The Poincaré Index and Its Applications

Abstract

We discuss an approach to the problem of calculating the local topological index of vector fields given on complex spaces or varieties with singularities developed by the author over the past few years. Our method is based on the study of the homology of a contravariant version of the classical Poincaré–de Rham complex. This idea allows not only simplifying the calculations, but also clarifying the meaning of the basic constructions underlying many papers on the subject. In particular, in the graded case, the index can be expressed explicitly in terms of the elementary symmetric polynomials. We also considered some useful applications in physics, mechanics, control theory, the theory of bifurcations, etc.

1. Introduction

In his famous book [1] Poincaré argues that mathematical theories are not intended to reveal to us the true nature of things, and their only purpose is to reconcile the physical laws that we have learned from experience, but which, without the help of mathematics, we could not even formulate. In his opinion, the most important goal of the coming Twentieth Century is to create the mathematics necessary for the future development of quantum physics and the theory of relativity. For the sake of completeness, it is interesting to note that even some very contradictory philosophical judgments of Poincaré turned out to be very fruitful in the development of the humanities; they gave a powerful impetus to Lenin to write (in 1908) a fundamental critical work outlining the foundations of the dialectical method and its application in the epistemology of the universe, where the name of Poincaré was mentioned more than 40 times (see [2]).

Indeed, in the 20th Century, Poincaré’s ideas were developed in various directions such as topology, geometry, the theory of differential equations, mathematical physics, mechanics, the ergodic theory of the chaos and of dynamical systems, and many others. Moreover, his ideas led to unprecedented progress in almost all areas of the natural sciences, including pure mathematics not necessarily related directly to other applied sciences.

Among other things, he invented powerful geometrical methods to understand “qualitatively” the behavior of solutions of ordinary differential equations. Questions such as the existence of closed integral curves, the appearance of limit cycles (closed phase curves), and the counting of the number of such cycles first appeared in Poincaré’s works on celestial mechanics and cosmogony, devoted to the study of the evolution of the solar system, its origin and stability, the rest of the universe and its ultimate fate, and so on (see [3,4]). It served as a starting point for his research on topology and geometry and helped to develop original and powerful methods of investigation. In fact, Poincaré was also the first to apply differential calculus and geometric methods, proposing rigorous mathematical analysis of related problems in various fields of science and technology. For example, it is well known that the qualitative theory of differential equations is a powerful tool for studying a wide range of problems arising in theoretical and applied physics, classical mechanics, chemistry, biology, and other areas of the natural sciences that can be described using differential systems. Thus, with his ideas, Poincaré foreshadowed many of the results that were subsequently obtained by other researchers, often even without mentioning the name of the pathbreaker. Moreover, his basic concepts have been rediscovered many times, again and again (see [5,6]). For instance, the notion of saddle-node bifurcation has become popular and has recently acquired various unusual new names. Poincaré was not only the founder of qualitative methods for the analysis of differential equations, but also began to study dissipative dynamical systems using methods other than those of Lagrangian dynamics (a topic to which he also contributed many outstanding ideas).

Among his creations, the knowledge of the topological index of a stationary point (i.e., an equilibrium point) of a vector field significantly simplifies the study of many problems of nonlinear analysis and the theory of dynamical systems. First of all, this concerns various existence, uniqueness, and non-uniqueness theorems for solutions, estimates, and the exact determination of the number of solutions, the study of the convergence of some approximate methods, the study of bifurcation points, and many other questions of theoretical and applied problems in the natural sciences (see [7,8,9,10]). Furthermore, as an application to a technique, Poincaré established (see [11]) that a necessary condition for the existence of a stable regime of maintained oscillations in the singing arc (a device, completely analogous to the triode, which was used in wireless telegraphy to generate electromagnetic waves) is the presence in the phase plane of a stable limit cycle. Similarly, one can explain the action of the memristor and other devices commonly used in modern technology (see [12,13]).

In this note, we discuss an approach to the problem of calculating the local topological index of vector fields given on complex spaces or varieties with singularities developed by the author in the past few years. The method is based on the study of the homology of a contravariant version of the classical Poincaré–de Rham complex. This idea allows one not only to simplify the calculations, but also to clarify the meaning of the basic constructions underlying many papers on the subject. In particular, in the graded case, the index can be expressed explicitly in terms of the elementary symmetric polynomials. We also considered some useful applications in physics, mechanics, control theory, the theory of bifurcations, etc.

2. Dynamical Systems and Nonlinear Self-Oscillations

First, we recall that one of the best-known problems investigated by Poincaré is the analysis of the system of differential equations:

$$\frac{dx}{dt}=P(x,y),\frac{dy}{dt}=Q(x,y),$$

(1)

where $P(x,y)$ and $Q(x,y)$ are continuous functions, given on an open domain G of the Euclidean plane ${\mathbb{R}}^2$ with coordinates x and y and having continuous partial derivatives in the domain. It should be remarked that such systems, whose right parts do not depend on the parameter t in an explicit way, are often called autonomous or dynamical systems on the plane.

In 1929, Soviet mathematician A. Andronov showed that these equations describe the most simple case of self-oscillations occurring in mechanics, as well as in physics in various systems with one degree of freedom; in chemical reactions between two substances; in biology when two types or kinds of animals coexist. In this case, stationary (steady-state) solutions of such a system may have two types: constant or periodic in t (see [14]).

Moreover, on the basis of studying actually observed phenomena of this type, it is required that the considered periodic motions be stable with respect to sufficiently small changes in the initial conditions and second elements of Equation (1). It is easy to show that isolated closed curves in the $xy$ plane correspond to periodic motions satisfying these conditions, to which neighboring solutions approach along a spiral from the inside or outside (with increasing t). As a result, self-oscillations occurring in the systems described by equations of type (1) correspond to stable Poincaré limit cycles.

3. The Poincaré Index of Vector Fields

It is not difficult to see that at each point of the domain G, one can define the vector whose components are values of the functions $P(x,y)$ and $Q(x,y)$ so that the corresponding dynamical system determines the vector field:

$$\mathcal{V}=P(x,y)\frac{\partial }{\partial x}+Q(x,y)\frac{\partial }{\partial y}.$$

If at least one coefficient of the field does not vanish at some point of the domain G, then the length of the vector $r=\sqrt{P^2+Q^2}$ at this point differs from zero. It is clear that the angle $\vartheta$ between the axis x and the vector at this point (in the standard orientation) is given by the following expressions:

$$sin\vartheta =\frac{Q}{\sqrt{P^2+Q^2}},cos\vartheta =\frac{P}{\sqrt{P^2+Q^2}}.$$

Otherwise, if $P(x,y)=Q(x,y)=0$, then the direction of the vector is undefined and the corresponding point is called the singular point of the vector field or the equilibrium point of the system.

