A generalization of Milnor’s formula

The Milnor number μf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _f$$\end{document} of a holomorphic function f:(Cn,0)→(C,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f :({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$\end{document} with an isolated singularity has several different characterizations as, for example: 1) the number of critical points in a morsification of f, 2) the middle Betti number of its Milnor fiber Mf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_f$$\end{document}, 3) the degree of the differential df\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {d}}f$$\end{document} at the origin, and 4) the length of an analytic algebra due to Milnor’s formula μf=dimCOn/Jac(f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _f = \dim _{\mathbb {C}}{\mathcal {O}}_n/{\text {Jac}}(f)$$\end{document}. Let (X,0)⊂(Cn,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X,0) \subset ({\mathbb {C}}^n,0)$$\end{document} be an arbitrarily singular reduced analytic space, endowed with its canonical Whitney stratification and let f:(Cn,0)→(C,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f :({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$\end{document} be a holomorphic function whose restriction f|(X, 0) has an isolated singularity in the stratified sense. For each stratum Sα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {S}}_\alpha $$\end{document} let μf(α;X,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _f(\alpha ;X,0)$$\end{document} be the number of critical points on Sα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {S}}_\alpha $$\end{document} in a morsification of f|(X, 0). We show that the numbers μf(α;X,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _f(\alpha ;X,0)$$\end{document} generalize the classical Milnor number in all of the four characterizations above. To this end, we describe a homology decomposition of the Milnor fiber Mf|(X,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{f|(X,0)}$$\end{document} in terms of the μf(α;X,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _f(\alpha ;X,0)$$\end{document} and introduce a new homological index which computes these numbers directly as a holomorphic Euler characteristic. We furthermore give an algorithm for this computation when the closure of the stratum is a hypersurface.


Summary of results
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. [29,Chapter 7] and [1,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 for some choice of a Hermitian metric on (C n , 0). 4) It is the length of the Milnor algebra where Jac( f ) = ∂ f ∂ x 1 , . . . , ∂ f ∂ x n 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 [23], or [14]. 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 5-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 Sect. 3.1. 2') the number of direct summands for α in the homology decomposition of the Milnor fiber M f |(X ,0) , see Proposition 1. 3') the Euler obstruction Eu d f (X α , 0) of the 1-form d f on (X α , 0), see Definition 8 and Corollary 3. 4') the derived homological index i.e. the Euler characteristic of a finite complex of coherent O X -modules, cf. Theorem 1 and Corollary 4.
A similar discussion for the generalizations of 1)-3) has been carried out by Seade et al. in [30]. The study of homology decomposition for the Milnor fiber has been initiated by Siersma [31] in the case that not only f |(X , 0) but also the space (X , 0) itself has isolated singularity. See also the closely related bouquet decomposition of the Milnor fiber [33] due to Tibȃr and the article [28] by Massey for a decomposition similar to Proposition 1 in terms of hypercohomology of constructible sheaves. The definition of the Euler obstruction of a 1-form goes back to MacPherson [27], but can also be adapted from the analogous notion for vector fields as described in [2] by Brasselet et al. We will use the definition by Ebeling and Gusein-Zade from [8].
Contrary to those previous topological considerations for the generalizations of 1)-3), we will describe the Euler obstructions Eu d f (X α , 0) as an analytic invariant in Theorem 1. This allows us to also generalize the characterization 4) of the Milnor number to 4'). For the particular case when S α is a hypersurface, an algorithm for the computation of the numbers μ f (α; X , 0) as a homological index is given in Sect. 5.

Example 1
The following example illustrates the setup considered in this article. Let X ⊂ C 3 be the Whitney umbrella given by the equation h = y 2 − xz 2 = 0 and endowed with the stratification S 0 = {0}, This stratification is known to satisfy the Whitney conditions A and B (Fig. 1).
As a function f : C 3 → C with isolated singularity on (X , 0) we consider  In particular, there are no critical points of the restriction f |S 1 = −x 2 of f to the first stratum S 1 ∼ = C * since 0 / ∈ S 1 . 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 2, 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.

