The Poincaré Index on Singular Varieties

Abstract

In this paper, we discuss a few simple methods for computing the local topological index and its various analogs for vector fields and differential forms given on complex varieties with singularities of different types. They are based on properties of regular meromorphic and logarithmic differential forms, of the dualizing (canonical) module and related constructions. In particular, we show how to compute the index on Cohen–Macaulay, Gorenstein and monomial curves, on normal and non-normal surfaces and some others. In contrast with known traditional approaches, we use neither computers, nor integration, perturbations, deformations, resolution of singularities, spectral sequences or other related standard tools of pure mathematics.

1. Introduction

It is widely known that the classical concept of the topological index of vector fields with isolated singularities given on real or complex planes goes back to H. Poincaré [1]. Somewhat later, his notion was generalized to the smooth higher-dimensional case by H. Hopf [2]. Then, many authors studied the index as a topological invariant in different contexts and various settings, including its infinite-dimensional analogs in complex and functional analysis, real and complex geometry, operator theory, etc.

However, being purely topological, the original definition of index essentially depends on concrete presentation of vector fields, on topological structure of manifolds, on geometrical and homotopical properties of varieties and many other details. Therefore, an extension of the classical approach to the case of singular varieties seems to be a highly nontrivial problem that essentially depends on structure (as a rule, nontrivial also) of the given singularities.

In 1991, X.Gómez-Mont, J. Seade and A. Verjovski [3] defined a “topological” index (which is often called GSV-index) for a holomorphic vector field on a complex hypersurface with an isolated singularity, generalizing the usual Poincaré–Hopf index in the smooth case. Then, the case of complete intersections was investigated in detail by a few authors (see, e.g., [4]). At last, in 1998, X. Gómez-Mont introduced an algebraic notion of the homological index for vector fields given on a reduced pure-dimensional complex analytic space (see [5]); it turns out that namely this notion is easy and well adapted for use in the theory of singular varieties of positive dimension. His fundamental idea is to study an algebraic and analytic invariant, the alternating sum of dimensions of homology groups of the truncated de Rham complex of holomorphic differential forms whose differential is defined by the contraction along a vector field given on a singular space or variety. In other words, this invariant is the Euler–Poincaré characteristic of this complex. Then, X. Gómez-Mont has proved that under suitable finiteness assumptions the homological index is equal to the GSV-index of a vector field up to a constant depending on the singularity of the variety, but not on the vector field. In its turn, the problem of computation of the Euler–Poincaré characteristic is naturally reduced to the computation of the homology of the complex, or, more generally, to the computation of its hypercohomology and related objects. Similarly, one can define the homological index of a differential 1-form with the use of the classical de Rham complex (see [6]), whose differential is defined by the exterior product of the given form.

At first, the homological index was computed for vector fields with isolated singularities tangent to a hypersurface embedded in a complex manifold with the aid of resolvents and spectral sequences [7]. However, in this situation, the author described another method of calculation of the homological index (see [8]). His main idea can be formulated as follows: the computation of the hyperhomology of any complex of holomorphic differential forms, whose differential determined either by contraction along a vector field or by exterior product by a differential 1-form, is simplified essentially if one uses resolvents consisting from meromorphic differential forms with logarithmic poles along a reduced hypersurface. The Euler–Poincaré characteristic of such a complex is an integer invariant called the logarithmic index of the corresponding vector field or differential form.

In the present paper, we exploit similar ideas for computing the homological index as well as its analogs on complex spaces or varieties with singularities of various types with the use of contracted complexes consisting from differential forms of distinct kinds. Thus, one can consider complexes consisting from either holomorphic, or logarithmic, regular meromorphic, weakly holomorphic (or even square integrable) differential forms, which are used (see [9,10,11,12]) in the context of the theory of residue, integration, duality, desingularization, etc. As a rule, in all these cases, the homological index and its analogs have transparent interpretations and significant applications. For example, it is possible to show that in the case of graded complete intersections with isolated singularities, the index can be expressed explicitly in terms of the elementary symmetric polynomials (see [13,14]).

At the beginning of the paper, we introduce some basic notions and definitions. Almost all of them are well-known and studied in a more general setting; they often appear in many subjects related to analytic geometry, complex analysis, residue theory, singularity theory, etc. Our aim here is only to unify them and to consider applications in some specific situations. Then, a series of simple methods of computation of the homological indices of various kinds are discussed; they are applied in different settings depending on concrete types of singularities. Thus, we subsequently show how to compute the index in the case of Gorenstein curves and graded surfaces, monomial varieties, non-normal surfaces and some others.

It should be worthy to underline that there are many works where the index and its analogs are computed by different topological, differential, analytic or algebraic methods based on residue theory (see, e.g., [15]), the theory of resolution of singularities (see [16]), the theory of characteristic and Chern classes (see, e.g., [17]) and others. It is also pertinent to note that, in the case of singular complete intersections, some natural analogs of local indices may even be nonintegers (see [17]). Furthermore, for singular varieties, any proof of global versions of the Poincaré–Hopf theorem requires more delicate observations and additional techniques, which essentially depend on types of singular varieties, methods of computation (see, e.g., [18,19]), and so on.

However, in contrast with known traditional approaches for computing local indices, we do not use either computers, or integration, perturbations, deformations, resolution of singularities, spectral sequences or other related standard tools of pure mathematics.

2. The Homological Index

Let $(X,\mathfrak{o})$ be the germ of complex space of dimension $n⩾1$ at a distinguished point $\mathfrak{o}$. We can choose one of its suitable representatives embedded in an open neighborhood U of the origin in ${\mathbb{C}}^m$ with coordinates $z_1,\dots ,z_m$; it will be denoted by $X.$ Then, X is defined by an ideal $\mathcal{I}\subset {\mathcal{O}}_U$ generated by a sequence of holomorphic functions $f_1,\dots ,f_k\in {\mathcal{O}}_U$. The coherent sheaves of regular differential p-forms on X are defined as follows:

$${\Omega }_X^p:={\Omega }_U^p/((f_1,\cdots ,f_k){\Omega }_U^p+d{f}_1\wedge {\Omega }_U^{p-1}+\cdots +d{f}_k\wedge {\Omega }_U^{p-1}){|}_X,$$

so that ${\Omega }_X^p=0$ whenever $p<0$ or $p>m$, and ${\Omega }_X^0\cong {\mathcal{O}}_X$. Then, the usual differential d endows all this family of sheaves with the structure of an increasing complex

$$0⟶{\mathcal{O}}_X\stackrel{d}{⟶}{\Omega }_X^1\stackrel{d}{⟶}{\Omega }_X^2⟶\cdots ⟶{\Omega }_X^m⟶0,$$

(1)

called the Poincaré (or Poincaré–de Rham) complex of X and denoted by $({\Omega }_X^{•},d)$; its co-homology groups are denoted by $H^{*}({\Omega }_X^{•},d)$.

Let $Der\left(X\right)={Hom}_{{\mathcal{O}}_X}({\Omega }_X^1,{\mathcal{O}}_X)$ be the ${\mathcal{O}}_X$-module of regular vector fields on X. Then, for any element $\mathcal{V}\in Der\left(X\right)$ the interior product (contraction) of vector fields and differential forms induces a homomorphism ${\iota }_{\mathcal{V}}:{\Omega }_X^p⟶{\Omega }_X^{p-1}$ of ${\mathcal{O}}_X$-modules. Since ${\iota }_{\mathcal{V}}^2=0$, a structure of a decreasing complex on the family ${\Omega }_X^{•}$ is well defined; we will call it the contracted de Rham complex associated with the vector field $\mathcal{V}$. Next, the truncated complex

$$0⟶{\Omega }_X^n\stackrel{{\iota }_{\mathcal{V}}}{⟶}{\Omega }_X^{n-1}\stackrel{{\iota }_{\mathcal{V}}}{⟶}{\Omega }_X^{n-2}⟶\cdots ⟶{\Omega }_X^1\stackrel{{\iota }_{\mathcal{V}}}{⟶}{\mathcal{O}}_X⟶0$$

(2)

is defined as the truncation of the initial complex in dimensions greater than n; it is denoted by $({\widehat{\Omega }}_X^{•},{\iota }_{\mathcal{V}})$. If all homology groups of the complex $({\widehat{\Omega }}_{X,\mathfrak{o}}^{•},{\iota }_{\mathcal{V}})$ are finite-dimensional vector spaces, then the Euler–Poincaré characteristic

$$\chi \left({\widehat{\Omega }}_{X,\mathfrak{o}}^{•},{\iota }_{\mathcal{V}}\right)=\sum _{i=0}^n{(-1)}^i{dim}_{\mathbb{C}}{H}_i\left({\widehat{\Omega }}_{X,\mathfrak{o}}^{•},{\iota }_{\mathcal{V}}\right)$$

is well defined; it is called the homological index of the vector field $\mathcal{V}$ at the point $\mathfrak{o}$ and denoted by ${Ind}_{hom,\mathfrak{o}}\left(\mathcal{V}\right)$. At nonsingular points of X, the homological index coincides with the classical topological index, which is often called the Poincaré or the Poincaré–Hopf local index (see [5]).

In the same manner, it is possible to define the homological index of differential 1-forms. More precisely, the exterior multiplication by any regular (or even meromorphic) differential 1-form $\omega \in {\Omega }_X^1$ defines an increasing complex $({\Omega }_X^{•},\wedge \omega )$:

$$0⟶{\mathcal{O}}_X\stackrel{\wedge \omega }{⟶}{\Omega }_X^1⟶\cdots ⟶{\Omega }_X^{m-1}\stackrel{\wedge \omega }{⟶}{\Omega }_X^m⟶0;$$

(3)

It was introduced by G. de Rham in [6]. We will also call it the de Rham complex associated with the differential form $\omega$. It is clear that both complexes (2) and (3) can be regarded as contravariant versions of the complex (1).

In this case, under similar finiteness conditions and in analogous notations, the Euler–Poincaré characteristic

$$\chi \left({\widehat{\Omega }}_{X,\mathfrak{o}}^{•},\wedge \omega \right)=\sum _{i=0}^n{(-1)}^i{dim}_{\mathbb{C}}{H}_i\left({\widehat{\Omega }}_{X,\mathfrak{o}}^{•},\wedge \omega \right)$$

(4)

is also defined; we will call it the homological index of the differential form $\omega$ at the point $\mathfrak{o}$ and denote it by ${Ind}_{hom,\mathfrak{o}}(\omega )$ (see [8]). As a rule, one can find for almost all statements concerning properties of homological index for vector fields the corresponding analogs (however, sometimes unexpected) for the index of differential 1-forms, and vice versa. For completeness, it should also be noted that even in the case of smooth manifolds, the question of the existence of holomorphic vector fields with isolated singularities and nontrivial differential forms is a very nontrivial one (see, e.g., [20]).

3. The Logarithmic Index

3.1. Logarithmic Differential Forms along Cartier Divisors

Let X be a reduced complex space or variety. An effective Cartier divisor $D\subset X$ is determined locally by a function germ $g\in {\mathcal{O}}_{X,\mathfrak{o}}$, which is a non-zero divisor. Thus, if $z=(z_1,\dots ,z_m)$ is a local system of coordinates in a neighborhood U of the origin, then the restriction $g\left(z\right){|}_X=0$ is a local equation of D at the distinguished point $\mathfrak{o}\in X$ and ${\mathcal{O}}_{D,\mathfrak{o}}\cong {\mathcal{O}}_{X,\mathfrak{o}}/\left(g\right)\cong {\mathcal{O}}_{{\mathbb{C}}^m,0}/(f_1,\dots ,f_k,g)$.

In what follows, we will assume that the given Cartier divisor has a positive depth along its singular locus. That is, in the standard notation, $depth(SingD,D)⩾1$. In particular, this means that the image of the Jacobi ideal $Jac\left(g\right)$ under the canonical surjection ${\mathcal{O}}_{X,\mathfrak{o}}\to {\mathcal{O}}_{D,\mathfrak{o}}$ contains at least one element, which is not a zero-divisor. Such Cartier divisors on singular varieties have no multiple components of maximal dimensions. However, in general, they are nonreduced, and embedded components of lower dimension may occur in their irredundant decompositions.

Anyway, if D is a Cartier divisor in a complex space such that D is a Cohen–Macaulay subspace of positive dimension, then there exists at least one non-zero divisor in the local ring ${\mathcal{O}}_{D,\mathfrak{o}}\cong {\mathcal{O}}_{{\mathbb{C}}^m,0}/(f_1,\dots ,f_k,g)$. Moreover, the inequality $codim(SingD,D)⩾1$ implies that D is reduced and vice versa; in this case, we also have the inequality $depth(SingD,D)⩾1$ without fail.