Let us now assume that the boundary of the domain G is a smooth closed curve $\gamma =\partial G$, that is $\gamma =\gamma (t)$, $t\in [0,2\pi ]$, $\gamma (0)=\gamma (2\pi )$, and the vector field $\mathcal{V}$ has no singular points on the curve, that is $\mathcal{V}(g)\ne 0$ for all $g\in \gamma$. One can regard the field $\mathcal{V}$ on the boundary as a periodical vector-function $\mathcal{V}(t)=\mathcal{V}(\gamma (t))$, depending on the parameter t on the curve $\gamma$. By definition, the Poincaré index of the vector field $\mathcal{V}$ on the boundary $\partial G$ is the number of rotations of the vector $\mathcal{V}(t)$, bypassing along the boundary when the parameter t varies from 0 till $2\pi$. In fact, it is the index of the curve $\gamma$, which can be represented as follows (see [3]):

$$Ind\gamma =\frac{1}{2\pi }{\oint }_{\gamma }d\phi ,$$

(2)

where:

$$\phi =\mathrm{arctg}\frac{Q}{P},d\phi =\frac{QdP-PdQ}{P^2+Q^2}.$$

In addition, if the vector field $\mathcal{V}$ has no singular points on $G\backslash \mathfrak{o}$, then $Ind\gamma$ is called the Poincaré index of $\mathcal{V}$ at the distinguished point $\mathfrak{o}$.

The celebrated Poincaré theorem asserts that, in the case when the index does not vanish, there is an equilibrium point of the dynamical system determined by Equation (1) in the interior part of G.

Conversely, if the index is zero, then we can extend the vector field $\mathcal{V}$ from the boundary inside the region G so that there are no equilibrium points or states. Further, the Poincaré index makes it possible to control the joint coexistence of equilibrium states of different types and limit cycles of dynamical systems.

The total increment of the angle of rotation of the vector when traversing a simple closed curve is $2\pi n$, where n is an integer that does not depend on the shape of the curve; it is called the index of the closed curve with respect to the given vector field.

Usually, the index of an equilibrium state (i.e., of a singular point of a vector field) is the index of a simple closed curve that encloses only this equilibrium state. Thus, the index of the node, focus, and center is 1, the index of the saddle is $-1$, and the indices of more complicated singular points can be other than 1.

If there is a closed integral curve on the phase plane, for example a limit cycle, then, since the phase velocity vector on such a curve is tangent to it, it rotates through an angle of $2\pi n$ when the loop is completely circumvented. Thus, inside the closed integral curve, there must be at least one singular point having an index. If the circuit has several simple equilibrium states (i.e., several singular points), then their number is always odd, and the number of saddles is one less than the number of other singular points (see [15]).

4. Methods of Calculation of the Index

As a rule, the index is easy to calculate only in those cases where the linear part of a vector field is non-degenerate, as well as for some classes of vector fields with a degenerate linear part. In general, various approaches and methods are known for calculating the index of an isolated singular point of a vector field on the plane, some of which are also suitable for the study of the multidimensional case. For example, we can mention some of them.

The method of small deformations: If an isolated point corresponds to a critical case, one can try to transform it into several isolated singular points corresponding to the regular case by means of small deformations or perturbations of the field components.

The geometric method: Its main idea can be explained as follows: The components of a plane vector field determine the curves on the plane. Thus, suppose that these components are polynomials of the second degree. Then, they determine two curves of the second order. By their type and relative position, one can evaluate the index. For example, if these curves are tangent circles, then the index at the singular point is equal to zero (since these circles can be moved apart using a small deformation). The usual intersection of second-order curves corresponds to the regular case, etc.

The homotopy method: This is where a vector field is replaced by a homotopically equivalent field, which is obtained by a non-degenerate deformation of the initial one and for which the calculation of the index of an isolated singular point is simplified; this uses the fact that homotopically equivalent fields have the same rotation.

The method for calculating rotation on a closed curve: This is based on the use of Poincaré’s formula (2).

Next, in the high-dimensional case, the method of calculating the degree of a mapping is highly efficient; it is based on use of multidimensional residue, various integral representations, etc. For example, given a vector field in ${\mathbb{R}}^n$ with an isolated singularity at the origin, consider the map:

$$h:S^{n-1}⟶S^{n-1},S^{n-1}=\{x\in {\mathbb{R}}^n:\left|x\right|=r\},$$

which is defined by the rule:

$$h(x)=r\frac{\mathcal{V}(x)}{|\mathcal{V}(x)|}$$

with a small enough $r>0$. Then, the index $\mathcal{V}$ at the point 0 is equal to ${deg}_0(h)$, the degree of the map h at the distinguished point.

It is easy to understand that the theory of differential equations naturally leads to the study of vector fields not only on the plane, but also on high-dimensional manifolds of various types. Therefore, even the simplest systems, consisting of many differential equations, can be geometrically considered as direction fields in high-dimensional Euclidean spaces or manifolds.

It is appropriate to note here that A. Andronov set up a problem to investigate three-dimensional and higher-dimensional cases using the phase space in the same detail as for the phase plane. However, he was not able to do so.

In this regard, we also note that any dynamical system corresponds to a high-dimensional manifold of its possible states and a system of differential equations, the integral curves of which, filling a given phase space, are possible motions of this system. Each of these motions is determined by certain initial conditions. Therefore, in this case as well, the main object of the study is the field of directions and the system of trajectories on the given manifold.

As a rule, in concrete applications, it is important to learn how to solve this kind of problem in many other cases, and—first of all—one needs to understand how to calculate the index of a vector field in the corresponding situation.

In fact, the methods mentioned above are not so simple, because they are based on a number of highly non-trivial ideas, concepts, and tools (such as homotopy, integration, differentiation) from topology, analysis, geometry, etc. Of course, there are many other methods, much more complicated (sometimes very cumbersome and too sophisticated). Their descriptions are easy to find in many textbooks, articles, and book collections. On the contrary, the author’s method described in the next sections is completely elementary.

5. The Homological Method

First of all, we note that one of the possible extensions of the concept of the index to another setting, which appeared in the theory of foliations, turned out to be quite suitable for use in general theory. The corresponding invariant is usually called the homological index; it is well defined for vector fields tangent to a manifold at its nonsingular points (see [16]). The homological index possesses the same properties of the classical topological index and can be represented in terms of the sum of the local Poincaré–Hopf indices at singular points of a vector field defined on any smooth perturbation or deformation of the given manifold.

More precisely, let $(X,\mathfrak{o})$ be the germ of the complex space. Choose one of its representatives in an open neighborhood U of the origin $0\in {\mathbb{C}}^m$ with coordinates $z_1,\dots ,z_m$, and denote it by X. Let ${\mathcal{O}}_U$ denote the sheaf of regular holomorphic functions on U. Then, the germ X is defined by the ideal $\mathcal{I}\subset {\mathcal{O}}_U$, which is generated by a sequence of functions $f_1,\dots ,f_k\in {\mathcal{O}}_U$. We denote the sheaves of germs of regular holomorphic differential p-forms on X by ${\mathsf{\Omega }}_X^p$, $p⩾0$, which is defined as follows:

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

In particular, ${\mathsf{\Omega }}_X^p=0$ for $p<0$ and $p>m$, ${\mathsf{\Omega }}_X^0={\mathcal{O}}_X$ and ${\mathsf{\Omega }}_X^p\cong {\wedge }^p{\mathsf{\Omega }}_X^1$ for $p⩾0$. In the pure algebraic setting, the elements of ${\mathsf{\Omega }}_X^1$ are often called Kähler differentials. Then, the standard de Rham differential d equips this family of sheaves (because $d^2=0$) with the structure of an increasing complex:

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

(3)

This complex is denoted by $({\mathsf{\Omega }}_X^{•},d)$ and called the Poincaré–de Rham complex on X; we denote by $H_{DR}^{*}({\mathsf{\Omega }}_X^{•})$ its cohomology groups (i.e., the quotients $\mathrm{Ker}(d)/\mathrm{Im}(d)$).

