. Then the reduced homology of the Milnor fiber decomposes as where d(α) = dim(S α ) is the complex dimension of the stratum S α and µ f (α; X, 0) the number of Morse critical points on S α in a morsification of f . Proposition 3.8 shows that the characterizations 1') and 2') of the numbers µ f (α; X, 0) in Section 1 coincide. We include a brief proof.
Proof. Choose ε > 0 sufficiently small so that the squared distance function to the origin r 2 : C n → R ≥0 does not have any critical points in the ball B ε neither on X nor on X ∩ f −1 ({0}). After shrinking ε > 0 once more, if necessary, we may assume that the space In any unfolding F = (f t , t) of f we may therefore identify the pairs . This thimble is given by the product of the tangential and the normal morse datum of f η at the critical point p i over c i . See [GM88] for a definition of these. Altogether, we obtaiñ Remark 3.9. The existence of the homology decomposition (6) also follows from the more general bouquet decomposition of the Milnor fiber due to M. Tibȃr [Tib95]. His proof, however, does not use morsifications and it requires further work to show that the numbers which play the corresponding role of the µ f (α; X, 0) in his homology decomposition coincide with the number of Morse critical points in a morsification.
Example 3.10. We continue with Example 3.7. For t = 1 the critical point of the morsified function f 1 on S 1 is (−1/2, 0, 0) T . On S 2 ⊂ X they are The complex links L(X, S α ) of X along the different strata are the following. For S 0 = {0}, it is the complex link of the Whitney umbrella (X, 0) itself, which is known to be the nodal cubic. Hence (X, S 0 ) ∼ =h S 1 is homotopy equivalent to a circle.
Along S 1 the normal slice of X consists of two complex lines meeting transversally. The complex link is therefore a pair of points L(X, For the third stratum S 2 , the normal slice is a single point and the complex link is empty. We adapt the convention that the real cone over the empty set C(∅) = {pt} is the vertex pt of the cone.
The homology decomposition for the Milnor fiber thus reads for the top dimensional stratum 2 S γ . The Euler obstruction of a function is defined using the gradient vector field grad f . For the purposes of this note, it is more natural to consider the 1-form df and its canonical lift to the dualΩ 1 of the Nash bundle as we will describe below. This provides the notion of the Euler obstruction Eu df (X, 0) of the 1-form df on (X, 0), as was first defined by W. Ebeling and S.M.
Gusein-Zade in [EGZ05]. In this section, we will follow their example and also consider the slightly more general case of an arbitrary 1-form ω on (X, 0).
Throughout this section, we let U ⊂ C n be an open domain and X ⊂ U a reduced, complex analytic space. Suppose that X is equidimensional of dimension d. On the set of nonsingular points X reg we can consider the map taking any point p to the class of its tangent space T p X as a subspace of T p C n by means of the embedding of X.
Definition 3.11. The Nash modification of X is the complex analytic closure of the graphX The restriction of the tautological bundle on U × Grass(d, n) toX will be referred to as the Nash bundleT . The dual bundle will be denoted byΩ 1 .
For the dual of the Nash bundle there is a natural notion of pullback of 1-forms on X which is defined as follows. We can think of a point (p, V ) ∈X as a pair of a point p ∈ X and a limiting tangent space V from X reg at p. The space V can be considered both as a subspace of T p C n and as the fiber of the Nash bundleT at the point (p, V ). Let us denote by ·, · the canonical pairing between a vector space and its dual. For a 1-form ω on C n , a limiting tangent space V at p and a vector v ∈ V we define Here we consider v as a point in the fiber of the Nash bundle over the point (p, V ) ∈ X on the left hand side and as a vector in V ⊂ T p C n on the right hand side.
In order to define the Euler obstruction of a 1-form, we need to adapt Definitions 3.1 and 3.4 in this setup. Since for 1-forms there is no associated Milnor fibration, we may drop the assumption that the stratification of X satisfies Whitney's condition B.
Let ω be a holomorphic 1-form on U .
Definition 3.12. Suppose S = {S α } α∈A is a complex analytic stratification of X satisfying Whitney's condition A. We say that ω|(X, p) is nonzero at a point p ∈ X in the stratified sense if ω does not vanish on the tangent space T p S β of the stratum S β containing p.