Background and motivation
Suppose the function f , the space (X , 0), and its stratification have been chosen as in 1') to 4') from Sect. 1. An application of [30,Proposition 2.3] by Seade et al. shows that where Eu f S α , 0 is the Euler obstruction of the function f on (S α , 0) from [2]. It is defined as follows.
Let ν :X → X be the Nash transformation 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 . The same procedure can then be applied to the closure of a stratum (S α , 0) in place of (X , 0).
Based on this notion, Seade et al. discuss a generalization of 1)-3) in [30] which is similar to the one presented here. To understand how our approach came about to also include 4), i.e. Milnor's formula, we have to consider the article [30] in the context of a series of papers 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 [9].
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 [13, 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 G SV (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 [12], Gómez-Mont introduces the homological index of a vector field v on (X , 0) as above in order to obtain an algebraic formula 1 for the GSV index. The homological index 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, Gómez-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). He proves in [12,Theorem 3.2] that with k(X , X ) a constant depending only on (X , 0) and the chosen smoothing, but independent of the vector field v. This is an important intermediate step for the computation of the GSV index from the homological index, but it remains to determine the constant k(X , X ).
For an isolated hypersurface singularity (X , 0) = (g −1 ({0}), 0) and its canonical smoothing X = B ε ∩ g −1 ({δ}), ε δ > 0 this is done in [12,Section 3.2]: in this case k(X , X ) = 0 so that the homological index and the GSV index really coincide. In the general case, however, the constant k(X , X ) is a non-trivial invariant of the singularity and its smoothing.
From our point of view, the main novelty in the approach by Gómez-Mont was the introduction of derived geometry in this setting and its comparison with topological invariants in Eq. (2). To prove it, he uses the fact that both indices obey the law of conservation of number which states, roughly speaking, that the total number of indices is preserved under small perturbations of the vector field. For the GSV index this is immediate from the definition since it only depends on the homotopy class of the non-zero section v|K on the boundary. For the homological index Ind hom (v, X , 0) this follows from a result in [11] which states that, more generally, the law of conservation of number holds for the holomorphic Euler characteristic of a complex of coherent sheaves with finite dimensional cohomology under arbitrary small perturbations of the (co-)boundary maps. The complex (Ω • X ,0 , v) in (1) with perturbations of v is a particular instance of this situation and, hence, for suitable representatives andṽ sufficiently close to v one has To conclude the proof of Eq. (2), observe that at smooth points p ∈ X reg the GSV index and the homological index coincide so that any deformation of v to someṽ necessarily changes the two indices at the origin by the same amount. Consequently, the difference of these two indices is a locally constant function in v. The general claim then follows from the fact that for the choice of a sufficiently small ball B ε around the origin, the set of those vector fields on B ε ∩ X with an isolated zero on (X , 0) form an open and connected set in the Banach space of continuous vector fields on B ε ∩ X which are holomorphic in the interior, cf. [12,Theorem 3.2].
In this article, we will not be dealing with vector fields, but with holomorphic 1forms. 2 This 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 a vector field. 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 transformation. 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 [10], Ebeling et al. define with differential given by the exterior multiplication with ω. Note that in case (X , 0) ∼ = (C n , 0) is smooth and ω = d f 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 (3) is the Koszul complex in the partial derivatives ∂ f ∂ x i 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 difference is also a constant, independent of the 1-form ω. Again, both invariants satisfy 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 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 so that the difference of these two indices at the origin is a locally constant function of ω. A similar reasoning as for the homological index for vector fields above can now be made to conclude that k (X , 0) is well defined invariant of (X , 0). The formula (4) can only be used to compute Eu ω (X , 0) from the homological index Ind ind (ω, X , 0) up to the constant k (X , 0) which is a sort of "residual homological index". Since this invariant of (X , 0) is unknown in general, we propose a modification of the homological index in Sect. 4 which directly computes the Euler obstruction, i.e. for which the residual homological index always vanishes by construction. This new index will be based on the Nash transformation ν :X → X of (X , 0), the complex of sheaves Ω • , ν * ω ∧ − onX , and the derived pushforward of this complex along the proper map ν. It will therefore be called the derived homological index and it is in general different from the homological index defined in [10] and also harder to compute. However, the derived homological index has the further advantage that it is defined for arbitrarily singular spaces (X , 0).