In fact, for smooth manifolds, the corresponding complex was first studied by Poincaré (it suffices to recall his classical lemma, which states that closed differential forms on smooth manifolds are exact). Moreover, as we already emphasized in the Introduction, the idea of using differential calculus in the qualitative theory of differential equations also belongs to Poincaré.

On the other hand, the family of sheaves of differential forms can be endowed with the structure of a complex in many other ways. For example, denote by $Der(X)$ the ${\mathcal{O}}_X$-module of regular vector fields on X, i.e., $Der(X)={Hom}_{{\mathcal{O}}_X}({\mathsf{\Omega }}_X^1,{\mathcal{O}}_X)$. Let us take element $\mathcal{V}\in Der(X)$. Then, the inner product (contraction) differential forms and of vector fields gives a homomorphism of ${\mathcal{O}}_X$-modules ${\iota }_{\mathcal{V}}:{\mathsf{\Omega }}_X^p\to {\mathsf{\Omega }}_X^{p-1}$, together with a structure of a decreasing complex on ${\mathsf{\Omega }}_X^{•}$ (because ${\iota }_{\mathcal{V}}^2=0$). This complex we call the contracted Poincaré–de Rham complex, and we denote it by $({\mathsf{\Omega }}_X^{•},{\iota }_{\mathcal{V}})$ (see [17]). The ${\iota }_{\mathcal{V}}$-homology groups of this complex are denoted by $H_{*}({\mathsf{\Omega }}_X^{•},{\iota }_{\mathcal{V}})$. In a more general context, this complex can be regarded also as a contravariant version of the classical Poincaré–de Rham complex (3).

Let us now consider a vector field $\mathcal{V}$ on an n-dimensional germ $X\subset {\mathbb{C}}^m$, and let:

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

(4)

be the truncated contracted Poincaré–de Rham complex. We denote this decreasing complex by $({\widehat{\mathsf{\Omega }}}_X^{•},{\iota }_{\mathcal{V}})$ (see [17] §2). Suppose also that all homology groups of $({\widehat{\mathsf{\Omega }}}_X^{•},{\iota }_{\mathcal{V}})$ are finite-dimensional vector spaces. Then, the following Euler–Poincaré characteristic is defined:

$$\chi ({\widehat{\mathsf{\Omega }}}_X^{•},{\iota }_{\mathcal{V}})={\displaystyle \sum _{i=0}^n}{(-1)}^i{dim}_k{H}_i({\widehat{\mathsf{\Omega }}}_X^{•},{\iota }_{\mathcal{V}});$$

it is called the homological index of $\mathcal{V}$ at the point $\mathfrak{o}\in X$ and denoted by ${Ind}_{hom,X,\mathfrak{o}}(\mathcal{V})$. In fact, this notion was introduced by X. Gómez-Mont (1998) for vector fields defined on a reduced complex analytic space X of dimension $n⩾1$ under the assumption that all homology groups $H_i({\widehat{\mathsf{\Omega }}}_X^{•},{\iota }_{\mathcal{V}})$ are finite-dimensional vector spaces (see [16]).

6. The Logarithmic Index

Since 2005, in a series of articles, we developed a new method for computing the homological index of vector fields on complex varieties based on the general theory of differential forms and vector fields. The key idea of our approach is that the index can be expressed in terms of the homology of complexes of regular and meromorphic differential forms defined on a variety, similarly to (3) and (4). Indeed, this method makes it possible to simplify the calculations, as well as to clarify the meaning of the basic constructions that underlie many works on this topic.

In particular, using this approach, we can calculate the index in many situations, including the case of curves $\gamma$ with singularities, hypersurfaces with nonisolated singularities, complete intersections, and many others, in terms of the elementary symmetric polynomials.

First, we discuss the method for computing the homological index of vector fields defined on hypersurfaces with arbitrary singularities; it is based on a representation of regular holomorphic differential forms on a hypersurface D in terms of meromorphic differential forms defined on the ambient manifold and having logarithmic poles along D.

Let us fix a divisor D. Then, one can define the coherent analytic sheaves ${\mathsf{\Omega }}_M^q(\log D),$$q\ge 0$, and ${Der}_M(\log D)$ as follows. The stalk ${\mathsf{\Omega }}_{M,x}^q(\log D)$ over the point $x\in M$ consists of those germs $\omega$ of meromorphic q-forms on M with poles (of the first order) along D that satisfy the following two conditions: $h\omega$ and $hd\omega$ are holomorphic at the point x. In other terms, $h·{\mathsf{\Omega }}_{M,x}^q(\log D)\subseteq {\mathsf{\Omega }}_{M,x}^q$ and $h\wedge d{\mathsf{\Omega }}_{M,x}^q(\log D)\subseteq {\mathsf{\Omega }}_{M,x}^{q+1}$. Similarly, the stalk ${Der}_{M,x}(\log D)$ will consist of germs $\mathcal{V}$ of holomorphic vector fields on M such that $\mathcal{V}(h)\in (h){\mathcal{O}}_{M,x}$. It is clear that the vector field $\mathcal{V}$ is tangent to D at its nonsingular points. We call ${\mathsf{\Omega }}_M^q(\log D)$ the sheaf of logarithmicq-forms, or, equivalently, the sheaf of meromorphic q-forms with logarithmic poles along the divisor D. We call ${Der}_M(\log D)$ the sheaf of logarithmic vector fields.

Let us take a vector field $\mathcal{V}\in {Der}_M(\log D)$. Then, the inner multiplication ${\iota }_{\mathcal{V}}$ defines the structure of a decreasing complex on ${\mathsf{\Omega }}_M^{•}(\log D)$ similarly to the above. Moreover, when $\mathcal{V}$ has only isolated singularities, then, for each $x\in M$, the following Euler–Poincaré characteristic of the complex of germs of logarithmic differential forms is well defined:

$$\chi \left({\mathsf{\Omega }}_{M,x}^{•}(\log D),{\iota }_{\mathcal{V}}\right)=\sum _{i=0}^{n+1}{(-1)}^{i}dim{H}_i\left({\mathsf{\Omega }}_{M,x}^{•}(\log D),{\iota }_{\mathcal{V}}\right).$$

(5)

We call this characteristic the logarithmic index of the vector field $\mathcal{V}$ at the given point x; it is denoted by ${Ind}_{\log D,x}(\mathcal{V}).$ From the above considerations it follows also that ${Ind}_{\log D,x}(\mathcal{V})=0$ if $\mathcal{V}(x)\ne 0.$