We say that a 1-form ω on U has an isolated zero on (X, p), if there exists an open neighborhood U of p such that ω is nonzero on X in the stratified sense at every point x ∈ U ∩ X \ {p}.
If in the following we do not specify a stratification, we again choose S to be the canonical Whitney stratification for a reduced, equidimensional complex analytic space X.
It is an immediate consequence of the Whitney's condition A that at every point p ∈ X such that the restriction ω|S α of ω to the stratum S α containing p is nonzero, also the pullback ν * ω is non-zero at any point (p, V ) ∈ ν −1 ({p}) in the fiber of ν :X → X over p. In particular, ν * ω is a nowhere vanishing section on the preimage of a punctured neighborhood U of p whenever ω has an isolated zero on (X, p) in the stratified sense.
Definition 3.13 (cf. [EGZ05]). Let (X, p) ⊂ (C n , p) be an equidimensional, reduced, complex analytic space of dimension d and ω the germ of a 1-form on (C n , p) such that ω|(X, p) has an isolated zero in the stratified sense. The Euler obstruction Eu ω (X, p) of ω on (X, p) is defined as the obstruction to extending ν * ω as a nowhere vanishing section of the dual of the Nash bundle from the preimage ν −1 (∂B ε ∩ X) of the real link ∂B ε ∩ X of (X, p) to the interior of ν −1 (B ε ∩ X) of the Nash transform. More precisely, it is the value of the obstruction class of the section ν * ω on the fundamental class of the pair ν −1 (B ε ∩ X), ν −1 (∂B ε ∩ X) : As we shall see below, the Euler obstruction of a 1-form ω with isolated singularity on (X, p) counts the zeroes on X reg of a generic deformation ω η of ω. In the case ω = df for some function f with isolated singularity on (X, p), these zeroes correspond to Morse critical points of f η on X reg in an unfolding. We have seen before that these are not the only critical points of f η .
Definition 3.14. Suppose S = {S α } α∈A is a complex analytic stratification of X satisfying Whitney's condition A. A point p ∈ X is a simple zero of ω|X, if the following holds. Let S β be the stratum containing p and σ(ω|S β ) the section of the restriction ω|S β as a submanifold of the total space of the vector bundle Ω 1 S β . Denote the zero section by σ(0).
i) The intersection of σ(ω|S β ) and the zero section in the vector bundle Ω 1 S β on S β is transverse at p. ii) ω does not annihilate any limiting tangent space V from a higher dimensional stratum at p.
Whenever ω = df for some holomorphic function f , this reduces precisely to the definition of a stratified Morse critical point p of f |X, Definition 3.4.
Proposition 3.15. Any 1-form ω with an isolated zero on (X, p) admits an unfolding W = (ω t , t) as above on some open sets U × T such that for a sufficiently small ball B ε ⊂ U around p and an open subset 0 ∈ T ⊂ T one has i) X ∩ B ε retracts onto the point p, ii) ω = ω 0 on U and ω has an isolated zero on X ∩ U , iii) for every t ∈ T , t = 0, the 1-form ω t has only simple isolated zeroes on X ∩ B ε and is nonzero on X ∩ U at all boundary points x ∈ X ∩ ∂B ε . Moreover, ω t can be chosen to be of the form ω t = ω − t · dl for a linear form l ∈ Hom(C n , C).
Definition 3.16. We define the multiplicity µ ω (α; X, p) of ω|(X, p) to be the number of simple zeroes of ω t on S α for t = 0 in an unfolding as in Proposition 3.15.
Again, we clearly have µ f (α; X, p) = µ df (α; X, p) in the case where ω = df is the differential of a function f with isolated singularity on (X, p). As a straightforward consequence we obtain: Corollary 3.17. For a holomorphic function f : U → C with an isolated singularity in the stratified sense at (X, p) a morsification F = (f t , t) of f |(X, p) can be chosen to be of the form f t = f − t · l for a linear form l ∈ Hom(C n , C).
Proof. (of Proposition 3.15) We will show using Bertini-Sard-type methods that there exists a dense set Λ ⊂ Hom(C n , C) of admissable lines such that the linear form l in Proposition 3.15 can be chosen to be an arbitrary linear form with [l] ∈ Λ.