Definition 1.

Let D be the Cartier divisor in a complex space X determined by a function $g\in {\mathcal{O}}_{X,\mathfrak{o}}$. Then, the ${\mathcal{O}}_X$-modules ${\Omega }_X^p(logD)$, $p⩾0$, of sheaves of logarithmic differential forms are locally defined (via the kernel of the operator of exterior multiplication) as follows:

$${\Omega }_{X,\mathfrak{o}}^p(logD):=\frac{1}{g}Ker\left(\wedge dg:{\Omega }_{X,\mathfrak{o}}^p⟶{\Omega }_{X,\mathfrak{o}}^{p+1}/g·{\Omega }_{X,\mathfrak{o}}^{p+1}\right),$$

where by $\wedge dg$ we denote the ordinary homomorphism of exterior multiplication by the total differential of the function g.

Remark 1.

In other words, the set of holomorphic forms annihilated (modulo $g·{\Omega }_{X,\mathfrak{o}}^{p+1}$) by the exterior multiplication by the total differential $dg$ coincides with $g·{\Omega }_{X,\mathfrak{o}}^p(logD)$. Hence, this definition can be also regarded as an analog of an important observation due to G. de Rham in the context of the theory of generalized functions. More precisely, he has proved that distributions T of degree zero, satisfying the condition $T\wedge \omega =0$, are equal to multiples of the Dirac distribution δ (see ([6], Equation (5))).

Remark 2.

Recall that if we consider a smooth variety X and the reduced divisor $D\subset X$ given by a regular function $g\in {\mathcal{O}}_X$, then the coherent analytic sheaves ${\Omega }_X^p(logD)$, $p\ge 0$, are defined traditionally as follows: the stalk ${\Omega }_{X,\mathfrak{o}}^q(logD)$ over the point $\mathfrak{o}\in X$ consists of those germs ω of meromorphic q-forms on X with simple poles along D that satisfy the following two conditions: $g\omega$ and $gd\omega$ are holomorphic at the distinguished point $\mathfrak{o}$ (see [21]).

3.2. Normal Varieties

Proposition 1.

Suppose that X is a normal variety. Then, the direct sum ${\oplus }_{p=0}^m{\Omega }_{X,\mathfrak{o}}^p(logD)$ is an ${\mathcal{O}}_{X,\mathfrak{o}}$-exterior algebra; it is usually denoted by ${\Omega }_{X,\mathfrak{o}}^{•}(logD)$.

Remark 3.

In general it is not true. However, if we set

$${Ker}_{X,\mathfrak{o}}^p(\wedge dg):=Ker\left(\wedge dg:{\Omega }_{X,\mathfrak{o}}^p⟶{\Omega }_{X,\mathfrak{o}}^{p+1}/g·{\Omega }_{X,\mathfrak{o}}^{p+1}\right),$$

then the direct sum ${\oplus }_{p=0}^m{Ker}_{X,\mathfrak{o}}^p(\wedge dg)$ is an ${\mathcal{O}}_{X,\mathfrak{o}}$-exterior algebra for any variety X. On the other hand, the algebra ${\Omega }_{X,\mathfrak{o}}^{•}(logD)$ is always closed under the exterior differentiation d and the exterior multiplication by $dg/g$.

Similarly to the case of divisors in a smooth ambient space studied in ([22], Section 1, Proposition), the following assertion can be used effectively in the computation of the sheaves of logarithmic differential forms given on varieties with singularities.

Corollary 1.

Let X be a normal variety of embedded dimension m and let $g\in {\mathcal{O}}_{X,\mathfrak{o}}$ be a non-zero divisor. Then, for all $p=1,\cdots ,m$, there are the following exact sequences of ${\mathcal{O}}_{X,\mathfrak{o}}$-modules

$$\begin{array}{ccc}& 0⟶{\Omega }_{X,\mathfrak{o}}^{p-1}/g·{\Omega }_{X,\mathfrak{o}}^{p-1}(logD)\stackrel{dg}{⟶}{\Omega }_{X,\mathfrak{o}}^p/g·{\Omega }_{X,\mathfrak{o}}^p⟶{\Omega }_{D,\mathfrak{o}}^p⟶0,& \hfill \end{array}$$

(5)

$$\begin{array}{ccc}& 0⟶{\Omega }_{X,\mathfrak{o}}^p/dg\wedge {\Omega }_{X,\mathfrak{o}}^{p-1}(logD)\stackrel{·g}{⟶}{\Omega }_{X,\mathfrak{o}}^p/dg\wedge {\Omega }_{X,\mathfrak{o}}^{p-1}⟶{\Omega }_{D,\mathfrak{o}}^p⟶0,& \hfill \end{array}$$

(6)

$$\begin{array}{ccc}& 0⟶{\Omega }_{X,\mathfrak{o}}^p+\frac{dg}{g}\wedge {\Omega }_{X,\mathfrak{o}}^{p-1}⟶{\Omega }_{X,\mathfrak{o}}^p(logD)\stackrel{·g}{⟶}Tors{\Omega }_{D,\mathfrak{o}}^p⟶0,& \hfill \end{array}$$

(7)

where $\wedge dg$ and $·g$ denote the homomorphisms of exterior and usual multiplications, respectively.

Example 1.

Let us consider the case where X is the Hilbert cubic. Thus, X is an affine cone over the rational curve of degree 3 in projective space ${\mathbb{P}}^3$ given by the embedding ${\mathbb{P}}^1\to {\mathbb{P}}^3$ by the sheaf ${\mathcal{O}}_{{\mathbb{P}}^1}\left(3\right)$. In usual terms, X is a determinantal normal surface singularity of codimension 2 and the local ring ${\mathcal{O}}_{X,\mathfrak{o}}$ is a domain.

We will denote the homogeneous coordinates in ${\mathbb{P}}^3$ by $(x:y:z:u)$. Let $\mathfrak{o}$ be the vertex of the cone. Then, ${\mathcal{O}}_{X,\mathfrak{o}}\cong \mathbb{C}\langle x,y,z,u\rangle /I$, where the prime ideal I is generated by the maximal minors of the Hankel matrix $\left[\genfrac{}{}{0pt}{}{xyz}{yzu}\right]$. That is,

$$f_{12}=xz-y^2,f_{13}=yz-xu,f_{23}=yu-z^2.$$

Let us set $g=x^2-u^2$ and denote by ${\vartheta }_i$, $i=1,\dots ,6$, the differential 2-forms $xdy-ydx$, $xdz-zdx$, $xdu-udx$, $ydz-zdy$, $ydu-udy$ and $udz-zdu$, respectively. In fact, these forms are obtained by the contraction of suitable coordinate 2-forms (such as $dx\wedge dy$, $dx\wedge dz$, etc.) along the Euler vector field. Then, we obtain an isomorphism

$${\Omega }_{X,\mathfrak{o}}^1(logD)\cong {\mathcal{O}}_{X,\mathfrak{o}}〈\frac{dg}{g},\frac{{\vartheta }_1}{g},\dots ,\frac{{\vartheta }_6}{g}〉.$$

It is not difficult to see that ${\Omega }_{X,\mathfrak{o}}^1(logD)$ is a nonfree ${\mathcal{O}}_{X,\mathfrak{o}}$-module of rank 7 and ${\vartheta }_i\in Tors\left({\Omega }_{D,\mathfrak{o}}^1\right)$ for all $i=1,\dots ,6$.

The following example of a non-normal surface is very useful for understanding our notion of logarithmic forms in the most general context.

Example 2.

Let $X\subset ({\mathbb{C}}^4,0)$ be the coproduct of two copies of a complex plane over the origin. It is clear that the structure local algebra of the corresponding germ X is isomorphic to the coproduct $\mathbb{C}\langle x,y\rangle {\times }_{\mathbb{C}}\mathbb{C}\langle z,u\rangle$. Hence, ${\mathcal{O}}_{X,\mathfrak{o}}\cong \mathbb{C}\langle x,y,z,u\rangle /I$, where $I=(xz,xu,yz,yu)$. It is easy to see, that the germ X is an isolated surface singularity and it is not a Cohen–Macaulay singularity. This implies that the germ X is a non-normal surface singularity. Moreover, X has no nontrivial infinitesimal deformations of the first order; therefore, it is a rigid singularity. The germ X is often called the two-dimensional fan.

In this case, the local ring ${\mathcal{O}}_{X,\mathfrak{o}}$ is no longer an integral domain. However, it contains enough non-zero-divisors. Let us consider $g=x^2+y^2+z^2+u^2$. It is not difficult to verify that the corresponding divisor $D\subset X$ is neither reduced, nor even equal-dimensional. Indeed, it contains an embedded zero-dimensional component at the distinguished point $\mathfrak{o}$.

Let us consider $g_1=x^2+y^2$, $g_2=z^2+u^2$, so that $g=g_1+g_2$. Then,

$${\Omega }_{X,\mathfrak{o}}^1(logD)\cong {\mathcal{O}}_{X,\mathfrak{o}}〈\frac{d{g}_1}{g_1},\frac{d{g}_2}{g_2},\frac{{\vartheta }_1}{g},\dots ,\frac{{\vartheta }_6}{g}〉$$

where ${\vartheta }_i$, $i=1,\dots ,6$, are the differential 2-forms from the above example. Again, it is not difficult to check that ${\Omega }_{X,\mathfrak{o}}^1(logD)$ is a nonfree ${\mathcal{O}}_{X,\mathfrak{o}}$-module of rank 8 and ${\vartheta }_i$, $i=1,\dots ,6$, are contained in the torsion modules $Tors\left({\Omega }_{D,\mathfrak{o}}^1\right)$.

3.3. The Logarithmic Index

Similarly to the smooth case, one can define the sheaf of logarithmic (that are tangent to the divisor D) vector fields on X. More precisely, ${Der}_X(logD)$ consists of germs $\mathcal{V}$ of holomorphic vector fields on X such that $\mathcal{V}\left(h\right)\in \left(h\right){\mathcal{O}}_{X,\mathfrak{o}}$. It is well-known (see [21]) that there is a natural perfect duality

$${Der}_X(logD)\times {\Omega }_X^1(logD)⟶{\mathcal{O}}_{X,\mathfrak{o}},$$

(8)

which is induced by inner product of differential forms and vector fields. Again, as in Section 2, the inner multiplication ${\iota }_{\mathcal{V}}$ defines the structure of a decreasing complex on ${\widehat{\Omega }}_X^{•}(logD)$:

$$0⟶{\Omega }_X^{n+1}(logD)\stackrel{{\iota }_{\mathcal{V}}}{⟶}{\Omega }_X^n(logD)⟶\cdots ⟶{\Omega }_X^1(logD)\stackrel{{\iota }_{\mathcal{V}}}{⟶}{\mathcal{O}}_X⟶0.$$

Definition 2.

Assume that $\mathcal{V}$ has only isolated singularities, then, for each $x\in X$, the Euler–Poincaré characteristic

$$\chi \left({\widehat{\Omega }}_{X,x}^{•}(logD),{\iota }_{\mathcal{V}}\right)=\sum _{i=0}^{n+1}{(-1)}^{i}dim{H}_i\left({\widehat{\Omega }}_{X,x}^{•}(logD),{\iota }_{\mathcal{V}}\right)$$

of the complex of germs of logarithmic differential forms is well defined. This characteristic is called the logarithmic index of the vector field $\mathcal{V}$ at the point x; it is denoted by ${Ind}_{logD,x}\left(\mathcal{V}\right)$.

It is not difficult to verify that ${Ind}_{logD,x}\left(\mathcal{V}\right)=0$ if $\mathcal{V}\left(x\right)\ne 0$ and there is a relation

${Ind}_{hom,D,\mathfrak{o}}\left(V\right)=dim{\mathcal{O}}_{X,\mathfrak{o}}/J_{\mathfrak{o}}\mathcal{V}-{Ind}_{logD,\mathfrak{o}}\left(V\right),$

where by $J_{\mathfrak{o}}\mathcal{V}=(v_0,\dots ,v_n){\mathcal{O}}_{M,\mathfrak{o}}$ we denote the ideal generated by the coefficients of the expansion $\mathcal{V}={\sum }_i{v}_i\partial /\partial z_i$ of the vector field $\mathcal{V}$.

Again, as in Section 2, one can define the logarithmic index of a logarithmic differential form $\omega \in {\Omega }_X^1(logD)$, and so on.

4. The Multi-Logarithmic Index

Indeed, Definition 2 of the logarithmic index can be extended to a more general setting. At first, we will introduce a notion of multi-logarithmic differential forms (see [12,23]).