It is not difficult to see that for $q=0,1,\dots ,n+1$, there are the following exact sequences:

$$0⟶{\mathsf{\Omega }}_{M,x}^{q-1}/h·{\mathsf{\Omega }}_{M,x}^{q-1}(\log D)\stackrel{\wedge dh}{⟶}{\mathsf{\Omega }}_{M,x}^q/h·{\mathsf{\Omega }}_{M,x}^q⟶{\mathsf{\Omega }}_{D,x}^q⟶0,$$

(6)

of ${\mathcal{O}}_{M,x}$-modules. Herein, by $\wedge dh$, we denote the exterior multiplication by the total differential $dh$. As a result, we obtain the exact sequence of complexes:

$$0\to \left({\mathsf{\Omega }}_{M,x}^{•}/h{\mathsf{\Omega }}_{M,x}^{•}(\log D),{\iota }_{\mathcal{V}}\right)[-1]\stackrel{\wedge dh}{⟶}\left({\mathsf{\Omega }}_{M,x}^{•}/h{\mathsf{\Omega }}_{M,x}^{•},{\iota }_{\mathcal{V}}\right)⟶\left({\mathsf{\Omega }}_{D,x}^{•},{\iota }_V\right).\to 0.$$

(7)

It should be remarked that the exterior multiplication by $\wedge dh$ induces a morphism of complexes in view of the identity:

$${\iota }_{\mathcal{V}}(\omega )\wedge dh={\iota }_{\mathcal{V}}(\omega \wedge dh)+{(-1)}^{q-1}\omega \wedge \mathcal{V}(h).$$

Indeed, the condition $\mathcal{V}(h)\in (h){\mathcal{O}}_{M,x}$ implies that the second summand on the right-hand side vanishes in the factor-complex ${\mathsf{\Omega }}_M^{•}/h{\mathsf{\Omega }}_M^{•}$. Further, the exact sequence of complexes:

$$0⟶({\mathsf{\Omega }}_{M,x}^{•},{\iota }_{\mathcal{V}})\stackrel{·h}{⟶}({\mathsf{\Omega }}_{M,x}^{•},{\iota }_{\mathcal{V}})⟶({\mathsf{\Omega }}_{M,x}^{•}/h{\mathsf{\Omega }}_{M,x}^{•},{\iota }_{\mathcal{V}})⟶0$$

(8)

implies that $\chi \left({\mathsf{\Omega }}_{M,x}^{•}/h{\mathsf{\Omega }}_{M,x}^{•},{\iota }_{\mathcal{V}}\right)=0.$ As a result, the exact sequence (7) yields the relation:

$${Ind}_{hom,D,x}(V)=-\chi (({\mathsf{\Omega }}_{M,x}^{•}/h{\mathsf{\Omega }}_{M,x}^{•}(\log D),{\iota }_{\mathcal{V}})[-1])=\chi ({\mathsf{\Omega }}_{M,x}^{•}/h{\mathsf{\Omega }}_{M,x}^{•}(\log D),{\iota }_{\mathcal{V}}).$$

(9)

Proposition 1.

Let $\mathfrak{o}\in D$ be an isolated singularity of a vector field $\mathcal{V}\in Der(\log D)$ and $J_{\mathfrak{o}}\mathcal{V}=({\mathcal{V}}_0,\dots ,{\mathcal{V}}_n){\mathcal{O}}_{M,\mathfrak{o}}$ the ideal generated by the coefficients of the expansion $\mathcal{V}={\sum }_i{\mathcal{V}}_i\partial /\partial z_i$ of the vector field $\mathcal{V}$. Then:

$${Ind}_{hom,D,\mathfrak{o}}(V)=dim{\mathcal{O}}_{M,\mathfrak{o}}/J_{\mathfrak{o}}\mathcal{V}-{Ind}_{\log D,\mathfrak{o}}(V).$$

Proof of Proposition 1. 

Consider the exact sequence:

$$0⟶\left({\mathsf{\Omega }}_{M,\mathfrak{o}}^{•}(\log D),{\iota }_{\mathcal{V}}\right)\stackrel{·h}{⟶}\left({\mathsf{\Omega }}_{M,\mathfrak{o}}^{•},{\iota }_{\mathcal{V}}\right)⟶\left({\mathsf{\Omega }}_{M,\mathfrak{o}}^{•}/h{\mathsf{\Omega }}_{M,\mathfrak{o}}^{•}(\log D),{\iota }_{\mathcal{V}}\right)⟶0.$$

(10)

First, we observe that the complex $({\mathsf{\Omega }}_{M,\mathfrak{o}}^{•},{\iota }_{\mathcal{V}})$ is isomorphic to the Koszul complex ${\mathbf{K}}_{•}\left(({\mathcal{V}}_0,\dots ,{\mathcal{V}}_n);{\mathcal{O}}_{M,\mathfrak{o}}\right)$ on the generators $e_i=d{z}_i$, $i=0,\dots ,m.$ Since $\mathfrak{o}\in D$ is an isolated singularity of $\mathcal{V}$, it follows that the sequence $({\mathcal{V}}_0,\dots ,{\mathcal{V}}_n)$ is ${\mathcal{O}}_{M,\mathfrak{o}}$-regular, and hence, the Koszul complex is acyclic in all dimensions except for zero, $H_0({\mathbf{K}}_{•})\cong {\mathcal{O}}_{M,\mathfrak{o}}/J_{\mathfrak{o}}\mathcal{V}.$ Thus, the relation:

$$\chi \left({\mathsf{\Omega }}_{M,\mathfrak{o}}^{•}/h{\mathsf{\Omega }}_{M,\mathfrak{o}}^{•}(\log D),{\iota }_{\mathcal{V}}\right)=dim{\mathcal{O}}_{M,\mathfrak{o}}/J_{\mathfrak{o}}\mathcal{V}-{Ind}_{\log D,\mathfrak{o}}(V)$$

for the Euler–Poincaré characteristics of complexes in the diagram (10) together with the identity (9) gives the desired assertion. □

Theorem 1.

Let Z denote the singular locus of D, i.e., $Z=SingD$, and let $c=\mathrm{codim}(Z,D)$ denote its codimension in the divisor D. Suppose that D is a normal hypersurface, that is $c⩾2$. Then:

$${Ind}_{\mathrm{hom},D,\mathfrak{o}}(\mathcal{V})={dim}_{\mathbb{C}}{\mathcal{O}}_{M,\mathfrak{o}}/(h,J_{\mathfrak{o}}\mathcal{V})+{\displaystyle \sum _{i=c^{\prime }}^{n+1}}{(-1)}^i{dim}_{\mathbb{C}}{H}_i({\mathsf{\Omega }}_{D,\mathfrak{o}}^{•},{\iota }_V),$$

where $c^{\prime }=2\left[\frac{c+1}{2}\right]+1$ and the square brackets denote the integer part of rational numbers. We also assume that the sum is zero if the lower bound of the sum is greater than the upper bound.

Proof of Theorem 1. 