Generalizations of the Milnor number
We briefly recall the necessary definitions of singularity theory on stratified spaces, cf. [24]. Let U ⊂ C n be an open domain, X ⊂ U a closed, reduced complex analytic set, and f : U → C a holomorphic function.
The existence of complex analytic Whitney stratifications was shown by Hironaka [21]. In [26, Corollaire 6.1.8] Lê and Teissier constructed a canonical Whitney stratification for reduced, equidimensional complex analytic spaces, and in [32] 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 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, cf. [24,Paragraphe (3.3) and Lemme 3.5]. Let B ε be the ball of radius ε around p in C n . There exists ε 0 > 0 such that for every ε 0 ≥ ε > 0 the intersections ∂ B ε ∩ X is transversal in the stratified sense, see e.g. [14, Part I, Section 1.4], [6], or [5]. Since f is a stratified submersion on (X , 0) away from the origin, the central fiber We can apply the same arguments so that also is a stratified transverse intersection for all ε 0 ≥ ε > 0. Fix one such ε = ε 0 > 0. Applying Thom's isotopy lemma, we find that 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).

Morsifications
For functions on stratified spaces the simplest 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 2. 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 : (C n , p) → (C, f ( p)) of f at p. An unfolding of f at p is a map germ such that f = f 0 . It is clear that whenever p ∈ X , any unfolding of f induces an unfolding F|(X , p) × (C, 0) of f |(X , p).  For the existence of morsifications and the density of Morse functions in the stratified setting see for example [14]. We will usually take f t (x) = f (x) − t · l(x) for a generic linear form l ∈ Hom(C n , C), cf. [25] and Corollary 2.
We may choose the open neighborhood V in Definition 4 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 5
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 [30, 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 1.

Example 2
We continue with Example 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 in the stratified sense.
On S 1 the function f η has exactly one Morse critical point for η = 0 (not pictured in Fig. 2). 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 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 the critical points of f t on (X , 0) sweep out seven branches in Γ under the variation of t. 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 that these branches do not appear in the real picture in Fig. 2. However, their behaviour 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 dots in Fig. 2 for t = 1. 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

Homology decomposition for the Milnor fiber
The  The classical theory of morsifications (see e.g. [1]) can be generalized to this setting using stratified Morse theory [14,Part II]. This leads to the following homology decomposition for the Milnor fiber: 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 1 shows that the characterizations 1') and 2') of the numbers μ f (α; X , 0) in Sect. 1 coincide. We will give a proof below, but first discuss some relations of Proposition 1 with various other results. For f and (X , 0) as in Proposition 1 the vanishing Euler characteristic of f |(X , 0) is defined to be the reduced topological Euler characteristic of the Milnor fiber χ M f |(X ,0) . Now the homology decomposition provides a formula for this invariant as a linear combination of the numbers μ f (α; X , 0):

Corollary 1 In the same setup as in Proposition 1 the vanishing Euler characteristic
Note that the coefficients (−1) d(α)−1 · (1 − χ(L(X , S α ))) depend only on the germ (X , 0), but not on the function f . In the case where (X , 0) is equidimensional of dimension d with an isolated singularity at the origin, the right hand side has two summands: The first one corresponds to the zero-dimensional stratum S 0 = {0} ⊂ X and the other one to the stratum S 1 = X reg of smooth points of X . Since S 0 is only a point, the number μ f (0; X , 0) is always equal to one and L(X , {0}) is the classical complex link of (X , 0). We wrote μ f (X , 0) for the only other number of Morse critical points, omitting the index α of the stratum.
We will see later on in Corollary 3 that μ f (X , 0) equals Eu d f (X , 0) and then Corollary 1 recovers a formula from [8,Proposition 4]: See also the discussion of the relation with the Lê-Greuel formula in Example 4 below for the case that (X , 0) is an isolated complete intersection singularity.