For a fixed α let X α = S α be the closure of the stratum S α , d(α) its dimension, and ν :X α → X α its Nash transform. Denote the fiber of ν over the point p ∈ X by E. Since the question is local in p, we may restrict our attention to arbitrary small open neighborhoods of E of the form ν −1 (U ) for some open set U p. Set N = (x, V, ϕ) ∈X α × Hom(C n , C) : ϕ|V = ν * ω(x, V ) and let π : N →X α and ρ : N → Hom(C n , C) be the two canonical projections. It is easy to see that N has the structure of a principle C n−d(α) -bundle overX α . In particular, the open subset S α = (ν • π) −1 (S α ) ⊂ N is a complex manifold of dimension n. Let Φ : N P(Hom(C n , C)) be the rational map sending a point (ϕ, x, V ) to the class [ϕ] ∈ P(Hom(C n , C)). Since ω had an isolated zero on (X, p), this map is regular on the dense open subset N \(π •ν) −1 ({p}) which in particular contains S α . In order to work with regular and proper maps, we may resolve the indeterminacy of Φ and obtain a commutative diagram Hom(C n , C) / / P(Hom(C n , C)).
Suppose L ∈ P(Hom(C n , C)) is a regular value ofΦ|Ŝ α , thenΦ −1 ({L}) ∩Ŝ α is a smooth complex analytic curve. If we let C ⊂ N be the image in N of its analytic closure inN , then evidently ρ|C : C → L is a finite, branched covering at 0 ∈ L. It follows a posteriori from the Curve Selection Lemma that ρ is a submersion at every point (x, V, ϕ) ∈ C ∩S α in a neighborhood of E. An inspection of the differential of ρ at such a point (x, V, ϕ) reveals that the transversality requirement i) in Definition 3.14 is satisfied for the 1-form ω − dϕ at x. Conversely, this means that for every nonzero linear form l ∈ L and every sufficiently small t = 0 the 1-form ω − t · dl has only isolated zeroes at those points x ∈ S α , for which (t · l, x, V ) ∈ C. Repeating this process for every stratum, we obtain a dense set Λ 1 ⊂ P(Hom(C n , C)) of preadmissable lines.
In order to verify also the requirement ii) in Definition 3.14, we proceed as follows. Let Y α = X α \ S α be the union of limiting strata of S α andS α , S α , and S α their preimages inX α , N , andN , respectively. The latter three spaces might have rather difficult geometry, but evidently dimŶ α < dimN = n and the map Y α → Y α is surjective.
There exists a dense subset Λ 2 ⊂ P(Hom(C n , C)) such that the restrictionΦ|Ŷ α has at most discrete fibers over Λ 2 . To see this, we may for example stratifyŶ α by finitely many locally closed complex submanifolds M i and choose Λ 2 as the set of all regular values ofΦ|M i . Since dim M i ≤ dimŶ α < n, the fiberQ = (Φ|Ŷ α ) −1 (L) of a point L ∈ Λ 2 is discrete and so is its image Q ⊂ N , becauseN → N is proper. This means that for a given l ∈ L there are only finitely many preimages (x, V, l) ∈ ρ −1 (L), i.e. the set of points x ∈ X, for which ω − dl annihilates a limiting tangent space V at x is finite in a neighborhood of p. We may choose U and B ε sufficiently small to avoid those points.
Proposition 3.18. For every 1-form ω on U with an isolated zero on (X, p) we have µ ω (α; X, p) = Eu ω (X α , p), where X α = S α is the closure of the stratum S α .

Proof. Choose a representative
of an unfolding of ω|(X, p) and a ball B ε ⊂ U as in Proposition 3.15. The Euler obstruction of ω at (X α , p) depends only on its obstruction class Being a homotopy invariant, this class does not change under small perturbations and it is therefore evident from the definitions that for every η ∈ T and every α ∈ A one has We may therefore select one η = 0 and use ω η instead of ω to compute the Euler obstruction. The evaluation of the obstruction class counts the number of zeroes of ω η .