First we remark that ${\mathsf{\Omega }}_{M,\mathfrak{o}}^q/h{\mathsf{\Omega }}_{M,\mathfrak{o}}^q(\log D)\cong {\mathsf{\Omega }}_{D,\mathfrak{o}}^q$ for $0\le q<c$. Therefore, it follows from the exact sequence (7) that that:

$$H_i({\mathsf{\Omega }}_D^q,{\iota }_V)\cong H_{i-1}({\mathsf{\Omega }}_D^q[-1],{\iota }_V)=H_{i-2}({\mathsf{\Omega }}_D^q,{\iota }_V)$$

for $i=3,\dots ,c+1.$ In particular, in this interval, the dimensions of the ${\iota }_{\mathcal{V}}$-homology groups of ${\mathsf{\Omega }}_{D,\mathfrak{o}}^{•}$ in both series $H_{2i}$ and $H_{2i-1}$ coincide.

Moreover, it is not difficult to prove that the dimensions of the first homology groups $H_1$ and $H_2$ coincide as well. This yields the desired formula. In order to distinguish the cases of even and odd codimension, the integer part in the lower bound of the above sum is needed. □

It should be emphasized that the above results one can also obtain with the use of an explicit construction of free resolutions of the modules ${\mathsf{\Omega }}_{D,\mathfrak{o}}^{•}$ and the computation of appropriate hypercohomology groups (see [16]). On the contrary, the notion of the logarithmic index (5) and Proposition 1 give us much simpler proofs, which do not require any specific technique.

7. The Quasihomogeneous Case

Let us now consider the case where the variety $X\subset {\mathbb{C}}^m$ of dimension $n⩾1$ is determined by a regular sequence of quasihomogeneous functions $f_1,\dots ,f_k$, whose weighted degrees are equal to $d_1,\dots ,d_k$ with respect to indeterminates $z_1,\dots ,z_m$ of weights $w_1,\dots ,w_m\in \mathbb{Z}$, respectively. Under these assumptions, the type of X is equal to $(d_1,\dots ,d_k;w_1,\dots ,w_m)\in {\mathbb{Z}}_{+}^k\times {\mathbb{Z}}_{+}^m$ and $dimX=m-k$.

Hence, the sheaves ${\mathsf{\Omega }}_X^p$, $p\ge 0$, and $Der(X)=Hom({\mathsf{\Omega }}_X^1,{\mathcal{O}}_X)$ are equipped with a natural grading in which $deg(d{f}_j)=deg(f_j)=d_j$, $j=1,\dots ,k$, and $deg(d{z}_i)=w_i$, $i=1,\dots ,m$. An element $\mathcal{V}$ of the homogeneous component $Der{(X)}_v$ is usually called the vector fields of weightv. In particular, the weight of $\partial /\partial z_i$ is equal to $-w_i$, $i=1,\dots ,m$, and the element ${\mathcal{V}}_0={\sum }_{i=1}^m{w}_i{z}_i\partial /\partial z_i\in Der{(X)}_0$ of zero weight is called the Euler vector field.

It should be noted that the type of an isolated complete intersection singularity of positive dimension is uniquely determined, except for the case of hypersurfaces of multiplicity 2 (see [18] (6.4)). The Poincaré series of modules ${\mathsf{\Omega }}_X^p$ of regular differential forms of degree $0\le p\le n$ on X can be represented (see [18] Lemma 3.2, or [17]) as follows:

$$\mathcal{P}({\mathsf{\Omega }}_X^p;x)=\mathcal{P}({\mathcal{O}}_X;x)·{res}_{t=0}{t}^{-p-1}\prod (1+t{x}^{w_i})/\prod (1+t{x}^{d_j}).$$

In order to formulate a more general result from [17], we introduce the following notations. Let $W_{\lambda }$ denote the elementary symmetric polynomials of degree $\lambda ⩾0$ in ℓ indeterminates $(y_1,\dots ,y_{\ell })$, given by the rule:

$$\prod _{i=1}^{\ell }(1+y_i\zeta )=\sum _{\lambda =0}^{\ell }{W}_{\lambda }(y_1,\dots ,y_{\ell }){\zeta }^{\lambda },$$

where $\zeta$ is a formal parameter. Similarly, let $D_{\lambda }(y_1,\dots ,y_{\ell })$ denote the symmetric polynomials of weight $\lambda ⩾0$ determined by the relation:

$$\prod _{i=1}^{\ell }{(1+y_i\zeta )}^{-1}=\sum _{\lambda =0}^{\infty }{(-1)}^{\lambda }{D}_{\lambda }(y_1,\dots ,y_{\ell }){\zeta }^{\lambda }.$$

Theorem 2.

Under the above assumptions, let $\mathcal{V}$ be a vector field on X of weight $v\in \mathbb{Z}$ with an isolated singularity at the distinguished point $\mathfrak{o}$. Then:

$$\begin{array}{ll}{Ind}_{hom,X,\mathfrak{o}}(\mathcal{V})={\displaystyle \sum _{{\lambda }_1+{\lambda }_2+{\lambda }_3=n}}{(-1)}^{{\lambda }_2}{x}^{(n-{\lambda }_1)v}& \times \\ & \times W_{{\lambda }_2}(x^{w_1},\dots ,x^{w_m})D_{{\lambda }_3}(x^{d_1},\dots ,x^{d_k})\mathcal{P}({\mathcal{O}}_X;1),\end{array}$$

where $\mathcal{P}({\mathcal{O}}_X;1)=\prod (1-x^{d_j})/\prod (1-x^{w_i}){|}_{x=1}$.

Proof of Theorem 2. 

We calculate the homological index of the decreasing contracted Poincaré–de Rham complex $({\widehat{\mathsf{\Omega }}}_X^{•},{\iota }_{\mathcal{V}})$ for a graded complete intersection with an isolated singularity, following the considerations from [17]. In fact, it is sufficient to substitute the expressions for the Poincaré polynomials $\mathcal{P}({\mathsf{\Omega }}_X^p;x)$ into the generating function of the complex $({\widehat{\mathsf{\Omega }}}_X^{•},{\iota }_{\mathcal{V}})$ and perform some elementary transformations. As a result, we obtain:

$$\begin{array}{c}G{H}_{\mathcal{P}}(({\widehat{\mathsf{\Omega }}}_X^{•},{\iota }_{\mathcal{V}});x,x^v)={\displaystyle {\sum }_{p\ge 0}}{(-1)}^p\mathcal{P}({\mathsf{\Omega }}_X^p;x)=\\ {\displaystyle ={(-1)}^n{x}^{nv}\mathcal{P}({\mathcal{O}}_X;x){res}_{t=0}{t}^{-n-1}{(1+t{x}^{-v})}^{-1}\prod (1+t{x}^{w_i})/\prod (1+t{x}^{d_j}),}\end{array}$$

and the proof is finished. □

Corollary 1.

Under the assumptions and in the notations of Theorem 2, the index of the Euler vector field ${\mathcal{V}}_0$ at the point $\mathfrak{o}$ is equal to:

$${Ind}_{hom,\mathfrak{o}}({\mathcal{V}}_0)=1+{(-1)}^n\mu (X),$$

where $\mu (X)$ is the Milnor number of X. In other words, it is equal to the Euler–Poincaré characteristic of the Milnor fiber of the singularity X.