Remark 2
We may also relate Corollary 1 to some results from [8] around the radial index for continuous 1-forms ω with isolated singularity on (X , 0) as defined by Ebeling and Gusein-Zade in [ Substituting this into Corollary 1 we recover the formula from [8,Theorem 4] Ind rad (d f , in the special case where the 1-form ω is of the form d f for some function f .

Proof of Proposition 1
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 for sufficiently small ε δ η > 0. After modifying f η a little more we may assume that all critical values is a C 0 -fiber bundle away from the points c i and the space 0) . 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 [14] for a definition of these. Altogether, we obtaiñ This finishes the proof of Proposition 1.

Remark 3
The existence of a homology decomposition as in (6) also follows from the bouquet decomposition of the Milnor fiber due to Tibȃr [33] which is stronger in the sense that it holds on a homotopy level. But since its proof is not built on morsifications, it is a priori not clear that in general the numbers which play the corresponding role of the μ f (α; X , 0) in the resulting homology decomposition coincide with the number of Morse critical points in a morsification. The interplay of Proposition 1 with Tibȃr's bouquet decomposition will be studied in a forthcoming note.

Example 3
We continue with Example 2. 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 ) ∼ =ht 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 (Fig. 4).
where we write Z[e] for a shift of Z by e in the homological degree. In combination with the bouquet decomposition theorem from [33], we may even infer that M f |(X ,0) is homotopy equivalent to a bouquet of seven circles.
At this point it is apt to compare our results to the well known Lê-Greuel formulas for isolated complete intersection singularities [16,22]. As we will see, the numerical invariants considered there are in general different from our numbers μ f (α; X , 0).
Then both (X , 0) and (X ∩ f −1 ({0}), 0) are isolated complete intersection singularities with Milnor fibers M h|(C n ,0) and M f |(X ,0) , respectively. It is well known that these Milnor fibers are homotopy equivalent to a bouquet of spheres of real dimension d equal to their complex dimensions, cf. [19], and we obtain the numbers i.e. the classical Milnor numbers of the ICIS.
If we were to compute these numbers using the Lê-Greuel formulas, [22, Theorem 3.7.1] and [16, Korollar 5.5], we would proceed as follows. For suitable representatives we consider the restriction of the function f to the Milnor fiber M h|(C 3 ,0) , i.e. the canonical smoothing of (X , 0). We may choose a small, generic perturbationf of f which has only Morse critical points on M h|(C 3 ,0) and deliberately identify M f |(X ,0) with the subspace for some 1 |δ| > 0. A part of the long exact sequence of the pair and it is easy to see using Morse theory that the term in the middle is a free Z-module of rank r equal to the number of Morse critical points off on M h|(C 3 ,0) . Now b h|(C 3 ,0) = 1 can be deduced directly from Milnor's classical formula. According to [16,Korollar 5.5], the number r can be computed as the length of an algebra: By the last term we mean the 2 × 2-minors of the Jacobian matrix of h and f . Then the exact sequence above yields b f |(X ,0) = 5.
Let us now compute the homology of M f |(X ,0) along the lines of Proposition 1. The difference is that we do not consider a morsification of f on the smoothing of (X , 0), but on the singular space itself. We choose this morsification to be and one finds that f t has Morse critical points on the two-dimensional stratum S 2 := X reg at only four points Thus, μ f (2; X , 0) = 4 in this example and there is a free direct summand of rank 4 in the homology decomposition (6) of M f | (X ,0) .
The part of the decomposition of H 1 (M f |(X ,0) ) that is yet missing stems from the critical point of f t on the zero dimensional stratum S 0 = {0} of X at the origin. It is now easy to see that the complex link L(X , {0}) is nothing but the Milnor fiber of the restriction of the function h to the hyperplane {x = 0} and therefore homotopy equivalent to a single sphere of dimension 1.
Altogether, this again yields H 2 (M f |(X ,0) ) ∼ = Z ⊕ Z 4 to be free of rank 5, but with an additional decomposition of this homology group as a direct sum. Note, however, that this decomposition depends on the particular choice of the space (X , 0) and the function f or-equivalent to that-it depends on the particular regular sequence (h, f ). It is not an invariant of the ICIS (X ∩ f −1 ({0}), 0) ⊂ (C 3 , 0) since the latter is defined by the ideal h, f which could also be given by any other set of generators.
Also note that in this example, the homological index as it was defined by W. Ebeling, S. Gusein-Zade, and J. Seade is in fact Ind hom (d f , X , 0)) = r = 6 due to [10, Theorem 3.2 (iii)]. That means, similar to the GSV index [12], it measures the number of critical points of a perturbationf of the function f on the smoothing of (X , 0) and might therefore be better suited for the study of functions on isolated hypersurface and complete intersection singularities.