Corollary 3.19. Whenever f : U → C is a holomorphic function with isolated singularity on (X, p), we have Example 3.20. We continue with Example 3.10. For α = 0 the real link of (S 0 , 0) is empty and the Euler obstruction is 1 by convention.
In the case α = 1 the closure X 1 = S 1 of the stratum S 1 is already a smooth line. Consequently, the Nash modification ν :X 1 → X 1 is an isomorphism and Ω 1 coincides with the usual sheaf of Kähler differentials. In this case, the Euler obstruction of df on (X 1 , 0) coincides with the degree of the map df | df | : ∂B ε ∩ X 1 → S 1 .
Since 0 ∈ X 1 is a classical Morse critical point, df has a simple, isolated zero on (X 1 , 0) and therefore In this particular case of a function on a complex line, the computation of the Euler obstruction reduces to Rouché's theorem. For α = 2 we really need to work with the Nash modification and the morsification F = (f t , t) of f |(X, 0). To this end, we identify Grass(2, 3) with its dual Grassmannian Grass(1, 3) ∼ = P 2 via In homogeneous coordinates (s 0 : s 1 : s 2 ) of P 2 the rational map Φ from (7) is given by the differential of h: The equations forX ⊂ P 2 × C 3 are rather complicated, but they simplify in the canonical charts of P 2 × C 3 . We will consider the chart s 0 = 0, leaving the computations in the other charts to the reader. The equations forX read In particular, we can use (z, s 1 ) as coordinates onX ∩ {s 0 = 0} ∼ = C 2 . The exceptional set E ⊂X, i.e. the set of points q ∈X, at which ν :X → X is not a local isomorphism, is the preimage of the x-axis in C 3 . In the above coordinates it is given by Let O(−1) be the (relative) tautological bundle on P 2 × C 3 . The dual bundle O(1) has a canonical set of global sections e 0 , e 1 , e 2 in correspondence with the homogeneous coordinates (s 0 : s 1 : s 2 ). With these choices the differential of We consider ν * df t as a section inΩ 1 , the dual of the Nash bundleT . Note thatT appears as part of the Euler sequence onX. The standard trivialization ofT in the chart s 0 = 0 is given by the sections and therefore the zero locus of ν * df t onX is given by the equations ν * df t (v 1 ) = ν * df t (v 2 ) = 0. Substituting all the above expressions we obtain It is easy to see that for t = 0 the exceptional set E = {z = 0} is contained in the zero locus of ν * df 0 . In particular, the zero locus is non-isolated and we can not use ν * df 0 to compute the Euler obstruction as in the proof of Proposition 3.18. For η = 0, however, the zero locus of ν * df η consists of only finitely many points. A primary decomposition reveals that there are seven branches in the local coordinates (z, s 1 ) ofX. They are precisely taken to the corresponding branches Γ i (t) from Example 3.7 by ν. Again, only the first five of them have limit points close to ν −1 ({0}) for t → 0, i.e. only the first five branches contribute to Eu df (X, 0) for sufficiently small ε η > 0. Therefore, Eu df (X, 0) = 5 = µ f (2; X, 0), as anticipated.
Remark 3.21. Definition 3.16 and Proposition 3.15 suggest yet another interpretation of the numbers µ ω (α; X, p), namely as microlocal intersection numbers. For a stratum S α of X and its closure X α one can define conormal cycle of X α as . This is a Whitney stratified subspace of the total space of the vector bundle Ω 1 U . The Whitney conditions imply that the fundamental class [Λ α ] ∈ H BM 2n (U ) is a well defined cycle in Borel-Moore homology. So is the class [σ(ω)] of the section σ(ω) of ω on U . In this context, Proposition 3.15 appears as a moving lemma, which puts the two cycles in a general position. Clearly, the number of intersection points of [Λ α ] and [σ(ω)] coincides with µ ω (α; X, p) = Eu ω (X α , p). See also [BMPS04,Corollary 5.4].

The Euler obstruction as a homological index
Throughout this section let again U ⊂ C n be an open domain and X ⊂ U a closed, equidimensional, reduced, complex analytic space.