Proof of Corollary 1. 

Taking the value $v=0$ in Theorem 2, we consider the following expression:

$$\phi (t)=t^{-n-1}{(1+t)}^{-1}\prod (1+t{x}^{w_i})/\prod (1+t{x}^{d_j})\mathcal{P}({\mathcal{O}}_X;x).$$

It is not difficult to see that:

$${\displaystyle \underset{t=0}{res}}\phi (t)=-{\displaystyle \underset{s=\infty }{res}}{s}^{-2}\tilde{\phi }(s)={\displaystyle \sum _{s_0\in \mathbb{C}\left\{x\right\}}}{\displaystyle \underset{s=s_0}{res}}{s}^{-2}\tilde{\phi }(s),$$

(11)

where $t=1/s$. Since:

$$s^{-2}\tilde{\phi }(s)={(1+s)}^{-1}\prod \frac{(s+x^{w_i})}{(1-x^{w_i})}/\prod \frac{(s+x^{d_j})}{(1-x^{d_j})},$$

then the residue of the function $s^{-2}\tilde{\phi }(s)$ at the point $s_0=-1$ is equal to ${(-1)}^n$. On the other hand, the remaining part of the sum (11) is equal to the Poincaré polynomial $\mathcal{P}(x)$ such that $\mathcal{P}(1)=\mu (X)$. Therefore, Theorem 2 yields the equality:

$${Ind}_{hom,\mathfrak{o}}({\mathcal{V}}_0)={(-1)}^n{\displaystyle \underset{t=0}{res}}\phi (t){|}_{x=1},$$

as required. □

In fact, the assertion of Corollary 1 is a direct consequence of the acyclicity of the truncated contracted Poincaré–de Rham complex in all dimensions not equal to 0 and n (the differential of this complex is induced by the contraction along the Euler field ${\mathcal{V}}_0$) with the isomorphism $H_n({\widehat{\mathsf{\Omega }}}_X^{•},{\iota }_{\mathcal{V}})\cong Tors({\mathsf{\Omega }}_X^n)$ taken into account (see details in [17] Assertion 1).

Example 1.

Let X be the line with coordinate $z=z_1$, determined by the equation $z_2=0$ on the plane ${\mathbb{C}}^2$, and let $\mathcal{V}=z^{v+1}\partial /\partial z$ be a vector field of weight v. Then, $n=1$, $d_1=1$, $w_1=w_2=1$, $W_0=D_0=1$, $W_1=2x$, $D_1=x$, so that:

$${Ind}_{hom,X,\mathfrak{o}}(\mathcal{V})=(1-x^{v+1})/(1-x){|}_{x=1}=v+1.$$

It should be noted that it is not difficult to examine this example using the methods from Section 4 which are based on the theory of ordinary differential equations, topology, analysis, algebra, and so on (see [19], Problem 1, § 36).

Example 2.

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

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

so that $d_1=d_2=2$ and $w_1=w_2=w_3=w_4=1$ (see [17]). Then, we obtain:

$${Ind}_{hom,X,\mathfrak{o}}(\mathcal{V})=\{1-x^v(4x-2x^2)+x^{2v}(6x^2-8x^3+3x^4)\}{(1-x^2)}^2{(1-x)}^{-4}{|}_{x=1}.$$

Consequently, if:

$$\begin{array}{cc}\hfill \mathcal{V}=(z_2{z}_3-z_3^2-z_4^2)\frac{\partial }{\partial z_1}+(z_2^2-z_1{z}_3+\frac{1}{2}{z}_2{z}_4)& \frac{\partial }{\partial z_2}+\hfill \\ \hfill & (z_1^2+z_2{z}_3+\frac{3}{4}{z}_3{z}_4)\frac{\partial }{\partial z_3}+(z_2{z}_4+\frac{1}{4}{z}_4^2)\frac{\partial }{\partial z_4},\hfill \end{array}$$

then $v=1$, and we have:

$${Ind}_{hom,X,\mathfrak{o}}(\mathcal{V})=(1+4x+4x^2-2x^3-2x^4+4x^5+3x^6){|}_{x=1}=12.$$

For completeness, we note that this example was analyzed in a very complicated work [20], where the authors constructed a cumbersome multidimensional complex of coherent sheaves, the picture of which takes up almost two pages. Then, they computed the hypercohomology of the complex with the use of a computer, a special computer system of algebraic computations, some additional programs (not included in the publication), and so on. Finally, the authors asserted that the value of the index is equal to 12.

Theorem 3.

In the notations of Theorem 2, the multiplicity $m_{\delta }$ of the discriminant δ associated with the miniversal deformation of X is calculated by the following formula:

$$m_{\delta }={(-1)}^{n+1}{x}^{-(n+1)d_k}{res}_{t=0}{t}^{-n-2}{(1+t{x}^{d_k})}^{-1}·\prod _{j=1}^m\frac{(1+t{x}^{w_i})}{(1-x^{w_i})}\prod _{j=1}^{k-1}\frac{(1-x^{d_j})}{(1+t{x}^{d_j})},$$

or, in a more explicit way, similarly to Theorem 2, we obtain:

$$m_{\delta }={\displaystyle \sum _{{\lambda }_1+{\lambda }_2+{\lambda }_3=n}}{(-1)}^{{\lambda }_2}{x}^{({\lambda }_1-n)d_k}·W_{{\lambda }_2}(x^{w_1},\dots ,x^{w_m})D_{{\lambda }_3}(x^{d_1},\dots ,x^{d_k})\mathcal{P}({\mathcal{O}}_X;1).$$

In other terms, this multiplicity is equal to the dimension of the quotient space ${\mathcal{O}}_{{\mathbb{C}}^m,0}/{\mathcal{C}}_k(X)$, where by ${\mathcal{C}}_k(X)$ we denote the critical ideal generated by the sequence $f_1,\dots ,f_{k-1}$ and the maximal minors of the Jacobian matrix associated with the sequence of functions $f_1,\dots ,f_k$ (see the details in [21]).

The further development of our approach allows us to compute the index of vector fields given on varieties of various types, which are determined by an arbitrary sequence of quasihomogeneous functions $f_1,\dots ,f_k$ (see [22]).

Example 3.

Let X be the cuspidal space-curve singularity in ${\mathbb{C}}^3$ determined by the following three polynomials:

$$f_1=xz-y^2,f_2=x^3-yz,f_3=x^{2}y-y^2,$$

where x, y, and z are suitable coordinate functions. It should be worthy to underline that these equations define the simplest noncomplete intersection (sometimes, it is called the Macaulay cusp) in three-dimensional space; this example goes back to F.S. Macaulay (see [23], p.53, Example (i)). There are many areas of modern algebra, geometry, and topology where this singularity plays a key role.