The Euler obstruction of a 1-form
In [ for the top dimensional stratum 3 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 d f 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 d f (X , 0) of the 1-form d f on (X , 0), as was first defined by Ebeling and Gusein-Zade in [8]. 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 6
The Nash transformation 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 1 and 3 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.

Definition 7
Let ω be a holomorphic 1-form on U ⊂ C n and suppose S = {S α } α∈A is a complex analytic stratification of X ⊂ U 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 8 (cf. [8]) 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 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 = d f 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 9
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 = d f for some holomorphic function f , this reduces precisely to the definition of a stratified Morse critical point p of f |X , Definition 3.
Analogous to morsifications we define an unfolding of a 1-form ω. Since Ω 1 U is trivial, we can consider ω as a holomorphic map U → C n . An unfolding of ω is then given by a holomorphic map germ Proposition 2 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 10
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 2.
Again, we clearly have μ f (α; X , p) = μ d f (α; X , p) in the case where ω = d f is the differential of a function f with isolated singularity on (X , p). As a straightforward consequence we obtain:

Corollary 2
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 for a linear form l ∈ Hom(C n , C).
A statement similar to Corollary 2 has been proven by Lê in [25]. In order to be self-contained, we include a proof of our version for morsifications of 1-forms.

Proof of Proposition 2
We will show using Bertini-Sard-type methods that there exists a dense set Λ ⊂ P(Hom(C n , C)) of admissible lines such that the linear form l in Proposition 2 can be chosen to be an arbitrary non-zero 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 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 (x, V , ϕ) 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 9 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 (x, V , t · l) ∈ C. Repeating this process for every stratum, we obtain a dense set Λ 1 ⊂ P(Hom(C n , C)) of pre-admissible lines.
In order to verify also the requirement ii) in Definition 9, we proceed as follows. Let Y α = X α \S α be the union of limiting strata of S α andỸ α , Y α , andŶ α their preimages inX α , N , andN , respectively. These three spaces might have rather difficult geometry, but evidently dimŶ α < dimN = n and the mapŶ 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.
To conclude the proof set Λ = Λ 1 ∩ Λ 2 , choose a linear form 0 = l ∈ L for some L ∈ Λ and adjust the choices of U and B ε accordingly. Since ∂ B ε ∩ X is compact there will be no zeroes of ω t = ω − t · dl on the boundary for small variations of t and T can be chosen so that this holds for all t ∈ T .
We are now prepared to show equivalence of 1') and 3'), in parallel to [30,Proposition 2.3].