Definition 3.

Let $C\subset X$ be a complete intersection subspace of pure codimension ℓ locally defined by a regular sequence of functions $g_1,\dots ,g_{\ell }$. Then, the sheaves ${\Omega }_X^p(logC)$, $p⩾\ell$, of germs of multi-logarithmic differential forms with respect to C, are locally defined as follows:

$${\Omega }_{X,\mathfrak{o}}^{{}^{•}}(logC):=\frac{1}{g}Ker\left(\mathcal{E}:{\Omega }_{X,\mathfrak{o}}^p⟶{\left({\Omega }_{X,\mathfrak{o}}^{p+1}/(g_1,\dots ,g_{\ell }){\Omega }_{X,\mathfrak{o}}^{p+1}\right)}^{\ell }\right),$$

where $g=g_1\cdots g_{\ell }$ and $\mathcal{E}(\eta )=(\eta \wedge d{g}_1,\dots ,\eta \wedge d{g}_{\ell })$.

Remark 4.

If $\ell =1$, then the multi-logarithmic differential forms are logarithmic in the sense of Definition 1. Indeed, we have $C=D$, so that ${\Omega }_X^p(logC)={\Omega }_X^p(logD)$, $p\ge 0$.

Proposition 2.

Let X be a normal complex space, and let D be the union of the Cartier divisors $D_j$ locally determined by regular functions $g_j$, $j=1,\dots ,\ell$. Then, the sheaves ${\Omega }_X^p(logC)$ consist of germs of meromorphic p-forms $\omega \in {\Omega }_X^p\left(D\right)$ with simple poles along D satisfying the conditions

$$d{g}_j\wedge \omega \in \sum _{i=1}^{\ell }{\Omega }_{X,\mathfrak{o}}^{p+1}\left({\check{D}}_i\right),j=1,\dots ,\ell ,$$

where ${\check{D}}_i=D_1\cup \dots \cup D_{i-1}\cup D_{i+1}\cup \dots \cup D_{\ell }$, $i=1,\dots ,\ell$.

It is clear that ${\Omega }_X^p(logC)$, $p\ge 0$, are coherent sheaves of ${\mathcal{O}}_X$-modules and there are natural inclusions $g_j·{\Omega }_X^p(logC)\subseteq {\Omega }_X^p\left({\check{D}}_j\right)$, $j=1,\dots ,\ell$, and $g·{\Omega }_X^p(logC)\subseteq (g_1,\cdots ,g_{\ell }){\Omega }_X^p$. Next, ${\Omega }_X^p(logD)\subseteq {\Omega }_X^p(logC),p\ge 0$, ${\Omega }_X^0(logD)\cong {\Omega }_X^0(logC)\cong {\mathcal{O}}_X$, and

$${\Omega }_X^m(logD)\cong {\Omega }_X^m(logC)\cong \frac{1}{g}{\Omega }_X^m.$$

There is also another description of the multi-logarithmic differential forms in terms of the multiple residue, which is often useful in computations. In fact, in the smooth case, this idea is also due to H.Poincaré (see [24]).

Proposition 3.

Under the assumptions and in the notation of Proposition 2, for any multi-logarithmic differential form $\omega \in {\Omega }_X^p(logC)$, $p⩾\ell$, there is a holomorphic function g, which is not identically zero on every irreducible component of the complete intersection C, a holomorphic differential form $\xi \in {\Omega }_U^{p-\ell }$ and a meromorphic p-form $\eta \in {\sum }_{i=1}^{\ell }{\Omega }_U^p\left({\check{D}}_i\right)$ such that there exists the following representation:

$$g\omega =\frac{d{g}_1}{g_1}\wedge \dots \wedge \frac{d{g}_{\ell }}{g_{\ell }}\wedge \xi +\eta .$$

Definition 4.

The restriction of the differential form $\xi /g$ to the complete intersection C is called the multiple residue form of ω (or the multiple residue of the differential form ω) and is denoted by ${Res}_C(\omega )$, i.e.,

$${Res}_C(\omega )=\frac{\xi }{g}{|}_C.$$

Hence, the order of the (meromorphic) residue form ${Res}_C(\omega )$ is equal to $p-\ell$. In fact, for smooth hypersurfaces in a smooth manifold the function g an invertible function. In general, it is clear that ${\Omega }_X^p(logD)\subseteq {\Omega }_X^p(logC)$, so that the restriction of ${Res}_C(\omega )$ to the submodule ${\Omega }_X^p(logD)$ is also well defined. Moreover, if $\omega \in {\Omega }_X^p(logD)$, then we can choose the above decomposition in such a way that the form $\eta$ is logarithmic along D as well.

Remark 5.

There are also some other explicit representations for the multiple residue p-form ${Res}_C(\omega )$. One of them exploits $\overline{\partial }$-closed meromorphic currents. For instance, if X is smooth, then there is an equality

$$〈\left[\frac{\xi }{g}\right]{|}_{{}_C},\phi 〉=\underset{\epsilon \to 0}{lim}{{\displaystyle \int }}_{C\cap \left\{\right|g|>\epsilon \}}\frac{\xi }{g}\wedge \phi ,$$

where $\phi \in {\mathcal{D}}_U^{2m-p-\ell }$ is a differential $C^{\infty }$-form on an open set U with compact support determined by the map $(g_1,\dots ,g_{\ell }):U\to {\mathbb{C}}^{\ell }$. Moreover, there are several other types of integral representations. The most known is based on a variant of the Grothendieck residue mapping in the case of complete intersections.

Relevant arguments for studying a more general case of divisors in Cohen-Macaulay varieties with a series of applications can be found in the references [8,13,14,23] and in subsequent papers by the author.

It is not difficult to see that, in the notations above, there are several ways to define the module of multi-logarithmic vector fields ${Der}_{X,\mathfrak{o}}(logC)$. For example, one can define this module as a dual to ${\Omega }_X^{\ell }(logC)$ in the following sense (cf. the expression (8)):

$${Der}_X(logC)\times {\Omega }_X^{\ell }(logC)⟶\frac{1}{g}{\mathcal{I}}_C,$$

where $g=g_1\cdots g_{\ell }$ and ${\mathcal{I}}_C=(g_1,\cdots ,g_{\ell })$ is an ideal of ${\mathcal{O}}_X$ (see details in [25]).

Analogously to Section 2, for any $\mathcal{V}\in {Der}_X(logC)$, the inner multiplication by ${\iota }_{\mathcal{V}}$ defines the structure of a decreasing complex on the family ${\Omega }_X^{•}(logC)$.

Definition 5.

Let $\mathcal{V}\in {Der}_X(logC)$ be a multi-logarithmic vector field. Under analogous finiteness conditions and notations as above, the Euler–Poincaré characteristic

$$\chi \left({\widehat{\Omega }}_{X,x}^{•}(logC),{\iota }_{\mathcal{V}}\right)=\sum _{i=0}^{n+1}{(-1)}^{i}dim{H}_i\left({\widehat{\Omega }}_{X,x}^{•}(logC),{\iota }_{\mathcal{V}}\right)$$

of the complex of germs of multi-logarithmic differential forms is defined correctly; it is called the multi-logarithmic index of the vector field $\mathcal{V}$ at the point x and is denoted by ${Ind}_{logX/C,x}\left(\mathcal{V}\right)$.

5. The Regular Meromorphic Index

In this section, we will suppose that X is a Cohen–Macaulay germ of complex space (not necessarily reduced) of dimension n. In this case, the Grothendieck dualizing module of X (or the module of dualizing differentials) is defined as follows:

$${\omega }_X^n={Ext}_{{\mathcal{O}}_{{\mathbb{C}}^m}}^{m-n}({\mathcal{O}}_X,{\Omega }_{{\mathbb{C}}^m}^m).$$

It is well-known that this module has no torsion, i.e., $Tors{\omega }_X^n=0$. Recall also that all 0-dimensional germs, 1-dimensional reduced germs and 2-dimensional normal germs are Cohen–Macaulay.

By definition (see [9,10,11]), the coherent sheaf of regular meromorphic differential forms of degree $p\ge 0$ on X is locally defined as the set of germs of meromorphic differential forms $\omega$ of degree p on X such that $\omega \wedge \xi \in {\omega }_X^n$ for any $\xi \in {\Omega }_X^{n-p}$. That is, for all $p\ge 0$, there are natural isomorphisms of ${\mathcal{O}}_X$-modules

$${\omega }_X^p\cong {Hom}_{{\mathcal{O}}_X}({\Omega }_X^{n-p},{\omega }_X^n)\cong {Ext}_{{\mathcal{O}}_{{\mathbb{C}}^m}}^{m-n}({\Omega }_X^{n-p},{\Omega }_{{\mathbb{C}}^m}^m).$$

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Elements of ${\omega }_X^p$ are called regular meromorphic differential forms on X of degree p. There are several equivalent definitions of these sheaves in terms of Noether normalization and trace (see [9,10]), in terms of residual currents (see [12]), and so on.

It is clear that ${\omega }_X^p=0$ for $p>n$ since ${\Omega }_X^{n-p}=0$. Next, we obtain ${\omega }_X^p=0$ for $p<0$ because ${\Omega }_X^p\cong Tors{\Omega }_X^p$ for $p>n$ and the dualizing module ${\omega }_X^n$ has no torsion. It is also easy to see that the de Rham differentiation d as well as the contraction ${\iota }_{\mathcal{V}}$ acting on ${\Omega }_X^{•}$ are naturally extended to the family of modules ${\omega }_X^p,0\le p\le n$; they endow this family with structures of increasing complex $({\omega }_X^{•},d)$ or decreasing complex $({\omega }_X^{•},{\iota }_{\mathcal{V}}^{*})$, respectively. In particular, the contraction ${\iota }_{\mathcal{V}}^{*}:{\omega }_X^p\to {\omega }_X^{p-1}$ is naturally defined as dual to the action ${\iota }_{\mathcal{V}}$ on the complex ${\Omega }_X^{•}$ in view of presentation (9).

Lemma 1

(see [9]). Let $Z=SingX$, $j:X\setminus Z⟶X$. Then, there are natural inclusions ${\omega }_X^p\subseteq j_{*}{j}^{*}{\Omega }_X^p$ for all $p\ge 0$. Moreover, if $c=codim(Z,X)\ge 1$, then ${\omega }_X^p$ has no Z-torsion; if $codim(Z,X)\ge 2$, then ${\omega }_X^p$ has no cotorsion for all $p\ge 0$.

Definition 6.

If all homology groups of $({\omega }_{X,\mathfrak{o}}^{•},{\iota }_{\mathcal{V}}^{*})$ are finite-dimensional vector spaces, then the Euler–Poincaré characteristic

$$\chi \left({\omega }_{X,\mathfrak{o}}^{•},{\iota }_{\mathcal{V}}^{*}\right)=\sum _{i=0}^n{(-1)}^i{dim}_{\mathbb{C}}{H}_i\left({\omega }_{X,\mathfrak{o}}^{•},{\iota }_{\mathcal{V}}^{*}\right)$$

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is well defined; it is called the regular meromorphic index of the vector field $\mathcal{V}$ at point $\mathfrak{o}$ and denoted by ${Ind}_{reg,\mathfrak{o}}\left(\mathcal{V}\right)$.

6. Cohen-Macaulay Curves

For convenience of notations, in this section and later, we will often denote the field of complex numbers by $\mathsf{k}$, the local dual analytical algebras of germs $({\mathbb{C}}^m,0)$ and $(X,\mathfrak{o})$ by P and by A, respectively. That is, A is a local ring with the maximal ideal ${\mathfrak{m}}_A$ and $A=P/I$, where the ideal I is generated by a sequence of elements $f_1,\dots ,f_k\in A$. Then, ${\Omega }_A^{•}\cong {\wedge }^{•}{\Omega }_A^1$, where ${\Omega }_A^1$ is the module of Kähler differentials, $Der\left(A\right)={Hom}_A({\Omega }_A^1,A)$ is the module of $\mathsf{k}$-differentiation of A, and so on.