Indeed, it is the germ of a curve having the rational parametrization $x=t^3$, $y=t^4$, and $z=t^5$ at the origin. Next, set $\mathcal{V}=v_1\frac{\partial }{\partial x}+v_2\frac{\partial }{\partial y}+v_3\frac{\partial }{\partial z}$ as the vector field on X. Then, we obtain:

$$\begin{array}{c}{Ind}_{hom,X,0}(\mathcal{V})={dim}_{\mathbb{C}}\left(\mathbb{C}\{x,y,z\}/(f_1,f_2,f_3,v_1,v_2,v_3)\mathbb{C}\{x,y,z\}\right)-5.\end{array}$$

It is easy to see that in this case, the obtained results correspond to the analysis of a three-dimensional analogue of System (1) in Section 2, the coordinates of which are connected by the three nontrivial relations defined by the functions $f_1$, $f_2$, and $f_3$.

8. Further Generalizations

One can apply the above-described approach for computing the index of holomorphic or meromorphic differential forms on manifolds with singularities of different types (cf. [15]). It is possible to show that in many interesting cases, this problem is reduced to the construction of resolutions of suitable types, which enable one to describe explicitly the structure of modules ${\mathsf{\Omega }}_X^p$ and the complex $({\widehat{\mathsf{\Omega }}}_X^{•},\wedge \omega )$, together with its cohomology groups. Moreover, it is often not necessary to require that the corresponding resolutions must be free or even finite. For instance, in the case of hypersurfaces with arbitrary singularities, it is convenient for the description of ${\mathsf{\Omega }}_X^p$ to use short (four-term) exact sequences containing the sheaves of logarithmic differential forms. Besides, the corresponding sequences consist of non-free modules (see [24]). Next, in the case of graded complete intersections, it is possible to construct an infinite resolution of the module of vector fields (see [18] (6.1)). Another interesting case realizes for a singular manifold satisfying the following property: its modules ${\mathsf{\Omega }}_X^1$ possess a resolution of finite length consisting of free ${\mathcal{O}}_X$-modules. This enables us to construct free resolutions for all modules ${\mathsf{\Omega }}_X^p$, $p>1$, and obtain an analog of Theorem 3. Using similar arguments and the theory of multi-logarithmic differential forms (see [25]), it is easy to compute the index of vector fields and differential forms given on various types of noncomplete intersections involving Cohen–Macaulay curves, normal surfaces, toric and determinantal varieties, etc.

Let us briefly discuss the case of varieties with arbitrary singularities. In this setting, our method can be also applied for computation of the homological index as follows (cf. [17] §5). Thus, let $Z=SingX$ denote the subset of singular loci of the manifold X. Then there exist a natural inclusion $j:X\backslash Z\to X$ and the following exact sequence of ${\mathcal{O}}_X$-modules for every $p\ge 0$:

$$0⟶{\mathcal{H}}_Z^0({\mathsf{\Omega }}_X^p)⟶{\mathsf{\Omega }}_X^p⟶j_{*}{j}^{*}{\mathsf{\Omega }}_X^p⟶{\mathcal{H}}_Z^1({\mathsf{\Omega }}_X^p)⟶0.$$

In the most general context, the sheaves $j_{*}{j}^{*}{\mathsf{\Omega }}_X^p$, $p\ge 0$, of differential meromorphic p-forms are non-coherent. However, there always exists (and as a rule, not unique) a coherent subsheaf ${\mathcal{F}}_X^p\subset j_{*}{j}^{*}{\mathsf{\Omega }}_X^p$ whose restriction to the open subset of regular points $U=X\backslash Z$ of X is isomorphic to ${\mathsf{\Omega }}_U^p$, i.e., ${\mathsf{\Omega }}_U^p\cong {\mathcal{F}}_X^p{|}_U$.

Let us assume that the restriction of a regular differential form $\omega \in {\mathsf{\Omega }}_X^1$ to ${\mathsf{\Omega }}_U^1$ can be extended on the subsheaf ${\mathcal{F}}_X^1$ meromorphic 1-forms in the following way: the exterior multiplication on the obtained differential form is compatible with the morphism of the exact sequence:

$$0⟶{\mathcal{H}}_Z^0({\mathsf{\Omega }}_X^p)⟶{\mathsf{\Omega }}_X^p⟶{\mathcal{F}}_X^p⟶{{}^{\#}\mathsf{\Omega }}_X^p⟶0,$$

where ${{}^{\#}\mathsf{\Omega }}_X^p\subseteq {\mathcal{H}}_Z^1({\mathsf{\Omega }}_X^p)$ for $p\ge 0$. Let $\tilde{\omega }\in {\mathcal{F}}_X^1$ denote the extension of the form $\omega$. Then, it is not difficult to verify that there exists a similar exact sequence of increasing complexes. Under the assumption that all cohomology groups are finite-dimensional vector spaces, then:

$$\chi ({\mathsf{\Omega }}_X^{•},\wedge \omega )=\chi ({\mathcal{F}}_X^{•},\wedge \tilde{\omega })+\chi ({\mathcal{H}}_Z^0({\mathsf{\Omega }}_X^{•}),\wedge \omega )-\chi ({{}^{\#}\mathsf{\Omega }}_X^{•},\wedge \tilde{\omega }).$$

In the case where an n-dimensional germ X has the dualizing complex $({\omega }_X^{•},d)$ (e.g., X is Cohen–Macaulay), one can set ${\omega }_X^p={Hom}_{{\mathcal{O}}_X}({\mathsf{\Omega }}_X^{n-p},{\omega }_X^n)$ as ${\mathcal{F}}_X^p$; it is a coherent sheaf of germs of regular meromorphicp-forms on X since ${\omega }_X^p\subseteq j_{*}{j}^{*}{\mathsf{\Omega }}_X^p$ for all p. For example, it is known that this inclusion is an equality in the case of normal varieties. Moreover, for Cohen–Macaulay curves, as well as for graded Gorenstein surfaces and for isolated singularities of complete intersections, we can prove that the Euler–Poincaré characteristic $\chi ({{}^{\#}\mathsf{\Omega }}_X^{•},\wedge \omega )$ vanishes. In all these cases, the homological index can be expressed in terms of cohomology groups of the torsion subcomplex $({\mathcal{H}}_Z^0({\mathsf{\Omega }}_X^{•}),\wedge \omega )$ together with the dualizing complex $({\omega }_X^{•},\wedge \omega )$, etc. It should be remarked that an explicit description of the dualizing complex is also known for some types of normal singularities, for toric varieties, and many others.

Sometimes, instead of sheaves ${\mathcal{F}}_X^p$, it is convenient to use the sheaves of weakly holomorphic differential forms or sheaves of germs of differential p-forms that extend in a special way from the regular locus of a variety to its resolution, normalization, and so on.

9. Applications

Herein, we briefly discuss some useful applications of the described ideas and methods.