Proposition 3 For every 1-form ω on U with an isolated zero on (X , p) we have
of an unfolding of ω|(X , p) and a ball B ε ⊂ U as in Proposition 2. 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 . Observe that by construction, ν * ω is nonzero at any point (x, V ) ∈X α \ν −1 (S α ), because ω η does not annihilate any limiting tangent space V at x. Thus, the zeroes of ν * ω η are located in ν −1 (S α ). At every such zero (x, V ) ∈ ν −1 (S α ) of ω η the intersection of σ (ω η |S α ) and the zero section in Ω 1 S α is transverse with positive orientation and therefore contributes an increment of 1 to the Euler obstruction. Consequently, Eu ω (X α , p) coincides with μ ω (α; X , p).
Corollary 3 Whenever f : U → C is a holomorphic function with isolated singularity on (X , p), we have Example 5 We continue with Example 3. 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 transformation ν :X 1 → X 1 is an isomorphism andΩ 1 coincides with the usual sheaf of Kähler differentials. In this case, the Euler obstruction of d f on (X 1 , 0) coincides with the degree of the map Since 0 ∈ X 1 is a classical Morse critical point, d f 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 transformation 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 f t = y 2 − (x − z) 2 − t(x + 2z) pulls back to We consider ν * d f 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 * d f t onX is given by the equations 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 * d f 0 . In particular, the zero locus is non-isolated and we can not use ν * d f 0 to compute the Euler obstruction as in the proof of Proposition 3. For η = 0, however, the zero locus of ν * d f 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 2 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 d f (X , 0) for sufficiently small ε η > 0. Therefore, as anticipated.
Remark 4 Definition 10 and Proposition 2 suggest yet another interpretation of the numbers μ ω (α; X , p), namely as microlocal intersection numbers-a point of view which has also been used in [28]. For a stratum S α of X and its closure X α one can define the conormal cycle of S α as This is an analytic subvariety of the total space of the vector bundle Ω 1 U . So is the section σ (ω) of ω on U . In this context, Proposition 2 appears as a moving lemma, which puts the two varieties in a general position. Clearly, the local intersection multiplicity (Λ α • σ (ω)) of the two varieties at ( p, 0) ∈ Ω 1 U coincides with μ ω (α; X , p) = Eu ω (X α , p). See also [2,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 and Corollary 3 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 d f (X α , p) of d f 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 d f (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 (d f , X , p) as in [10] is defined. The comparison of Eu d f (X , p) with Ind hom (d f , 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 d f η (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 (d f η , X , p) is The number k (X , p) is an invariant of the space (X , p), but unknown in general. Therefore, the homological index Ind hom (d f , X , p) can not be used to count the number of Morse critical points on X reg in a morsification; it only separates 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: ( †) I (ω, X , p) coincides with Eu ω (X , p) at any smooth point p of X . ( ‡) 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 ( †) 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 the derived homological index in Theorem 1 which then satisfies ( †) and ( ‡) 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 observation. In local coordinates x 1 , . . . , x n of (X , p), the complex (3) becomes a Koszul complex on the local ring O X , p in the components of 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 at a smooth point p of X and this 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 where ω does not vanish.
From this viewpoint, the difficulty in comparing 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 928 M. Zach 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.
The following Lemma establishes the requirement ( ‡) for the derived homological index in Theorem 1 as motivated from Lemma 1 ii).

Lemma 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 transformation. For any point p ∈ X the stalk at p of the complex of sheaves 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 topdimensional 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 transformation over p. If ν * ω is nonzero then, according to Lemma 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.

Theorem 1 Suppose U ⊂ C n is an open domain, X ⊂ U a reduced, equidimensional complex analytic subspace of dimension d, endowed with a complex analytic stratification satisfying Whitney's condition A. Let ω be a holomorphic 1-form on U with an isolated zero on X in the stratified sense at a point p. Then
Eu where ν :X → X is the Nash transformation and (Ω • , ν * ω ∧ −) is the complex of coherent sheaves onX given by the exterior powers of the Nash bundle and multiplication with ν * ω. is a holomorphic function with an isolated singularity on (X , p). For α ∈ A let ν :X α → X α be the Nash transformation 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 1 to the space X α = S α and the restriction of the 1-form d f to it.

Proof of Theorem 1
The sheaves in the complex Rν * (Ω • , ν * ω ∧ −) are finite O nmodules since the morphism ν is proper. By assumption, ω has an isolated zero on (X , p) in the stratified sense and hence Lemma 2 implies that the cohomology of this complex is supported at the origin. In particular, its Euler characteristic is finite. t) is an unfolding of ω|(X , p) as in Proposition 2 and-possibly after shrinking U -let 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 [12]: 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 2 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 Rν * Ω , ν * ω ∧ − x ∼ =qis 0 according to Lemma 2. Whenever x ∈ X reg ∩ B ε is a point with a simple zero of ω η at x we find the following. The Nash transformation ν is a local isomorphism around x and therefore is the Koszul complex on the modules Ω k X ,x . Lemma 1 allows us to compute the Euler characteristic The statement now follows from the principle of conservation of number.