Herein, we will also assume that A is 1-dimensional Cohen–Macaulay ring corresponding to a reduced curve singularity X of embedding dimension $m\ge 2$, where $m={dim}_k{\mathfrak{m}}_A/{\mathfrak{m}}_A^2$. In this case,

$${\omega }_A^0\cong {Hom}_A({\Omega }_A^1,{\omega }_A^1),{\omega }_A^1\cong {Ext}_{\mathcal{P}}^{m-1}(A,{\Omega }_P^m)\cong {Hom}_A(A,{\omega }_A^1).$$

In standard terms, let $\mathfrak{I}\left(k\right)$ be the injective hull of the residue field $k=A/{\mathfrak{m}}_A$, and $K_A={Hom}_A(H_{\mathfrak{m}}^1\left(A\right),\mathfrak{I}\left(k\right))$ be the canonical module of A. Then, $K_A\cong {\omega }_A^1\subset {\Omega }_A^1{\otimes }_{A}F/A$, where F is the total ring of fractions of A (see [26]). Next, there exists an exact sequence of A-modules

$$0⟶Tors{\Omega }_A^1⟶{\Omega }_A^1\stackrel{c_A}{⟶}{\omega }_A^1⟶{\left({\omega }_A^1\right)}^t⟶0,$$

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where $c_A$ is the canonical map (see [27]). It follows from definition that $Ker\left(c_A\right)=Tors{\Omega }_A^1\cong H_{\mathfrak{m}}^0\left({\Omega }_A^1\right)$ is the torsion submodule of ${\Omega }_A^1$, while the module $Coker\left(c_A\right)={\left({\omega }_A^1\right)}^t$, contained in $H_{\mathfrak{m}}^1\left({\Omega }_A^1\right)$, is called the cotorsion of the curve singularity. In particular, torsion and cotorsion modules are concentrated at the singularity $(A,{\mathfrak{m}}_A)$, and, consequently, they are finite-dimensional vector spaces.

In the notations of Section 5, the module ${\omega }_X^1$ is isomorphic to the Grothendieck dualizing module of the curve singularity, and there is an embedding ${\omega }_X^1\subset j_{*}{j}^{*}{\Omega }_X^1$, where $j:X\setminus Z\hookrightarrow X$ is the natural inclusion, and $Z=SingX=\left\{\mathfrak{o}\right\}$ (cf. Lemma 1). Next, there is an isomorphism $Tors{\Omega }_X^1\cong {\mathcal{H}}_{\left\{\mathfrak{o}\right\}}^0\left({\Omega }_X^1\right)$, an embedding ${\left({\omega }_X^1\right)}^t\subset {\mathcal{H}}_{\left\{\mathfrak{o}\right\}}^1\left({\Omega }_X^1\right)$, and $c_X$ is the canonical map induced by the multiplication by the fundamental class of the germ X.

Claim 1.

Let $\tilde{A}$ be the normalization of a 1-dimensional local ring A in its total ring of fractions F, and let $\mathfrak{c}=Ann(\tilde{A}/A)$ be the conductor of $\tilde{A}$ in A. Then, there are two natural isomorphisms ${\omega }_{\tilde{A}}^1\cong {\Omega }_{\tilde{A}}^1$ and ${\omega }_{\tilde{A}}^1\cong \mathfrak{c}·{\omega }_A^1$ and the inclusion $\mathfrak{m}·{\omega }_{\tilde{A}}^0\subseteq \mathfrak{c}·{\omega }_A^0$.

Proof. 

Following the argumentation in ([28], 3.2), one can obtain the existence of both isomorphisms, while the inclusion is evident. □

Proposition 4.

In the same notations, let $\mathcal{V}\in {Der}_k\left(A\right)$ be a $\mathsf{k}$-differentiation of a reduced 1-dimensional analytical algebra A. Then, there is a natural extension $\tilde{\mathcal{V}}$ of $\mathcal{V}$ on the normalization $\tilde{A}$ of A such that the following diagram with exact rows

$$\begin{array}{cccccccc}0⟶& Tors{\Omega }_A^1& ⟶& {\Omega }_A^1& \stackrel{c_A}{⟶}& {\omega }_A^1& ⟶{\left({\omega }_A^1\right)}^t& ⟶0\\ & & & \downarrow {\iota }_{\mathcal{V}}& & \downarrow {\iota }_{\tilde{\mathcal{V}}}& \downarrow & \\ & 0& ⟶& {\Omega }_A^0\cong A& ⟶& {\omega }_A^0& ⟶{\left({\omega }_A^0\right)}^t& ⟶0,\end{array}$$

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is commutative. In the notations of Section 5, there is an identification ${\omega }_A^0={Hom}_A({\Omega }_A^1,{\omega }_A^1)$, the canonical inclusion $A\hookrightarrow {\omega }_A^0$ and a natural isomorphism ${\left({\omega }_A^0\right)}^t\cong {Ext}_A^1({\left({\omega }_A^1\right)}^t,{\omega }_A^1)$. The vertical arrow ${\iota }_{\mathcal{V}}$ is the contraction along $\mathcal{V}$ acting on ${\Omega }_A^1$, while the mapping ${\iota }_{\tilde{\mathcal{V}}}$ is induced by an extension of $\mathcal{V}$.

Proof. 

First note that any vector field $\mathcal{V}$ on the curve singularity A can be extended to its normalization $\tilde{A}$ as well as to the total ring of fractions F of A because $\mathrm{char}\left(k\right)=0$ (see, e.g., [29], Lemma 2.33); this extension is denoted by $\tilde{\mathcal{V}}\in {Der}_k\left(\tilde{A}\right)$. Thus,

$${\iota }_{\tilde{\mathcal{V}}}\left({\Omega }_{\tilde{A}}^1\right)={\iota }_{\tilde{\mathcal{V}}}\left({\omega }_{\tilde{A}}^1\right)={\iota }_{\tilde{\mathcal{V}}}(\mathfrak{c}·{\omega }_A^1)=\mathfrak{c}·{\iota }_{\tilde{\mathcal{V}}}\left({\omega }_A^1\right),$$

and the vertical arrow ${\iota }_{\tilde{\mathcal{V}}}$ of the diagram is well defined. Next, under our assumptions, A is a Cohen–Macaulay local ring of Krull dimension 1. Then, ${Hom}_A(k,{\omega }_A^1)=0$. Therefore, ${Hom}_A(M,{\omega }_A^1)=0$ for any A-module M of finite type with $SuppM\subseteq \left\{{\mathfrak{m}}_A\right\}$.

Let us apply the functor ${Hom}_A(•,{\omega }_A^1)$ to the following exact sequences

$$0\to Tors{\Omega }_A^1\to {\Omega }_A^1\to {\tilde{\Omega }}_A^1\to 0,$$

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$$0\to {\tilde{\Omega }}_A^1\stackrel{c_A}{⟶}{\omega }_A^1\to {\left({\omega }_A^1\right)}^t\to 0,$$

(14)

which are obtained by splitting of the sequence (11). As a result, we get a natural isomorphism ${Hom}_A({\tilde{\Omega }}_A^1,{\omega }_A^1)\cong {Hom}_A({\Omega }_A^1,{\omega }_A^1)$ together with the first four terms of the corresponding long exact sequence

$$0\to {Hom}_A({\omega }_A^1,{\omega }_A^1)⟶{Hom}_A({\Omega }_A^1,{\omega }_A^1)⟶{Ext}_A^1({\left({\omega }_A^1\right)}^t,{\omega }_A^1)⟶{Ext}_A^1({\omega }_A^1,{\omega }_A^1),$$

since supports of $Tors{\Omega }_A^1$ and ${\left({\omega }_A^1\right)}^t$ are contained in the singular point $\left\{{\mathfrak{m}}_A\right\}$. At last, from ([26], (6.1 d)) it follows that ${Ext}_A^1({\omega }_A^1,{\omega }_A^1)=0$, ${Hom}_A({\omega }_A^1,{\omega }_A^1)\cong A$, and there is a canonical exact sequence

$$0⟶A⟶{Hom}_A({\Omega }_A^1,{\omega }_A^1)⟶{Ext}_A^1({\left({\omega }_A^1\right)}^t,{\omega }_A^1)⟶0.$$

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The inclusion $A⟶{Hom}_A({\Omega }_A^1,{\omega }_A^1)$ is determined by the correspondence $1_A\mapsto c_A$ (see details in [28], Section 3). It is not difficult to verify that the fundamental mapping $c_A$ is compatible with the contraction ${\iota }_{\mathcal{V}}$ as well as with its extension ${\iota }_{\tilde{\mathcal{V}}}$. As a result, the diagram (12), which is a combination of the latter exact sequence and (11), is commutative. □

Corollary 2

(cf. [13], Claim 1). Under the same assumptions, suppose also that the vector field $\mathcal{V}$ has an isolated singularity. Then, $Tors{\Omega }_A^1=Ker\left({\iota }_{\mathcal{V}}\right)$.

Proof. 

The inclusion $Tors{\Omega }_A^1\subseteq Ker\left({\iota }_{\mathcal{V}}\right)$ follows directly from the commutativity of the diagram (12). Set $K^1=Ker({\iota }_{\mathcal{V}}:{\Omega }_A^1\to A)$. Then, the support of the module $K^1$ is concentrated at the singular point of the reduced germ $(A,{\mathfrak{m}}_A)$. The coherence condition then implies that A-module $K^1$ is a vector space of finite dimension over the ground field, that is, $K^1$ is a torsion module. In particular, $K^1\subseteq H_{\left\{\mathfrak{m}\right\}}^0\left({\Omega }_A^1\right)\cong Tors{\Omega }_A^1$. This completes the proof. □

Corollary 3

(cf. [11], 4.4). The dimensions (or lengths) of both cotorsion modules of the 1-dimensional local ring A are equal, that is, ${dim}_{\mathsf{k}}{\left({\omega }_A^0\right)}^t={dim}_{\mathsf{k}}{\left({\omega }_A^1\right)}^t$.

Remark 6.

In a similar manner, one can check that ${\left({\omega }_A^1\right)}^t\cong {Ext}_A^1({\left({\omega }_A^0\right)}^t,{\omega }_A^1)$ (see [11], 4.4). Indeed, applying ${Hom}_A(•,{\omega }_A^1)$ to the bottom row of the diagram (12), we obtain the exact sequence

$$0\to {Hom}_A({\omega }_A^0,{\omega }_A^1)⟶{\omega }_A^1⟶{Ext}_A^1({\left({\omega }_A^0\right)}^t,{\omega }_A^1)\to 0.$$

By definition, the leftmost module is isomorphic to ${Hom}_A({Hom}_A({\Omega }_A^1,{\omega }_A^1),{\omega }_A^1)$, while the rightmost one is isomorphic to ${Hom}_A({Hom}_A({\tilde{\Omega }}_A^1,{\omega }_A^1),{\omega }_A^1)\cong {\tilde{\Omega }}_A^1$. This gives us the exact sequence (14).

Remark 7.

It should be worthy to underline that ${\omega }_A^0\supseteq A$. Moreover, ${\omega }_A^0\ne A$ in general. In particular, ${\omega }_A^0$ contains all the so-called weakly holomorphic function on the 1-dimensional germ X corresponding to A. In other words, the module ${\omega }_X^0$ contains submodule ${\pi }_{*}\left({\mathcal{O}}_{\tilde{X}}\right)$ consisting of all meromorphic function germs on the curve X, which become holomorphic on its normalization $\pi :\tilde{X}\to X$.

Now we are able to compute the regular meromorphic index of vector fields given on Cohen–Macaulay curves in the following way.

To simplify notations, we will write ${\iota }_{\mathcal{V}}$ instead of ${\iota }_{\tilde{\mathcal{V}}}$. Then, the contracted complex $({\omega }_A^{•},{\iota }_{\mathcal{V}})$ of regular meromorphic differential forms on A

$$0⟶{\omega }_A^1\stackrel{{\iota }_{\mathcal{V}}}{⟶}{\omega }_A^0⟶0$$

is well defined (see Proposition 4).

Proposition 5.

Under assumptions and in notations of Proposition 4, the first homology group of the complex $({\omega }_A^{•},{\iota }_{\mathcal{V}})$ vanishes, that is,

$$H_1({\omega }_A^{•},{\iota }_{\mathcal{V}})=0.$$

Proof. 

By definition, ${\iota }_{\mathcal{V}}\left({\omega }_{\tilde{A}}^1\right)={\iota }_{\mathcal{V}}(\mathfrak{c}·{\omega }_A^1)=\mathfrak{c}·{\iota }_{\mathcal{V}}\left({\omega }_A^1\right)$. Since $\tilde{\mathcal{V}}\left(\mathfrak{c}\right)\subseteq \mathfrak{c}$, then $\tilde{\mathcal{V}}\left({\mathfrak{m}}_A\right)\subseteq {\mathfrak{m}}_A$ (cf. [29]), and, consequently, $Ker({\iota }_A:{\omega }_{\tilde{A}}^1\to {\omega }_{\tilde{A}}^0)=0$. On the other hand, the conductor$\mathfrak{c}$ is the maximal element of the set of ideals of A, which are ideals in the ring $\tilde{A}$ (the latter is a ring of principal ideals). It is easy to check that $\mathfrak{c}=(\theta )\tilde{A}$, where $\theta \in {\mathfrak{m}}_A$ is a non-zero divisor (see [28], (3.1.b)). Hence, $Ker({\iota }_{\mathcal{V}}:{\omega }_A^1\to {\omega }_A^0)=0$. □

Claim 2.