9.1. Hamiltonian Mechanics

First, recall that the Poisson bracket of two functions in ${\mathbb{R}}^{2r}$ with coordinates $\{p_1,\dots ,p_r,q_1,\dots ,q_r\}$ is determined (due to D. Poisson, 1809) by the following expression:

$$\{f,g\}=\sum _{i=1}^r\left(\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}-\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}\right).$$

Using this bracket, the Hamiltonian differential equations (equivalent to the equations of classical Newton mechanics) can be written as a system in which the coordinates $q_i$ and the impulses $p_i$ are contained symmetrically:

$$\dot{q_i}=\{H,q_i\},\dot{p_i}=\{H,p_i\},$$

where H denotes the total energy of the system. Poisson also discovered that if f and g are constants of motion, then $\{f,g\}$ is also a constant of motion. The reason for this phenomenon was found by Jacobi, who showed that $\{f,g\}$ satisfies the identity:

$$\left\{\right\{f,g\},h\}+\left\{\right\{g,h\},f\}+\left\{\right\{h,f\},g\}=0.$$

(12)

Using this relation, we immediately obtain the well-known definition of a Poisson algebra equipped with a skew-symmetric biderivation $\mathcal{W}$ satisfying the Jacobi identity (12) (see [26]).

Let us now consider a Poisson manifold X and a skew-symmetric bivector derivation $\mathcal{W}$ of the ring of regular functions on X. Analogous to the case of complex (4) considered in Section 5, we can introduce the decreasing complex $({\mathsf{\Omega }}_{X/k}^{•},{\iota }_{\mathcal{W}})$ on X using the contraction along the bivector field $\mathcal{W}$. It can be shown that both homology groups and the Euler–Poincaré characteristic of this complex contain important information about the topological, differential, and analytic structure of systems in question given on the Poisson manifold X. This simplifies essentially the calculation of the basic invariants of Poisson varieties (see [27]). Above all, in a similar manner, one can study the corresponding systems arising in many applications in physics, mechanics, optics, the calculus of variations, and in other fields of the natural sciences, which can be described in terms of Hamiltonian equations or their analogs and generalizations.

9.2. Quantum Physics and Mechanics

In quantum mechanics, where Hermitian operators play the role of classical dynamical quantities, the corresponding equations are written using the Poisson quantum brackets as follows:

$${\{\widehat{f},\widehat{g}\}}_q=\frac{i}{\hslash }(\widehat{f}\widehat{g}-\widehat{g}\widehat{f}),$$

so that the Heisenberg equation can be regarded as an analog of the Hamiltonian equations.

The essence of the formalism of Hamilton and his followers is that one can work with Poisson brackets without using differential equations, since they correspond to the Lie brackets in the Poisson algebra. In particular, such an algebraic approach allows one to find conserved quantities, such as the motion integrals, without solving the corresponding differential system explicitly. Above all, the homology groups of a quantum analog of the Poincaré–de Rham complex (3) contain important information about basic invariants of the system in question.

9.3. Topological Quantum Numbers

Using the notion of the index of vector fields, one can compute the topological quantum number, which characterizes instanton configurations of the Yang–Mills theory with gauge group $\mathrm{SU}(2)$ that are homotopically nonequivalent (see [28]). In general, instantons that are topologically nontrivial solutions of the Yang–Mills equations can be used in the analysis of tunneling processes arising in different systems described by non-relativistic quantum mechanics [29]. The topological characterization of monopoles and instantons is very useful in applications. For example, the “monopole problem” was a direct motivation for the inflationary models in cosmology [30]. Moreover, monopoles can catalyze the decay of a proton [31], while the index of vector fields can be exploited for a description and classification of vector field models in curved spacetime [32], etc.

9.4. The Index of Differential Forms

The Poincaré index can also be determined for a wide class of differential forms given on arbitrary manifolds. The methods described by the author for computing the index of vector fields with minor changes can be easily extended to this theory; this allows significantly simplifying and generalizing a series of known results from this part of mathematical analysis (see [21]).

9.5. General Index Theory

There are various generalizations of the concept of the index in very different theoretical and applied problems, including the notion of the global Poincaré–Hopf index and its real analogues, the GSV-index (see [33]), Morse and Conley indices (see [34]), their infinite-dimensional analogues, etc. In each case, the homological method is used to analyze the complex of differential forms of the appropriate type, and the calculation of the global index is reduced to the analysis of hyperhomology of the corresponding complex, etc.

9.6. Deformations and Bifurcations

The theory of deformations originated by B. Riemann studies families of objects that depend on a parameter. In particular, various kinds of perturbations can be considered as a special type of deformation. In particular, in the context of mathematical control theory, the corresponding deformation is represented as a family of smooth functions defined on the direct product of the control space and the behavior space. These families are studied using a detailed analysis of the homology and cohomology of the Poincaré–de Rham complex, on which the method for calculating the Poincaré index described above is based.

In many problems in mechanics, one often has to study the properties of singular points of the Hamiltonian depending on parameters. On the other hand, it is well known that a non-degenerate (in the sense of Morse theory) singular point is stable with respect to small variations of the parameter; it also remains non-degenerate, and its index does not change. However, for bifurcation values of the parameter, as a rule, degeneracies of various types arise, and when passing through them, the index of the singular point may change. In such cases, it is important to know which bifurcations are possible and which are not. Recall that the theory of bifurcations of dynamical systems describes qualitative “jump” changes in phase portraits of differential equations under deformations, i.e., with the continuous change of parameters. At that, in the case of the loss of the stability of a singular point, a limit cycle can occur, and in the case of the loss of the stability of the limit cycle, a complex attractor can arise (in chaotic dynamics). Namely, such changes are called bifurcations. For example, the Poincaré–Andronov–Hopf bifurcation means (see [9]) that the singular point changes the stability so that a pair of eigenvalues intersects the imaginary axis, and in addition, a limit cycle arises in a small neighborhood of the singular point. Thus, knowledge of the bifurcation of a singular point helps to find the oscillatory regimes that occur in the system when the parameters are changed.

Indeed, one of the most important applications is the occurrence of self-oscillations, i.e., the loss of the stability of the stationary regime and the birth of a limit cycle. The concept of the Poincaré–Andronov–Hopf bifurcation clarifies this phenomenon. Moreover, it is known that a loss of stability can occur in a “soft” or “hard” way. In the first case, a stable cycle, which is born near a stationary regime, attracts closed trajectories, and then, instead of the stationary regime, a self-oscillating regime is observed in the system. The amplitude of oscillations near the bifurcation is small; approximately, it is equal to the square root of the deviation of the parameter from the bifurcation value. In the second case, the unstable limit cycle merges with a stable stationary solution, and therefore, there are no stable regimes near the stationary points, so the system jumps abruptly to another regime: a steady-state, a periodic, or a more complex regime, far from the studied equilibrium (see [15]).

9.7. Other Applications

For completeness, we can mention also other areas of applications closely related to the results described above such as the theory of holomorphic foliations, the theory of symplectic manifolds, the theory of normal forms, the theory of Picard–Fuchs differential equations (see [35]), the theory of evolutionary processes, nonlinear control theory (see [36]), and so on.

10. Conclusions

We discussed an approach to the problem of computing the Poincaré index of vector fields on varieties with singularities that is based on the fact that the index can be described in terms of the homology of a contravariant version of the classical Poincaré–de Rham complex of regular and meromorphic differential forms. This idea allows not only simplifying the calculations, but also makes it possible to clarify the meaning of the basic constructions, which have been exploited in many previous works related to this subject. We also discussed a series of important, but little-known applications in physics, mechanics, control theory, deformation theory, bifurcation theory, and others.