Explicit computations for a function on a singular hypersurface
The following section will be phrased in purely algebraic terms. This is due to the fact that the complex numbers do not form a computable field and also the ring of convergent power series is often unavailable in computer algebra systems for symbolic computations. For these reasons, we will assume that both (X , 0) ⊂ (C n , 0) and either f or ω as in Theorem 1 or Corollary 4 are algebraic and defined over some finite extension field K of Q. More generally, we will work with proper maps of algebraic spaces and coherent algebraic sheaves F onX which are given in terms of some finitely presented, graded module M over the ring In our applications in Sect. 5.2 the map π will be the projection of the Nash transformation and F should be thought of as one of the exterior powers of the dual of the Nash bundle. It is well known that for any coherent algebraic sheaf F onX the sheaves R p π * (F) are O X -coherent. Let O h X , O hX , and F h be the respective analytifications. Grauert's theorem on direct images [15] assures that also the direct images R p π * (F h ) are O h Xcoherent and usingČech cohomology we obtain a natural morphism of cohomology sheaves ε : R p π * (F) → R p π * (F h ).

for every p.
Whenever F is given by a graded module M as above, one can express the higher direct images of F in terms of the cohomology of the relative twisting sheaves O(−w) on P r × (C n , 0) (cf. Propositions 4, Proposition 5 below) and vice versa for the respective analytifications. Now the formal completions of the rings C{x 1 , . . . , x n } and C[x 1 , . . . , x n ] x 1 ,...,x n are isomorphic and so are the formal completions of for all p and w.
In the particular cases we will be considering, the sheaf F-or, more generally, a complex of sheaves F • -will always be of a form such that the modules R p π * F and R p π * (F h ), or, respectively, the cohomology sheaves of Rπ * F • and Rπ * (F • ) h , have at most isolated support at the origin. Thus, in either one of the settings the canonical maps to the formal completions of the cohomology modules are isomorphisms of vector spaces and it follows that the comparison morphisms ε above are isomorphisms in this case as well. With the given restrictions on F or F • , we may therefore carry out all the computations of the Euler characteristics of the coherent analytic sheaves derived from F h or (F • ) h on X h in the purely algebraic setting over the field K .

Example 6
We continue with Example 5 and prepare for the explicit computation. As previously discussed, the only interesting stratum of X is S 2 = X reg . To compute the number μ f (2; X , 0) we will describe a complex of graded S-modules representing (Ω • , ν * d f ∧ −). 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 transformX 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 be the tautological section. Together with 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 * d f ∧ −) onX is thus represented by the complex of graded modules As we shall see in the next section, Proposition 5, we can compute the derived pushforward Rν * (Ω • , ν * d f ∧ −) via a truncatedČech-double-complex on the complex of modules (M • , ν * d f ∧ −).

Derived pushforward on relative projective space
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 a finitely generated graded S-module M let M be the corresponding 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 . To this end, 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 on M is obtained fromČ p (M) and the differentialsď :Č p (M) → C p+1 (M) taking an element to the element inČ p+1 (M) with its ( j 0 , . . . , j p+1 )-th component given by (−1) k s d j k a j 0 ,...,ĵ k ,..., j p+1 .
As usual, the hat· indicates that the index is to be omitted. We will writě and set The last term has a structure as a direct limit of S-modules via the maps 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 quasiisomorphic to theČech 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 theČech double complex forM • up to quasi isomorphism analogous to the case of a single sheaf. The result now follows from (16)