Under the same assumptions, let us additionally suppose that the vector field $\mathcal{V}$ has an isolated singularity on the germ $(X,\mathfrak{o})$. Then,

$${dim}_k{H}_0({\omega }_A^{•},{\iota }_{\mathcal{V}})={dim}_k{H}_0({\Omega }_A^{•},{\iota }_{\mathcal{V}}),$$

and this dimension is equal to ${dim}_{k}A/J_{\mathfrak{o}}\mathcal{V}$, where $J_{\mathfrak{o}}\mathcal{V}$ is the ideal of A generated by the coefficients of $\mathcal{V}$. As a result, we obtain a chain of identities:

$$\chi ({\omega }_A^{•},{\iota }_{\mathcal{V}})={dim}_k{H}_0({\omega }_A^{•},{\iota }_{\mathcal{V}})-{dim}_k{H}_1({\omega }_A^{•},{\iota }_{\mathcal{V}})={dim}_{k}A/J_{\mathfrak{o}}\mathcal{V}$$

that is,

${Ind}_{reg,\mathfrak{o}}\left(\mathcal{V}\right)={dim}_{k}A/J_{\mathfrak{o}}\mathcal{V}.$

Proof. 

The snake lemma for diagram (12) gives the long exact sequence

$$0\to Ker({\left({\omega }_A^1\right)}^t\to {\left({\omega }_A^0\right)}^t)⟶Coker\left({\iota }_{\mathcal{V}}\right)⟶Coker\left({\iota }_{\tilde{\mathcal{V}}}\right)\to Coker({\left({\omega }_A^1\right)}^t\to {\left({\omega }_A^0\right)}^t)\to 0.$$

The difference of lengths of the leftmost and rightmost modules of this sequence does not depend on the vector field $\mathcal{V}$; it is equal to the difference of dimensions (lengths) of ${\left({\omega }_A^1\right)}^t$ and ${\left({\omega }_A^0\right)}^t$, which is equal to zero (see [11], 4.4). Therefore, the dimensions of both modules in the middle of the sequence are also equal. Because these modules are concentrated at the singular point, they are finite-dimensional spaces of the same dimension. It remains to be remarked that $Coker\left({\iota }_{\tilde{\mathcal{V}}}\right)\cong H_0({\omega }_A^{•},{\iota }_{\tilde{\mathcal{V}}})$, while $Coker\left({\iota }_{\mathcal{V}}\right)\cong H_0({\Omega }_A^{•},{\iota }_{\mathcal{V}})\cong A/J_{\mathfrak{o}}\mathcal{V}$. This completes the proof. □

Now we will show how the index and the lengths of torsion modules are related. It is evident that ${\iota }_{\mathcal{V}}(Tors{\Omega }_A^1)=0$ and the diagram (12) yields the following exact sequence of contracted complexes:

$$0⟶(Tors{\widehat{\Omega }}_A^{•},{\iota }_{\mathcal{V}})⟶({\widehat{\Omega }}_A^{•},{\iota }_{\mathcal{V}})\stackrel{c_A}{⟶}({\omega }_A^{•},{\iota }_{\tilde{\mathcal{V}}})⟶{\left({\omega }_A^{•}\right)}^t⟶0.$$

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Claim 3.

Under the assumptions of Claim 2, there is the following relation:

${Ind}_{hom,\mathfrak{o}}\left(\mathcal{V}\right)={Ind}_{reg,\mathfrak{o}}\left(\mathcal{V}\right)-{dim}_{k}Tors{\Omega }_A^1,$

that is,

$${Ind}_{hom,\mathfrak{o}}\left(\mathcal{V}\right)={dim}_{k}A/J_{\mathfrak{o}}\mathcal{V}-{dim}_{k}Tors{\Omega }_A^1.$$

Proof. 

By Corollary 3, the Euler–Poincaré characteristic of the cotorsion complex ${\left({\omega }_A^{•}\right)}^t$ is equal to zero. Therefore, the exact sequence (16) yields the equality

$$\chi ({\widehat{\Omega }}_A^{•},{\iota }_{\mathcal{V}})=\chi ({\omega }_A^{•},{\iota }_{\mathcal{V}})-\chi (Tors{\widehat{\Omega }}_A^{•},{\iota }_{\mathcal{V}}).$$

On the other hand, it is clear that $\chi (Tors{\widehat{\Omega }}_A^{•},{\iota }_{\mathcal{V}})=-{dim}_{k}Tors{\Omega }_A^1$. Hence, the assertion of the Claim 2 completes the proof. □

As a result, the computation of homological index of a vector field $\mathcal{V}$ given on curves is reduced to the computation of dimension of torsion module of Kähler differentials and the quotient algebra $A/J_{\mathfrak{o}}\mathcal{V}$, whose dimension is equal to the regular meromorphic index of $\mathcal{V}$.

Remark 8.

It is well-known that for a reduced Gorenstein 1-dimensional germ, the dimension of the torsion module $Tors{\Omega }_A^1$ is equal to the so-called Tjurina number $\tau \left(A\right)$ of the germ. The latter is equal to the dimension of the space $T^1\left(A\right)$ of the first order infinitesimal deformations of the germ, or to the dimension of the Zariski tangent space to the base of its minimal versal deformation.

In the general case, the dimension of $Tors{\Omega }_A^1$ can be expressed in terms of the Noether and Dedekind differents (see [30]) with the aid of a formula due to R.Berger [31], there are also many papers where the torsion module of Kähler differentials is computed explicitly for various types of curves.

Corollary 4.

Under the assumptions of Claim 2, we have

${Ind}_{hom,\mathfrak{o}}\left(\mathcal{V}\right)={dim}_{k}A/J_{\mathfrak{o}}\mathcal{V}-\tau \left(A\right),$

so that in the graded case, the index of the Euler vector field satisfies the following equality:

$${Ind}_{hom,\mathfrak{o}}\left(\mathcal{E}\right)=1-\tau \left(A\right).$$

Proof. 

In the graded case $J_{\mathfrak{o}}\mathcal{E}={\mathfrak{m}}_A\subset A$. □

7. The Graded Case

Suppose that A is a $\mathbb{Z}$-graded local analytical algebra corresponding to a quasihomogeneous germ X. It is clear that in the graded case (under suitable finiteness conditions) the homological index is equal to the sum of integers each of which is equal to the alternating sum of dimensions of the corresponding graded pieces of homology groups, or, equivalently, to the alternating sum of dimensions of the corresponding graded pieces of all terms of the contracted de Rham complex (cf. [13]).

More precisely, let us define the generating function $G{H}_P$ of the homology of any decreasing bounded complex $({\mathcal{C}}_{•},\partial )=\left\{{\mathcal{C}}_i\right\},i\ge 0$, of graded A-modules as follows:

$$G{H}_P({\mathcal{C}}_{•};x)=\sum _{i\ge 0}{(-1)}^i\mathcal{P}(H_i\left({\mathcal{C}}_{•}\right);x),$$

where $\mathcal{P}(H_i\left({\mathcal{C}}_{•}\right);x)$ are Poincaré series of the corresponding homology groups. Evidently, if the differential ∂ has degree v, then

$$G{H}_P({\mathcal{C}}_{•};x)=\sum _{i\ge 0}{(-1)}^i{x}^{iv}\mathcal{P}({\mathcal{C}}_i;x);$$

if all homology groups are finite dimensional vector spaces, then

$$\chi \left({\mathcal{C}}_{•}\right)=G{H}_P({\mathcal{C}}_{•};1)=\left(G{H}_P({\mathcal{C}}_{•};x)\right){|}_{x=1}.$$

Let $A=P/I$ be a graded analytical algebra so that the ideal I is generated by a sequence of quasihomogeneous functions $f_1,\dots ,f_k$ of weighted degrees$d_1,\dots ,d_k$ in m variables $z_1,\dots ,z_m$ of weights $w_1,\dots ,w_m$, respectively. In other terms, the type of the singularity is equal to $(d_1,\dots ,d_k;w_1,\dots ,w_m)\in {\mathbb{Z}}^k\times {\mathbb{Z}}^m$. Then, all modules ${\Omega }_A^p$ and ${\omega }_A^p$, $p\ge 0$, as well as $Der\left(A\right)$ are endowed with a natural grading induced by relations $deg{f}_j=degd{f}_j=d_j$, $j=1,\dots ,k$, $deg{z}_i=degd{z}_i=w_i$, $i=1,\dots ,m$. In particular, the weighted degree of $\partial /\partial z_i$ is equal to $-w_i$, $i=1,\dots ,m$. Thus, if $\mathcal{V}$ is a homogeneous element of $Der\left(A\right)$, then its extension $\tilde{\mathcal{V}}$ does. In particular, all homology groups of complexes $({\Omega }_A^{•},{\iota }_{\mathcal{V}})$ and $({\omega }_A^{•},{\iota }_{\mathcal{V}})$ are graded vector spaces.

Claim 4.

Let A be a reduced graded 1-dimensional analytical algebra and $\mathcal{V}\in {Der}_k{\left(A\right)}_v$ be a $\mathsf{k}$-differentiation of weighted degree v. Then,

$${Ind}_{hom,\mathfrak{o}}\left(\mathcal{V}\right)=\left(\mathcal{P}({\Omega }_A^0;x)-x^v\mathcal{P}({\Omega }_A^1;x)\right){|}_{x=1},$$

$${Ind}_{hom,\mathfrak{o}}\left(\mathcal{V}\right)=\left(\mathcal{P}({\omega }_A^0;x)-x^v\mathcal{P}({\omega }_A^1;x)\right){|}_{x=1}-{dim}_{k}Tors{\Omega }_A^1.$$

Proof. 

Similarly to the proof of Claim 3. □

Recall now that if A is a 1-dimensional Gorenstein analytical k-algebra, then the dualizing module ${\omega }_A^1$ is free of rank 1, so that ${\omega }_A^1\cong A(\eta )$, where $\eta \in {\omega }_A^1$ is a generator of ${\omega }_A^1$. Hence, exact sequence (15) yields the following inclusion:

$$A⟶{Hom}_A({\Omega }_A^1,{\omega }_A^1)\cong {Der}_k\left(A\right).$$

The image of A does not depend on the generator $\eta$; it is denoted by $D_A$ and its elements are called trivial derivations (or, equivalently, trivial differentiations) of A over k. Then, $D_A$ is a free A-module of rank 1, and ${Der}_k\left(A\right)/D_A\cong J_A^{-1}/A$, where $J_A\cong c_A\left({\Omega }_A^1\right)$ is the Jacobi ideal of A. In this case, the local algebra A is graded if and only if $A/J_A$ or, equivalently, ${Der}_k\left(A\right)/D_A$ are cyclic A-modules (see [28]). The generator ${\mathcal{V}}_0$ of the latter module has a weighted degree equal to 0; it is called the Euler differentiation of A, or the Euler vector field and denoted often by $\mathcal{E}$.

Corollary 5.

Let A be a reduced graded 1-dimensional Gorenstein analytical algebra, and $\mathcal{V}\in {Der}_k{\left(A\right)}_v$ be a $\mathsf{k}$-differentiation of weighted degree v. Then,

$${Ind}_{hom,\mathfrak{o}}\left(\mathcal{V}\right)=\left(x^{-c}\mathcal{P}({Der}_k\left(A\right);x)-x^{v-c}\mathcal{P}(A;x)\right){|}_{x=1}-\tau \left(A\right),$$

where $c=-deg(\eta )$, $deg(\eta )$ is the weighted degree of the canonical generator η of the dualizing module ${\omega }_A^1$, and $\tau \left(A\right)={dim}_{\mathsf{k}}{T}^1\left(A\right)$ is the Tjurina number of A (see Remark 8). In particular, for the Euler vector field

$${Ind}_{hom,\mathfrak{o}}\left(\mathcal{E}\right)=x^{-c}\left(\mathcal{P}({Der}_k\left(A\right);x)-\mathcal{P}(A;x)\right){|}_{x=1}-\tau \left(A\right).$$

Proof. 