For a holomorphic function f : U → C with an isolated singularity on X at a point p ∈ X, Proposition 3.18 and Corollary 3.19 suggest the following interpretation of the Euler obstruction: In a morsification F = (f t , t) of f |(X, p) the singularities of f |(X, p) become Morse critical points on the regular strata S α . In this sense, a morsification separates the singularities of the function f |(X, p) from the singularities of the space (X, p) itself. The Euler obstructions Eu df (X α , p) of df on the closures X α = S α of the strata know the outcome of this separation beforehand and even without a given concrete morsification. A particular, but remarkable consequence of these considerations is that Eu df (X α , p) = 0 for all α ∈ A whenever f does not have a singularity on (X, p) -independent of the singularities of the germ (X, p) itself.
Suppose for the moment that also the space (X, p) has itself only an isolated singularity so that the homological index Ind hom (df, X, p) as in [EGZS04] is defined. The comparison of Eu df (X, p) with Ind hom (df, X, p) is based on the fact that both the Euler obstruction and the homological index satisfy the law of conservation of number and that they coincide at Morse critical points. In an arbitrary unfolding F = (f t , t) of f |(X, p) we can therefore use both the Euler obstruction and the homological index to count the number of Morse critical points on X reg arising from f |(X, p). But for a fixed unfolding parameter t = η only the Euler obstruction Eu dfη (X, p) can be used to measure whether f η is still singular at (X, p) or whether all singularities of f have left from the point p for t = η = 0. If the latter is the case -as for example in a morsification -the homological index Ind hom (df η , X, p) is Ind hom (df η , X, p) = Ind hom (df, X, p) − Eu df (X, p) = −k (X, p).
The number k (X, p) is an invariant of the space (X, p), but unknown in general. Therefore, the homological index Ind hom (df, X, p) can not be used to count the number of Morse critical points on X reg in a morsification; it only seperates the singularities of the function f from the singularities of X up to an unknown quantity.
We return to the more general setting of an arbitrarily singular X ⊂ U . Suppose ω is a holomorphic 1-form on U and let p ∈ X be a point for which ω has an isolated zero on (X, p). Then Eu ω (X α , p) is counting the number of simple zeroes on S α close to p in a generic perturbation ω η of ω. It is evident from the construction that we may restrict our attention to the case where X = X α = S α is irreducible and reduced and we only need to consider isolated zeroes of ω η on X reg . Translating the previous discussion to this setting we see that -conversely -a homological index I(ω, X, p) has to coincide with the Euler obstruction Eu ω (X, p) whenever the following two conditions are met: 1. I(ω, X, p) coincides with Eu ω (X, p) whenever p ∈ X is a smooth point of X. 2. For every singular point p of X one has I(ω, X, p) = 0 whenever ω is a 1-form such that ω|(X, p) is nonzero or has at most a simple zero at p in the stratified sense.
It is therefore worthwhile to investigate once again the structural reasons as to why 1. is satisfied for Ind hom (ω, X, p) at smooth points and why Eu ω (X, p) = 0 whenever ω has at most a simple zero on X at a point p on a lower dimensional stratum. We will exploit these reasons for the construction of a homological index I(ω, X α , p) which satisfies 1. and 2. simultaneously.
The fact that the homological index of a 1-form ω with an isolated zero at a smooth point (X, p) ∼ = (C n , p) coincides with its Euler obstruction and its topological index is based on the following fact. In local coordinates x 1 , . . . , x n of (X, p), the complex (2) becomes a Koszul complex on the local ring O X,p in the components of ω = n i=1 ω i dx i . Since O X,p is Cohen-Macaulay and the zero locus of ω is isolated, the ω i must form a regular sequence on O X,p and the following lemma applies, cf. [BH93, Corollary 1.6.19].
the Koszul complex associated to v. We consider R = 0 M to be situated in degree zero, M = 1 M in degree one, etc.
i) Whenever (v 1 , . . . , v r ) is a regular sequence on R as an R-module, then (9) is exact except for the last step where we find ii) Whenever v / ∈ mM , the Koszul complex is exact.
Consequently, Ind hom (ω, X, p) = dim C O X,p / ω 1 , . . . , ω n which evaluates to 1 on simple zeroes of ω. Part ii) of this lemma explains why the homological index of ω is zero at all smooth points q ∈ X where ω does not vanish.