An algorithm for the derived homological index
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 with an isolated zero on (X , 0) in the stratified sense and defined over the same field. We briefly describe how to use Proposition 5 in order to compute the Euler obstruction Eu ω (X , 0) as in (11) from Theorem 1. Set A = K [x 1 , . . . , x n ] x 1 ,...,x n and let S = A[s 1 , . . . , s n ] be the graded ring in the s-variables. Let J be the homogeneous ideal of S defining the Nash transform X ⊂ 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 joint with the morphisms given by the pullbacks ν * ω as in Example 6. To compute the Euler obstruction Eu ω (X , 0) as a homological index we proceed as follows.
1) We compute a partial graded free resolution of every one of the M q using standard 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, cf. [17]. 2) From this we obtain the bound d on the pole order for theČech double complex and we can build the truncatedČech 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 ∧ − and the maps of theČech complexes over A from the corresponding maps defined over S. 4) Since ω had an isolated zero, the cohomology of the resulting total complex must be finite over K . We can proceed by the usual standard basis methods for the computation of Euler characteristics.
These computations apply in particular to the case X = S α and ω = d f as in Corollary 4. Remark 5 Note that in Proposition 5 we do not need to compute a quasi-isomorphism of M • with a complex of free S-modules by means of a double complex, but only resolutions of the individual terms M q . With a view towards the application of Proposition 5 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 an isolated zero on (X , 0).

Example 7
We continue with Example 6 and compute μ f (2; X , 0) by means of Formula (12) in Corollary 4. The algorithm above has been implemented in the computer algebra system Singular [7] and the computations were carried out over the rings ring S = 0 ,(s (0..2) ,x,y, z) ,(dp (3) ,ds (3)); with a mixed ordering and the ring ring A = 0 ,(x,y, z) ,ds ; with a local ordering. As a first step, we need to determine the bound d in Proposition 5 and therefore compute graded resolutions of the modules M q in Example 6. The shifts w p,i p as in Proposition 5 are gathered in Table 1 where we write w k whenever the shift w appears exactly k times among the generators of the respective module.
We deduce from Table 1 that choosing d = 2 is sufficient for our computations and proceed to assemble the truncatedČech double complex with termš  This complex now takes the form with the horizontal maps being induced by exterior multiplication with ν * d f on the first factor and the vertical maps by theČech differentials on the second factor of the respective tensor products. Note that these maps are both merely those of the Koszul complex in the components of ν * d f and the sequence (s 2 0 , s 2 1 , s 2 2 ), respectively. The associated total complex is again a complex of finite S-modules. The number of generators of the modules involved is listed in the second row of Table 2. In order to extract the degree-0-part, i.e. the corresponding complex of A-modules, we choose the appropriate monomial bases in the s-variables. As the reader can see from the third row of Table 2, this immediately pushes the number of generators and the size of the involved matrices towards sizes which are impossible to treat without the aid of a computer. Nevertheless, the computations are feasible and a modern desktop computer takes only fractions of a second to compute the cohomology of this complex. The result is where we write h k (·) for the vector space dimension of the k-th homology of a complex. Comparing this with Formula (12) confirms the result: μ f (2; X , 0) = 5.
We conclude this section with two more remarks.

Remark 6
As the previous example suggests, it is essential to use mixed orderings with a local block for the affine variables for the computation of the homological index. Indeed, if one chooses a global ordering on the variables x, y, z, one will obtain the value 7 for k = 2 in (17), i.e. one again also counts the two critical points of f on X away from the origin.

Remark 7
In this section we have restricted ourselves to the case of a function f on a hypersurface (X , 0) ⊂ (C n , 0). The reason for this is that in this case the Grassmannian involved in the computation of the Nash blowup can be identified with P n−1 and there is a well established theory and notation for the twisting sheaves O(−e) and their cohomology. Theoretically, it would also be possible to do similar computations for varieties (X , 0) of arbitrary codimension c in (C n , 0) by means of the Plücker embedding Grass(c, n) → P ( n c )−1 .
In practice, however, this introduces a very inconvenient amount of homogeneous variables and new equations coming from the Plücker relations. Furthermore, the description of the tautological bundle over the Grassmannian and its cohomology in terms of the twisting sheaves is more complicated for general c. This leads us to exclude the general case from our discussion for now.