In virtue of the above definitions, there are natural isomorphisms of graded modules

$${\omega }_A^1\cong A[-c],{\omega }_A^0\cong {Hom}_A({\Omega }_A^1,{\omega }_A^1)\cong {Der}_k\left(A\right)[-c],$$

and the following identities for Poincaré polynomials:

$$\mathcal{P}({\omega }_A^1;x)=x^{-c}\mathcal{P}(A;x),\mathcal{P}({\omega }_A^0;x)=x^{-c}\mathcal{P}({Der}_k\left(A\right);x).$$

Moreover, it is well-known that ${dim}_{k}Tors{\Omega }_A^1=\tau \left(A\right)={dim}_k{T}^1\left(A\right)$ in the case of Gorenstein curves. As a result, Claim 4 yields the desired equality. □

Remark 9.

For completeness it should be also noted that for quasihomogeneous Gorenstein curves, there are equalities ${dim}_{\mathsf{k}}{\left({\omega }_A^1\right)}^t={dim}_{\mathsf{k}}({Der}_k\left(A\right)/D_A)=\mu \left(A\right)$. That is, the dimension (length) of the first cotorsion module is equal to the Milnor number of the germ (see [28], Satz 1). By definition, the Milnor number of an isolated singularity of dimension n is equal to the homotopy type of its Milnor fibre, which is homotopy equivalent to a wedge of n-spheres. More precisely, μ is equal to the number of n-spheres. Moreover, if A is a graded complete intersection, then Tjurina and Milnor numbers coincide, that is, $\tau \left(A\right)=\mu \left(A\right)$.

8. Normal Surfaces

In a similar way, it is possible to analyze graded normal two-dimensional singularities. As an illustration, we discuss a simple generalization of Claim 2 to higher dimensions.

Proposition 6.

Let A be the local algebra associated with a normal Cohen–Macaulay germ of dimension $n⩾2,$ and let $\mathcal{V}\in {Der}_{\mathsf{k}}\left(A\right)$ be a vector field with an isolated singularity. Then, there exists a natural isomorphism $H_0({\tilde{\Omega }}_A^{•},{\iota }_{\mathcal{V}})\cong H_0({\omega }_A^{•},{\iota }_{\tilde{\mathcal{V}}})$, where by ${\tilde{\Omega }}_A^{•}$ we denote the quotient complex ${\Omega }_A^{•}/Tors{\Omega }_A^{•}$.

Proof. 

Since A is normal, then ${\tilde{\Omega }}_A^0\cong {\Omega }_A^0\cong {\omega }_A^0\cong A.$ Next, there is the following commutative diagram with exact rows

$$\begin{array}{cccccccc}& 0& ⟶& {\tilde{\Omega }}_A^2& \stackrel{c_A}{⟶}& {\omega }_A^2& ⟶{\left({\omega }_A^2\right)}^t& ⟶0\\ & & & \downarrow {\iota }_{\mathcal{V}}& & \downarrow {\iota }_{\tilde{\mathcal{V}}}& \downarrow & \\ & 0& ⟶& {\tilde{\Omega }}_A^1& \stackrel{c_A}{⟶}& {\omega }_A^1& ⟶{\left({\omega }_A^1\right)}^t& ⟶0\\ & & & \downarrow {\iota }_{\mathcal{V}}& & \downarrow {\iota }_{\tilde{\mathcal{V}}}& \downarrow & \\ & 0& ⟶& {\Omega }_A^0\cong A& ⟶& {\omega }_A^0\cong A& ⟶0;\end{array}$$

(17)

it is similar to the diagram (12).

The snake lemma gives us the long exact sequence associated with the two bottom rows of diagram (17):

$$0\to Ker({\tilde{\Omega }}_A^1\to A)\to Ker({\omega }_A^1\to A)\to {\left({\omega }_A^1\right)}^t\to H_0({\tilde{\Omega }}_A^{•},{\iota }_{\mathcal{V}})\to H_0({\omega }_A^{•},{\iota }_{\tilde{\mathcal{V}}})\to 0.$$

(18)

It suffices to show that the dimensions of modules $H_0({\tilde{\Omega }}_A^{•},{\iota }_{\mathcal{V}})$ and $H_0({\omega }_A^{•},{\iota }_{\tilde{\mathcal{V}}})$ coincide. It is possible to prove this using an equivalent description of the regular meromorphic forms in terms of the Noether normalization and the trace map (see ([11], (2.1)) or [32])). □

Remark 10.

In the case of normal complete intersections of dimension $n⩾3$ the proof of Proposition 6 is easy, because for all $0⩽p<n-1$ there is an isomorphism between ${\Omega }_A^p$ and the bidual modules ${\Omega }_A^p\cong {{\Omega }_A^p}^{\vee \vee }$. In particular, both ${\Omega }_A^1$ and ${\Omega }_A^0$ have no torsion and cotorsion, so that the exact sequence (18) splits into two isomorphisms.

Claim 5.

Assume that A is a graded normal two-dimensional singularity, and $\mathcal{V}\in {Der}_{\mathsf{k}}\left(A\right)$ is a vector field with an isolated singularity. Then,

${Ind}_{hom,\mathfrak{o}}\left(\mathcal{V}\right)={Ind}_{reg,\mathfrak{o}}\left(\tilde{\mathcal{V}}\right)-{dim}_{\mathsf{k}}Tors{\Omega }_A^1+{dim}_{\mathsf{k}}Tors{\Omega }_A^2.$

Proof. 

It is well-known that a normal two-dimensional germ satisfies Serre’s conditions $R_1$ and $S_2.$ In particular, it is an isolated Cohen–Macaulay singularity. Hence, ${\omega }_A^p$ is well defined for all $p⩾0$. Next, such a germ may have only two non-trivial cotorsion modules in dimensions 1 and 2. Moreover, they are isomorphic: ${\left({\omega }_A^2\right)}^t\cong {\left({\omega }_A^1\right)}^t$ (see [11], (4.8), Bemerkung (1)). Making use of diagram (17), we complete the proof. □

Corollary 6.

Under the assumptions and in the notations of Claim 5, assume additionally that $\mathcal{V}$ is a vector field of weight $v\in \mathbb{Z}.$ Then,

$$\begin{array}{c}{Ind}_{hom,\mathfrak{o}}\left(\mathcal{V}\right)=\left(\mathcal{P}({\Omega }_A^0;x)-x^v\mathcal{P}({\Omega }_A^1;x)+x^{2v}\mathcal{P}({\Omega }_A^2;x)\right){|}_{x=1}\\ =\left(\mathcal{P}({\omega }_A^0;x)-x^v\mathcal{P}({\omega }_A^1;x)+x^{2v}\mathcal{P}({\omega }_A^2;x)\right){|}_{x=1}-{dim}_{\mathsf{k}}Tors{\Omega }_A^1+{dim}_{\mathsf{k}}Tors{\Omega }_A^2.\end{array}$$

Assertion 1.

Let A be a normal 2-dimensional local Gorenstein algebra, and let $\mathcal{V}$ be a vector field of weight $v\in \mathbb{Z}.$ Then, there is the following equality:

$$\begin{array}{c}{Ind}_{hom,\mathfrak{o}}\left(\mathcal{V}\right)=\left(\mathcal{P}(A;x)-x^{v-c}\mathcal{P}({Der}_{\mathsf{k}}\left(A\right);x)+x^{2v-c}\mathcal{P}(A;x)\right){|}_{x=1}\\ -{dim}_{\mathsf{k}}Tors{\Omega }_A^1+{dim}_{\mathsf{k}}Tors{\Omega }_A^2,\end{array}$$

where c is equal to the weight of the generator η of the dualizing module ${\omega }_A^2\cong A(\eta ).$

Proof. 

Under the above assumptions, there are natural isomorphisms:

$${\omega }_A^2\cong A[-c],{\omega }_A^1\cong {Hom}_A({\Omega }_A^1,{\omega }_A^2)\cong {Der}_{\mathsf{k}}\left(A\right)[-c],{\omega }_A^0\cong A,$$

and the following relations for Poincaré series:

$$\mathcal{P}({\omega }_A^2;x)=x^{-c}\mathcal{P}(A;x),\mathcal{P}({\omega }_A^1;x)=x^{-c}\mathcal{P}({Der}_{\mathsf{k}}\left(A\right);x),\mathcal{P}({\omega }_A^0;x)=\mathcal{P}(A;x).$$

This completes the proof. □

Example 3.

Let us compute the Poincaré polynomials of the homology groups $H_i({\omega }_A^{•},{\iota }_{\tilde{\mathcal{V}}})$ for a graded two-dimensional complete intersection of the type $(d_1,\dots ,d_{m-2};w_1,\dots ,w_m)$, $c=\sum d_j-\sum w_i.$

Since A is a normal complete intersection, then $Tors{\Omega }_A^1=0$, ${\omega }_A^1\cong {{\Omega }_A^1}^{\vee \vee }$ and ${\omega }_A^0\cong {{\Omega }_A^0}^{\vee \vee }\cong A$. Next, by ([33], Proposition 6.1), one has

$$\mathcal{P}({Der}_{\mathsf{k}}\left(A\right);x)=\mathcal{P}(A;x)+x^c\mathcal{P}(A;x)-x^c$$

and, consequently, for the Euler vector field, Assertion 1 implies

$${Ind}_{hom,\mathfrak{o}}\left(\mathcal{E}\right)=1+\tau \left(A\right),$$

(19)

because ${dim}_{\mathsf{k}}Tors{\Omega }_A^2=\tau \left(A\right)$ in virtue of the local duality for Gorenstein surfaces (cf. Corollary 5).

Furthermore, $\mathcal{P}({\omega }_A^1;x)=x^{-c}\mathcal{P}(A;x)+\mathcal{P}(A;x)-1.$ It is not difficult to see that $Ker({\iota }_{\tilde{\mathcal{V}}}:{\omega }_A^2\to {\omega }_A^1)=0$. Hence, ${\omega }_A^2\cong {\iota }_{\tilde{\mathcal{V}}}\left({\omega }_A^2\right)$ and we obtain

$$\mathcal{P}({\iota }_{\tilde{\mathcal{V}}}\left({\omega }_A^2\right);x)=x^v\mathcal{P}({\omega }_A^2;x)=x^{v-c}\mathcal{P}(A;x).$$

Next, set now $K_A^1=Ker({\iota }_{\tilde{\mathcal{V}}}:{\omega }_A^1\to {\omega }_A^0)$. Then, the exact sequence

$$0\to K_A^1⟶{\omega }_A^1\stackrel{{\iota }_{\tilde{\mathcal{V}}}}{⟶}A⟶A/{\iota }_{\tilde{\mathcal{V}}}\left({\omega }_A^1\right)\to 0$$

implies the following chain of relations

$$\begin{array}{c}\mathcal{P}(K_A^1;x)=\mathcal{P}({\omega }_A^1;x)-x^{-v}\mathcal{P}(A;x)+x^{-v}\mathcal{P}(A/{\iota }_{\tilde{\mathcal{V}}}\left({\omega }_A^1\right);x),\hfill \\ \mathcal{P}(K_A^1;x)=x^{-c}\mathcal{P}(A;x)+\mathcal{P}(A;x)-1-x^{-v}\mathcal{P}(A;x)+x^{-v}\mathcal{P}(A/{\iota }_{\tilde{\mathcal{V}}}\left({\omega }_A^1\right);x),\hfill \\ \mathcal{P}(K_A^1/{\iota }_{\tilde{\mathcal{V}}}\left({\omega }_A^2\right);x)=x^{-c}\mathcal{P}(A;x)+\mathcal{P}(A;x)-1-x^{-v}\mathcal{P}(A;x)\hfill \\ +x^{-v}\mathcal{P}(A/{\iota }_{\tilde{\mathcal{V}}}\left({\omega }_A^1\right);x)-x^{v-c}\mathcal{P}(A;x),\\ \mathcal{P}(H_1({\omega }_A^{•},{\iota }_{\tilde{\mathcal{V}}});x)=-1+(1+x^{-c}-x^{-v}-x^{v-c})\mathcal{P}(A;x)+x^{-v}\mathcal{P}(A/{\iota }_{\tilde{\mathcal{V}}}\left({\omega }_A^1\right);x),\hfill \end{array}$$

where the latter polynomial is, in fact, equal to $x^{-v}\mathcal{P}(A/J_{\mathfrak{o}}\mathcal{V};x)$ by Proposition 6 and $\mathcal{P}(A/J_{\mathfrak{o}}\mathcal{V};x)=\mathcal{P}(H_0({\Omega }_A^{•},{\iota }_{\mathcal{V}});x).$