From this viewpoint, the difficulty in comaring the Euler obstruction of a 1-form ω at a singular point p of X with its homological index at p stems from the fact that the restriction ω|(X, p) is not anymore an element of a free module, but of the module of Kähler differentials Ω 1 X,p . The key idea is to address this issue by replacing Ω 1 X,p and ω with the Nash bundleΩ 1 and the section ν * ω. In order to work with finite O X -modules we need to consider the derived pushforward of the associated bundles. Analogous to Lemma 4.1 ii) we find the following.
Lemma 4.2. Let U ⊂ C n be an open domain, X ⊂ U an irreducible and reduced closed analytic subspace of dimension d, and ν :X → X its Nash modification. For any point p ∈ X the stalk at p of the complex of sheaves Rν * Ω • , ν * ω ∧ − p is exact, whenever ω does not annihilate any limiting tangent space V from X reg at p.
Proof. The statement that ω does not annihilate any limiting tangent space V of a top-dimensional stratum at p is equivalent to saying that ν * ω is nonzero at every point (p, V ) ∈X in the fiber ν −1 ({p}) of the Nash modification over p.
If ν * ω is nonzero then, according to Lemma 4.1 ii), the complex of sheaves is exact along ν −1 ({p}) and therefore quasi-isomorphic to the zero complex. Consequently, also the stalk at p of the derived pushforward of this complex has to vanish.  is a holomorphic function with an isolated singularity on (X, p). For α ∈ A let ν :X α → X α be the Nash modification of the closure X α = S α andΩ k α the k-th exterior power of the dual of the Nash bundle onX α . Then Proof. We may apply Theorem 4.3 to the space X α = S α and the restriction of the 1-form df to it.
Proof. (of Theorem 4. 3) The sheaves in the complex Rν * (Ω • , ν * ω ∧ −) are finite O n -modules since the morphism ν is proper. By assumption, ω has an isolated zero on (X, p) in the stratified sense and hence Lemma 4.2 implies that the cohomology of this complex is supported at the origin. In particular, its Euler characteristic is finite. Suppose W = (ω t , t) is an unfolding of ω|(X, p) as in Proposition 3.15 andpossibly after shrinking U -let W : U × T → C n × T be a suitable representative thereof. Denote by π : U × T → T the projection to the parameter t. The unfolding of ω induces a family of complexes of sheaves Ω • , ν * ω t ∧ − on the Nash transformX and hence also on the derived pushforward. This furnishes a complex of coherent sheaves on U × T which becomes a family of complexes over T via the projection π. Clearly, every sheaf R k ν * Ω r is π-flat. We may apply the main result of [GM98]: There exist neighborhoods p ∈ U ⊂ U and 0 ∈ T ⊂ T such that for every η ∈ T we have i.e. the Euler characteristic satisfies the law of conservation of number.
Suppose U , T and B ε have also been chosen as in Proposition 3.15 and fix η ∈ T , η = 0. By construction, ω η has only simple, isolated zeroes on the interior of X ∩ B ε and none on the boundary. Whenever x ∈ (X \ X reg ) ∩ B ε is such a point, at which ω η has a simple zero outside X reg , the restriction of ω η to any limiting tangent space V of X reg at x is nonzero and consequently Whenever x ∈ X reg ∩ B ε is a point with a simple zero of ω η at x we find the following. The Nash modification ν is a local isomorphism around x and therefore is the Koszul complex on the modules Ω k X,x . Lemma 4.1 allows us to compute the Euler characteristic (−1) d · χ Ω • X,x , ω η ∧ − = 1. The statement now follows from the principle of conservation of number.
Example 4.5. We continue with Example 3.20. As previously discussed, the only interesting stratum of X is S 2 = X reg . To prepare for the computations of µ(2; f, 0) we will describe a complex of graded S-modules representing (Ω • , ν * df ∧−). We set A = C[x, y, z], S = A[s 0 , s 1 , s 2 ] and consider S as a homogeneous coordinate ring of P 2 A over A. The ideal J ⊂ S of homogeneous equations for the Nash transform X is obtained from the equations for the total transform by saturation: Denote by L the ideal of 2 × 2-minors of the matrix Over X reg these equations describe the graph of the rational map Φ underlying the Nash blowup (7). Now where y, z is the ideal defining the singular locus of X on which Φ is not defined. Let Q p be the module representing p Q with Q the tautological quotient bundle on P 2 A . A graded, free resolution of the Q p is given by appropriate shifts of the Koszul complex in the s-variables. Let θ = s 0 · e 0 + s 1 · e 1 + s 2 · e 2 ∈ H 0 (P 2 A , O(1) 3 ) ∼ = S 3 1 be the tautological section. Together with ν * df = −2(x − z) · e 0 + 2y · e 1 + 2(x − z) · e 2 ∈ H 0 (P 2 A , O 3 ) ∼ = S 3 0 we obtain the following double complex.
For every q the module M q representing the restriction qΩ 1 of Q q toX is given by Q q ⊗ S/J. The complex of sheaves (Ω • , ν * df ∧ −) onX is thus represented by the complex of graded modules As we shall see in the next section, Proposition 5.2, we can compute the derived pushforward Rν * (Ω • , ν * df ∧ −) via a truncatedČech-double-complex on the complex of modules (M • , ν * df ∧ −).

5.
How to compute µ f (α; X, 0) for S α a hypersurface The following section will be phrased in purely algebraic terms. This is due to the fact that the complex numbers are not a computable field and also the ring of convergent power series is usually not available in computer algebra systems for symbolic computations. If we were working in the projective setting, Chow's theorem [Cho49] and the GAGA-principles due to Serre [Ser56] allow us to restrict to the algebraic case. In the local context we can not do so. For these reasons, we will assume that both (X, 0) ⊂ (C n , 0) and either f or ω as in Theorem 4.3 or Corollary 4.4 are algebraic and defined over some finite extension field K of Q.
Thus we will -with a view towards Theorem 4.3 -work with proper maps π : X → Y of algebraic spaces. Let F be a coherent algebraic sheaf on X and F h its analytification. It is well known that the sheaves R p π * (F) are O Y -coherent. Grauert's theorem on direct images [Gra60] assures that also the direct images R p π * (F h ) are O h Y -coherent and using Cech cohomology we obtain a natural morhism of cohomology sheaves ε : R p π * (F) → R p π * (F h ).
for every p.
We will see below that whenever π is the restriction of a projection π : P r × (C n , 0) → (C n , 0), as we may assume for the purpose of this article by virtue of the Plücker embedding, one can express the direct images of a coherent algebraic sheaf F in terms of the cohomology of the relative twisting sheaves O(−w) and vice versa for their analytifications. Now the formal completions of the rings C{x 1 , . . . , x n } and C[x 1 , . . . , x n ] x1,...,xn are isomorphic and so are the formal completions of R p π * (O(−w)) and R p π * (O h (−w)) for all p and w. In what follows, the sheaf F -or, more generally, the complex of sheaves F • -will always have Rπ * (F) and Rπ * (F h ) with isolated support at the origin. Thus, their Euler characteristics both have to coincide with the Euler characteristic of their isomorphic formal completions. In particular, the comparison morphism ε above is an isomorphism in this case and we may therefore carry out all computations in the algebraic setting.
Let A be a commutative Noetherian ring. We set S = A[s 0 , . . . , s r ] and consider S as a graded A-algebra. On the geometric side let π : P r A → Spec A be the associated projection. Let O =S be the structure sheaf of P r A and O(−w) the relative twisting sheaves for w ∈ Z. Given any finitely generated graded Smodule M there is a corresponding sheafM of O-modules on P r A . We will first describe how to compute Rπ * (M ) as a complex of finitely generated A-modules up to quasi-isomorphism and then generalize these results for complexes of finite, graded S-modules (M • , D • ) and their associated complexes of sheaves on P r A . We may useČech cohomology with respect to the canonical open covering of P r A . For a graded S-module M leť These modules are not finitely generated over S, but they have a natural structure as a direct limit of finite S-modules given by the submoduleš TheČech-complex of twisted sections inM is obtained from theČ p (M ) together with the differentialď :Č p (M ) →Č p+1 (M ) taking an element a i1,...,ip (s i0 · · · s ip ) d , a i1,...,ip ∈ M to the element inČ p+1 (M ) with component (j 0 , . . . , j p+1 ) given by 1 (s j0 · · · s jp+1 ) d p+1 k=0 (−1) k s d j k a j0,...,ĵ k ,...,jp+1 .