9. The Intersection of Two Quadrics

Let X be the intersection of the following two homogeneous quadrics in ${\mathbb{C}}^4$:

$$\left\{\begin{array}{c}{z}_1^2+z_2^2+z_3{z}_4=0,\hfill \\ z_1{z}_2+z_3^2+z_4^2=0.\hfill \end{array}\right.$$

Indeed, it is the germ of the normal surface that is a graded complete intersection of type $(2,2;1,1,1,1)$ (see [13,14,34]). Then, $c=0$, and we have the following identities (see [33]):

$$\begin{array}{cc}\hfill \mathcal{P}(A;x)& ={(1-x^2)}^2/{(1-x)}^4={(1+x)}^2/{(1-x)}^2,\hfill \\ \hfill \mathcal{P}({Der}_{\mathsf{k}}\left(A\right);x)& =2\mathcal{P}(A;x)-1=(1+6x+x^2)/{(1-x)}^2,\hfill \\ \hfill \mathcal{P}(H_1({\omega }_A^{•},{\iota }_{\mathcal{V}});x)& =-1+(2-x^{-v}-x^v)\mathcal{P}(A;x)+x^{-v}\mathcal{P}(A/J_{\mathfrak{o}}\mathcal{V};x),\hfill \\ \hfill \mathcal{P}(Tors\left({\Omega }_A^2\right);x)=\mathcal{P}(H_{\mathfrak{m}}^0\left({\Omega }_A^2\right);x)& =-1+\mathcal{P}(A;x){res}_{t=0}{t}^{-3}{(1+t)}^{-1}{(1+tx)}^4/(1+t{x}^2)\hfill \\ \hfill & =-1+{(1+x)}^2(1-2x+3x^2)=4x^3+3x^4,\hfill \end{array}$$

so that ${dim}_{\mathsf{k}}Tors\left({\Omega }_A^2\right)=7.$

Because the weight of the Euler vector field is equal to zero, then

$$\mathcal{P}(A/J_{\mathfrak{o}}\mathcal{E};1)=\mathcal{P}(H_0({\omega }_A^{•},{\iota }_{\tilde{\mathcal{E}}});1)=1$$

since $J_{\mathfrak{o}}\mathcal{E}={\mathfrak{m}}_A$. As a result, we obtain

$$\mathcal{P}(H_1({\omega }_A^{•},{\iota }_{\tilde{\mathcal{E}}});x)=-1+\mathcal{P}(A/J_{\mathfrak{o}}\mathcal{E};x)=0,\chi ({\omega }_A^{•},{\iota }_{\tilde{\mathcal{E}}})=1,$$

$${Ind}_{hom,\mathfrak{o}}\left(\mathcal{E}\right)=\chi ({\widehat{\Omega }}_A^{•},{\iota }_{\mathcal{E}})=1+\tau \left(A\right)=8$$

(cf. the relation (19)). Next, for any vector field of weight $v=1$ we have $\mathcal{P}(A/J_{\mathfrak{o}}\mathcal{V};x)=1+4x+4x^2$. Hence,

$$\begin{array}{c}\mathcal{P}(H_1({\omega }_A^{•},{\iota }_{\tilde{\mathcal{V}}});x)=-1+(2-x^{-1}-x){(1+x)}^2/{(1-x)}^2+x^{-1}\mathcal{P}(A/J_{\mathfrak{o}}\mathcal{V};x)\hfill \\ =-1-x^{-1}{(1+x)}^2+x^{-1}+4+4x=-(x^{-1}+3+x)+x^{-1}+4+4x=1+3x.\hfill \end{array}$$

We conclude immediately that

$${Ind}_{reg,\mathfrak{o}}\left(\tilde{\mathcal{V}}\right)=\chi ({\omega }_A^{•},{\iota }_{\tilde{\mathcal{V}}})=9-4=5$$

and, consequently,

$${Ind}_{hom,\mathfrak{o}}\left(\mathcal{V}\right)={Ind}_{reg,\mathfrak{o}}\left(\tilde{\mathcal{V}}\right)+{dim}_{\mathsf{k}}Tors{\Omega }_A^2=5+7=12.$$

For completeness, one can also analyze the homology of the cotorsion complex ${\left({\omega }_A^{•}\right)}^t$. In fact, two nontrivial terms of the cotorsion complex have equal dimensions. Moreover, both modules are isomorphic, in view of the usual isomorphisms ${\left({\omega }_A^1\right)}^t\cong H_{\mathfrak{m}}^1\left({\Omega }_A^1\right)$ and ${\left({\omega }_A^2\right)}^t\cong H_{\mathfrak{m}}^1\left({\Omega }_A^2\right)\cong H_{\mathfrak{m}}^1\left({\Omega }_A^1\right)$ induced by the local Grothendieck duality. That is why one can use general formulas for the Poincaré polynomials of the local cohomology groups (see [33], (3.2)).

In addition, one can also compute the Poincaré series of both cotorsion modules directly as follows. Similar to (12), there is a commutative diagram

$$\begin{array}{cccccccc}0⟶& Tors{\Omega }_A^2& ⟶& {\Omega }_A^2& \stackrel{c_A}{⟶}& {\omega }_A^2& ⟶{\left({\omega }_A^2\right)}^t& ⟶0\\ & & & \downarrow {\iota }_{\mathcal{V}}& & \downarrow {\iota }_{\tilde{\mathcal{V}}}& \downarrow & \\ & 0& ⟶& {\Omega }_A^1& \stackrel{c_A}{⟶}& {\omega }_A^1& ⟶{\left({\omega }_A^1\right)}^t& ⟶0,\end{array}$$

(20)

The two rows of the diagram imply the identities

$$\begin{array}{c}\mathcal{P}({\left({\omega }_A^1\right)}^t;x)=\mathcal{P}({\omega }_A^1;x)-\mathcal{P}({\Omega }_A^1;x),\hfill \\ \mathcal{P}({\left({\omega }_A^2\right)}^t;x)=\mathcal{P}({\omega }_A^2;x)-\mathcal{P}({\Omega }_A^2;x)+\mathcal{P}(Tors\left({\Omega }_A^2\right);x)),\hfill \end{array}$$

which yield a chain of relations

$$\begin{array}{cc}\hfill \mathcal{P}({\left({\omega }_A^1\right)}^t;x)& =2\mathcal{P}(A;x)-1-\mathcal{P}({\Omega }_A^1;x)\hfill \\ \hfill & =\mathcal{P}(A;x)(2-4x+2x^2)-1=1+4x+2x^2,\hfill \\ \hfill \mathcal{P}({\left({\omega }_A^2\right)}^t;x)& =\mathcal{P}(A;x)\{1-(6x^2-8x^3+3x^4)\}+(4x^3+3x^4)\hfill \\ \hfill & ={(1+x)}^2(1-x)(1+3x)+(4x^3+3x^4)=1+4x+2x^2.\hfill \end{array}$$

Again, for the Euler vector field, the homology groups of the cotorsion complex ${\left({\omega }_A^{•}\right)}^t$ vanish. If $v=1,$ then

$$\mathcal{P}(Ker({\iota }_{\mathcal{V}}:{\left({\omega }_A^2\right)}^t\to {\left({\omega }_A^1\right)}^t);x)=2x+2x^2;\mathcal{P}(Coker({\iota }_{\mathcal{V}}:{\left({\omega }_A^1\right)}^t\to {\left({\omega }_A^0\right)}^t);x)=1+3x,$$

so that ${dim}_{\mathsf{k}}{H}_2\left({\left({\omega }_A^{•}\right)}^t\right)={dim}_{\mathsf{k}}{H}_1\left({\left({\omega }_A^{•}\right)}^t\right)=4.$ Next, if $v=2$ then

$$\mathcal{P}(Ker({\iota }_{\mathcal{V}}:{\left({\omega }_A^1\right)}^t\to {\left({\omega }_A^0\right)}^t);x)=4x+2x^2;\mathcal{P}(Coker({\iota }_{\mathcal{V}}:{\left({\omega }_A^1\right)}^t\to {\left({\omega }_A^0\right)}^t);x)=1+4x+x^2,$$

so that ${dim}_{\mathsf{k}}{H}_2\left({\left({\omega }_A^{•}\right)}^t\right)={dim}_{\mathsf{k}}{H}_1\left({\left({\omega }_A^{•}\right)}^t\right)=6.$ Finally, for all $v⩾3$ one has

$${dim}_{\mathsf{k}}{H}_2\left({\left({\omega }_A^{•}\right)}^t\right)={dim}_{\mathsf{k}}{H}_1\left({\left({\omega }_A^{•}\right)}^t\right)=7,$$

so that ${\iota }_{\tilde{\mathcal{V}}}$ is a zero homomorphism!

Similarly to Claim 5, one can easily compute the index of differential forms on normal surfaces (see (4) in Section 2).

Proposition 7.

Let A be the local algebra corresponding to a graded reduced normal surface singularity. Assume that $\omega \notin Tors{\Omega }_A^1.$ Then,

${Ind}_{hom,\mathfrak{o}}(\omega )={Ind}_{reg,\mathfrak{o}}(\omega )-{dim}_{\mathsf{k}}Tors{\Omega }_A^1+{dim}_{\mathsf{k}}Tors{\Omega }_A^2.$

Proof. 

Indeed, such germs are Cohen–Macaulay. It remains to use the diagram (20) and the equality ${dim}_{\mathsf{k}}{\left({\omega }_A^1\right)}^t={dim}_{\mathsf{k}}{\left({\omega }_A^2\right)}^t$ (see [11], (4.8), Bemerkung (1)). □

Remark 11.

If ω has an isolated singularity, then $\chi ({\widehat{\Omega }}_A^{•},\wedge \omega )={dim}_{\mathsf{k}}{H}^2({\widehat{\Omega }}_A^{•},\wedge \omega )$. Moreover, the local Grothendieck duality implies that $Tors{\Omega }_A^1\cong {Ext}_A^2({\Omega }_A^1,{\omega }_A^2).$ In particular, for any 2-dimensional Gorenstein germ, we have ${dim}_{\mathsf{k}}Tors{\Omega }_A^1={dim}_{\mathsf{k}}{T}^2\left(A\right),$ where $T^2\left(A\right)$ is the second cotangent cohomology of the germ.

10. Monomial Curves

Analogously to the previous discussion one can easily compute the index in other similar situations.

Example 4.

Assume that $P=\mathbb{C}[z_1,\cdots ,z_m]$, $\{f_1,\cdots ,f_{m-1}\}$ is a regular sequence and $A=P/I$. Then, A is a complete intersection of dimension 1. Suppose that A is graded of type $(d_1,\dots ,d_{m-1};w_1,\dots ,w_m)$. Then, $\eta =d{z}_1\wedge \dots \wedge d{z}_m/d{f}_1\wedge \dots \wedge d{f}_{m-1}$ is a generator of the dualizing module ${\omega }_A^1$ and $c=-deg(\eta )=\sum d_j-\sum w_i$. There are the following identities (see [33], (6.1), (6.4)):

$$\begin{array}{c}\mathcal{P}(A;x)=\prod (1-x^{d_j})/\prod (1-x^{w_i}),\hfill \\ \mathcal{P}(Tors{\Omega }_A^1;x)=1+\mathcal{P}(A;x)·{res}_{t=0}{t}^{-2}{(1+t)}^{-1}\prod (1+t{x}^{w_i})/\prod (1+t{x}^{d_j}),\hfill \\ {dim}_{k}Tors{\Omega }_A^1=\mu \left(A\right)=\tau \left(A\right),\hfill \\ \mathcal{P}({\omega }_A^0;x)=x^{-c}P({Der}_k\left(A\right);x)=1+x^{-c}\mathcal{P}(A;x),\hfill \\ \mathcal{P}({\omega }_A^1;x)=x^{-c}P(A;x),\mathcal{P}({\left({\omega }_A^0\right)}^t;x)=\mathcal{P}({\left({\omega }_A^1\right)}^t;x)=x^{-1}+3+x.\hfill \end{array}$$

As a result, we obtain

$${Ind}_{hom,\mathfrak{o}}\left(\mathcal{V}\right)=\left(1+(x^{-c}-x^{v-c})P(A;x)\right){|}_{x=1}-\tau \left(A\right),$$

$${Ind}_{hom,\mathfrak{o}}\left(\mathcal{E}\right)=1-\tau \left(A\right).$$

Thus, if A is the intersection of two homogeneous quadrics in 3-dimensional space of type $(2,2;1,1,1)$, we have $\mathcal{P}(Tors{\Omega }_A^1;x)=3x^2+2x^3$, i.e., $\tau \left(A\right)=\mu \left(A\right)=5$, and

$${Ind}_{hom,\mathfrak{o}}\left(\mathcal{V}\right)=(1+x^{-1}{(1+x)}^2(1+x+\dots +x^{v-1})){|}_{x=1}-5=4(v-1),$$

and so on.