As usual,· indicates that the index is to be omitted. We will writě for the p-th cohomology of theČech complex on a module M and its truncations. The modules S(−w) and the corresponding twisting sheaves O(−w) have a well known cohomology, see [Har77,Chapter III.5]. We deliberately identify The last term has a structure as a direct limit of S-modules via the maps The pairing of monomials provides us with an identification for all w ∈ Z. Note that this pairing is compatible with the natural S-module structure on both sides.
Proposition 5.1. Let M be a graded S-module and Proof. The statements follow from a diagram chase in the double complex (15). Note that in (15) all columns but the last one are exact by construction. The same holds for all rows but the first one. Since taking cohomology commutes with direct sums, the complex is identical with the last column of (15), while the first row is theČech-complex on M .
We can use Proposition 5.1 to describe Rπ * (M ) as a complex of finite A-modules. Choose any d ≥ max{w −k,i −k : 0 ≥ −k ≥ −r − 1} − r and let be the inclusions of finite S-modules as before. The restriction on the choice of d assures that the degree zero part of every E(−w −k,i −k ) is fully contained in the image of Ψ −k . Consequently, the homomorphism of complexes in degree zero is an isomorphism of complexes of finite A-modules.
In other words, there is a short exact sequence of free finite A-modules In terms ofČech-cohomology this implies the following. We may replace every Cech complexČ • (K −p ) in (15) by its truncationČ • ≤d (K −p ) and restrict to the degree zero strands in each term. Another diagram chase reveals a quasi-isomorphism Proof. The right derived pushforward of a single sheafM on P r A is usually defined via injective resolutions ofM and it is well known that the resulting complex is quasi-isomorphic to the Cech-complex onM for the affine covering above. For a complex of sheavesM • the derived pushforward can be computed as the total complex of a double complex I •,• of injective sheaves which forms an injective resolution ofM • . There is a corresponding spectral sequence identifying this total complex with the total complex of the Cech-double complex forM • up to quasi isomorphism analogous to the case of a single sheaf. The result now follows from (16): On the first page of the spectral sequence of the Cech-double complexČ • (M • ) we may replace each term H p (Č • (M q )) by the truncation H p (Č • ≤d (M • )). We conclude with a brief description of how to use Proposition 5.2 in order to compute (11). Let (X, 0) ⊂ (C n , 0) be a reduced algebraic hypersurface defined over some finite extension K of Q and ω an algebraic 1-form as in Theorem 4.3 defined over the same field.
Set A = K[x 1 , . . . , x n ] x1,...,xn and let S = A[s 1 , . . . , s r ] be the homogeneous ring in the s-variables. Let J be the homogeneous ideal of S defining the Nash transformX ⊂ P n−1 × (C n , 0) and M q the graded modules presenting the duals of the exterior powers of the Nash bundleΩ q onX together with the morphisms given by the pullbacks ν * ω as in Example 4.5. 1) We can compute a partial graded free resolution of every one of the M q using Gröbner bases and a mixed ordering whose first block is graded and global in the s-variables and whose second block is local in the x-variables. 2) From this we obtain the bound d on the pole order for the Cech-double complex and we can build the truncated Cech-double complexČ • ≤d (M • ) as a double complex of finite S-modules.
3) The degree-0-strands ofČ • ≤d (M • ) are finite A-modules generated by monomials in the s-variables. We can choose generators and relations accordingly and extract the induced matrices for ν * ω ∧ − over A from the maps defined over S. 4) Since ω had an isolated zero, the cohomology of the resulting complex must be finite over K. We can proceed by the usual Groebner basis methods for the computation of Euler characteristics. These computations apply in particular to the case X = S α and ω = df as in Corollary 4.4.
Remark 5.3. Note that in Proposition 5.2 we do not need to compute a graded free resolution of the whole complex M • by means of a double complex of free, graded S-modules, but only resolutions of the individual terms M q . With a view towards the application of Proposition 5.2 for the computation of (11) this entails that the number d can be chosen once and for all for a given space (X, 0) and then used for every 1-form ω with isolated zero on (X, 0).