Remark 12.

It is very useful to compare this identity with the corresponding expression in [13], where the general case of quasihomogeneous complete intersections was examined by a different method.

Example 5.

It is not difficult to prove that every irreducible component of a quasihomogeneous curve has a monomial parametrization; in other terms, these components are monomial curves. For any monomial curve, let us denote by $H\subseteq \mathbb{N}$ its value or, equivalently, numerical semigroup, so that the dual analytical algebra is generated by monomials $t^h,h\in H$, that is, $A\cong k\langle H\rangle$. Then,

$$\mathcal{P}(A;x)=\sum _{\nu \in H}{x}^{\nu }=1+\sum _{\nu \in H_{+}}{x}^{\nu },\mathcal{P}(Der\left(A\right);x)=\sum _{\nu \in End\left(H\right)}{x}^{\nu }=1+x^{\mathfrak{c}}+\dots ,$$

where $H_{+}=H\setminus \left\{0\right\}$, $End\left(H\right)=\{\nu \in \mathbb{N}:\nu +H_{+}\subset H\}$, so that $H\subseteq End\left(H\right)$ and $\mathfrak{c}=Ann(\tilde{A}/A)$ is the conductor of A in $\tilde{A}$ (see [35]). For Gorenstein monomial curves $[End(H):H]=1$, and ${dim}_{\mathsf{k}}\mathfrak{c}=c$. Then, Corollary 5 implies the following relation:

$${Ind}_{hom,\mathfrak{o}}\left(\mathcal{E}\right)=1-\tau \left(A\right)$$

for the index of the Euler vector field $\mathcal{E}$ on any monomial Gorenstein curve.

One can also explicitly compute the dimensions of all graded components of the first cotangent cohomology $T^1\left(A\right)\cong {Ext}_A^1({\Omega }_A^1,A)$ in terms of the value semigroup H (see, e.g., [35]).

Example 6.

Now let A be a Cohen–Macaulay curve of codimension two (cf. [36]). Then, $A=P/I$ and the classical Hilbert–Burch theorem asserts that the ideal I is generated by the minors of maximal order of some $(n+1)\times n$-matrix N, such that there is an exact sequence

$$0⟶P^n\stackrel{N}{⟶}{P}^{n+1}⟶P⟶A⟶0.$$

Under our assumptions, the height of the ideal I equals two. Hence, we have

$${Hom}_P(A,P)={Ext}_P^1(A,P)=0.$$

Therefore the dual sequence turns into a free resolvent of the canonical module $K_A={Ext}_P^2(A,P)\cong {\omega }_A^1$ of the germ A:

$$0⟶P^{\vee }⟶{P^{n+1}}^{\vee }\stackrel{N^{\vee }}{⟶}{P^n}^{\vee }⟶K_A⟶0,$$

where $M^{\vee }={Hom}_P(M,P)$ for any A-module M, and $N^{\vee }$ is the transposed matrix. Taking into account the natural grading of all modules, it is possible to compute the Poincaré series $\mathcal{P}({\omega }_A^1;x)$ in terms of weighted degrees of entries of N, and to write out an explicit expression for the index similarly to the above example.

11. Non-Normal Surfaces

Now we will continue the analysis of Example 2 in Section 3 and show how to compute the homological index by elementary calculations. In this case, the surface germ $X\subset ({\mathbb{C}}^4,0)$ is isomorphic to the coproduct of two copies of the complex plane over the origin. Let us denote the local coordinates of the ambient space by $\{x,y,z,u\}$. Then, $A=\mathbb{C}\langle x,y\rangle {\times }_{\mathbb{C}}\mathbb{C}\langle z,u\rangle$ is the dual analytic algebra of the germ X, that is, $A\cong P/I$, where $P=\mathbb{C}\langle x,y,z,u\rangle$ and the ideal $I=(xz,xu,yz,yu)$ is prime. It is clear that X is an isolated surface singularity. As already remarked, the germ X is non-normal because its dual local ring A is not Cohen–Macaulay. In view of ([27], Lemma 3.5), the germ X has no non-trivial (flat) deformations, that is, the first cotangent cohomology of X vanishes, i.e., $T^1\left(X\right)=0$ or, equivalently, $\tau \left(X\right)=0$. In the standard terminology, the germ X is called rigid.

Using evident symmetries, one can construct a free resolution of the ideal I over the ambient ring P:

$$0⟶P\stackrel{{\phi }_2}{⟶}{P}^4\stackrel{{\phi }_1}{⟶}{P}^4\stackrel{{\phi }_0}{⟶}I⟶0,$$

where ${\phi }_0={[f_1,f_2,f_3,f_4]}^{\vee }$, $f_1=xz$, $f_2=xu$, $f_3=yz$, $f_4=yu$, and the transposed matrix

$${\phi }_1=\left[\begin{array}{cccc}-u& 0& -y& 0\\ z& 0& 0& -y\\ 0& -u& x& 0\\ 0& z& 0& x\end{array}\right]$$

is determined by the four syzygies of the first order between the generators $f_1,f_2,f_3,f_4$. In fact, there is one syzygy ${\phi }_2=[-y,x,u,-z]$ of the second order only. In the usual homogeneous grading, we have $deg\left({\phi }_0\right)=2$, $deg\left({\phi }_1\right)=deg\left({\phi }_2\right)=1$ and $\mathcal{P}(P;t)=1/{(1-t)}^4$. Hence, $\mathcal{P}(I;t)=(4t^2-4t^3+t^4)/{(1-t)}^4$ and

$$\mathcal{P}(A;t)=1/{(1-t)}^4-\mathcal{P}(I;t)=(1+2t-t^2)/{(1-t)}^2.$$

Since $f_1{f}_4=f_2{f}_3=xyzu$, the square $I^2$ of the ideal I is generated by nine elements $f_1^2$, $f_2^2$, $f_3^2$, $f_4^2$, $f_1{f}_2$, $f_1{f}_3$, $f_1{f}_4$, $f_2{f}_4$, $f_3{f}_4$. Then, we can derive 12 syzygies of the first order from the matrix ${\phi }_1$ between all subsets of the nine generators with common factors, and so on.

As a result, we obtain

$$0⟶P^4\stackrel{{\psi }_2}{⟶}{P}^{12}\stackrel{{\psi }_1}{⟶}{P}^9\stackrel{{\psi }_0}{⟶}{I}^2⟶0,$$

so that $deg{\psi }_0=4$, $deg{\psi }_1=deg{\psi }_2=1$. Hence, $\mathcal{P}(I^2;t)=t^4(9-12t+4t^2)/{(1-t)}^4$.

In our case (see [32], Section 6), the first fundamental exact sequence takes the form

$$0⟶I/I^2⟶{\Omega }_A^1{\otimes }_{A}P⟶{\Omega }_A^1⟶0,$$

which implies the identity

$$\mathcal{P}({\Omega }_A^1;t)=4t\mathcal{P}(A;t)-\mathcal{P}(I;t)+\mathcal{P}(I^2;t)=4t(1+t-2t^2+t^3)/{(1-t)}^2.$$

In order to compute $\mathcal{P}({\Omega }_A^2;t)$, we will use the following simple trick. First, recall that for positively graded isolated singularities, the contracted de Rham complex associated with the Euler vector field $\mathcal{E}={\mathcal{V}}_0$

$$({\Omega }_A^{•},{\iota }_{\mathcal{E}}):0⟶{\Omega }_A^4\stackrel{{\iota }_{\mathcal{E}}}{⟶}{\Omega }_A^3\stackrel{{\iota }_{\mathcal{E}}}{⟶}{\Omega }_A^2\stackrel{{\iota }_{\mathcal{E}}}{⟶}{\Omega }_A^1\stackrel{{\iota }_{\mathcal{E}}}{⟶}A⟶0,$$

is acyclic in all positive dimensions and $H_0({\Omega }_A^{•},{\iota }_{\mathcal{E}})\cong \mathbb{C}$.

This implies the identity ${\sum }_{p⩾0}{(-1)}^p\mathcal{P}({\Omega }_A^p;t)=1$. Next, easy calculations show that the module ${\Omega }_A^4$ is generated by the differential form $dx\wedge dy\wedge dz\wedge du$, so that $\mathcal{P}({\Omega }_A^4;t)=t^4$. Similarly, the module ${\Omega }_A^3$ is generated by four 3-forms $dx\wedge dy\wedge dz$, $dx\wedge dy\wedge du$, $dx\wedge dz\wedge du$, $dy\wedge dz\wedge du$ and ${\iota }_E(dx\wedge dy\wedge dz\wedge du)$, that is, $\mathcal{P}({\Omega }_A^3;t)=4t^3+t^4$. Since

$$\mathcal{P}({\Omega }_A^2;t)=1-\mathcal{P}(A;t)+\mathcal{P}({\Omega }_A^1;t)+\mathcal{P}({\Omega }_A^3;t)-\mathcal{P}({\Omega }_A^4;t),$$

we obtain

$$\mathcal{P}({\Omega }_A^2;t)=2t^2(3-2t-2t^2+2t^3)/{(1-t)}^2.$$

In particular, for a differential form of weight 1 we obtain

$${Ind}_{hom,\mathfrak{o}}(\omega )=(\mathcal{P}(A;t)-t^{-1}\mathcal{P}({\Omega }_A^1;t)+t^{-2}\mathcal{P}({\Omega }_A^2;t)){|}_{t=1}=3,$$

while for any vector field ${\mathcal{V}}_0$ on X of weighted degree zero (not necessarily the Euler vector field), we obtain the following expression for the index:

$${Ind}_{hom,\mathfrak{o}}\left({\mathcal{V}}_0\right)=(\mathcal{P}(A;t)-\mathcal{P}({\Omega }_A^1;t)+\mathcal{P}({\Omega }_A^2;t)){{|}_{t=1}=(1+4t^3)|}_{t=1}=5.$$

In conclusion, it should be remarked that one can compute the Poincaré polynomial $\mathcal{P}(H_1({\Omega }_A^{•},{\iota }_{{\mathcal{V}}_0});t)$ in a similar manner with the use of two relations $\mathcal{P}(Tors{\Omega }_A^1;t)=4t^2$ and $\mathcal{P}(Tors{\Omega }_A^2;t)=8t^3$. In particular, these relations imply an analog of the relation of Assertion 1 for the Euler vector field

${Ind}_{hom,\mathfrak{o}}\left(\mathcal{E}\right)=1-{dim}_{\mathsf{k}}Tors{\Omega }_A^1+{dim}_{\mathsf{k}}Tors{\Omega }_A^2$

instead of the equality ${Ind}_{hom,\mathfrak{o}}\left(\mathcal{E}\right)=1+\tau \left(A\right)$, which is valid for normal surfaces (see the relation (19)). As remarked in the beginning of this section in the case under consideration, one has $\tau \left(A\right)=0$.

Analogously one can compute the index of vector fields and differential forms given on Gorenstein curves of codimension 3, on cones over zero-dimensional projective schemes, on almost complete intersections, and many others.

12. Conclusions

We discuss a few simple methods for computing the local topological index and its various analogs for vector fields and differential forms given on complex varieties with singularities of different types. Our methods are based on basic properties of contravariant versions of the classical Poincaré–de Rham complex, on the theory of regular meromorphic and logarithmic differential forms, on properties of the dualizing (canonical) module and related constructions. In particular, we introduce several types of local indices such as logarithmic, multi-logarithmic, meromorphic and regular meromorphic, discuss their interconnections with homological index and other known concepts and notions. The key point of our approach lies in the fact that computation of the homology of the contracted de Rham complex can be simplified essentially in concrete situations with the aid of complexes of differential forms of distinct kinds such as holomorphic, logarithmic, regular meromorphic, and weakly holomorphic (or even square integrable), which are used in the theory of residue and duality. Shortly speaking, the main idea of our approach is the following: for efficient calculation it is necessary to choose an appropriate complex of sheaves or modules. Anyway, in all considered cases, the homological index and its analogues have transparent interpretations and useful applications. As an illustration, we also discuss a number of specific examples and show how to calculate all the indices on Cohen–Macaulay, Gorenstein and monomial curves, on normal and non-normal surfaces and some others. In contrast with many works on this subject, we use neither computers, nor integration, deformations, spectral sequences and other related traditional tools of pure mathematics. This gives a real possibility to investigate some problems that arose in mechanics, physics and technical sciences through almost elementary ways, not resorting to complicated mathematical constructions, calculations, etc.