The Homological Index and Algebraic Formulas

Abstract

We have already defined and studied several indices of vector fields on singular varieties, each of them being related to some property of the index of Poincaré–Hopf, or to some extension of the tangent bundle to the case of singular varieties. There is another line of research with remarkable works by various authors, that originates in the well-known fact (cf. Example 1.6.2) that for a holomorphic vector field v in Cⁿ with an isolated singularity at 0, the local Poincaré–Hopf index satisfies:

$${\rm Ind}_{{\rm PH}} (v,{\rm }0){\rm } = {\rm }\dim {\rm }{\mathcal O} _{{C}^n ,0} /(a_1 ,{\rm }\cdot{\rm }\cdot{\rm }\cdot{\rm },{\rm }a_n ),$$

where (a₁, … , a_n) is the ideal generated by the components of v. In the real analytic setting, the equivalent statement is given by the formula of EisenbudLevin–Khimshiashvili, expressing the local Poincaré–Hopf index through the signature of a certain quadratic form.

These facts motivated the search for algebraic formulas for indices of vector fields on singular varieties. A major contribution in this direction was given by V. I. Arnold for gradient vector fields. There are also significant contributions by various authors, such as X. Gòmez–Mont, S. Gusein–Zade, W. Ebeling and others.

In this chapter we give a glance of some of the research in this direction, and we refer to the literature for more on that topic. We discuss first the homological index for holomorphic vector fields, introduced by X. Gómez–Mont and further studied by himself in collaboration with Ch. Bonatti, P. Mardešić, L. Giraldo, H.-C. G. von Bothmer and W. Ebeling. In the last section of this chapter we discuss briefly the Eisenbud–Levin–Khimshiashvili formula for the index of real analytic vector fields, and its generalization to singular varieties.

The homological index has the important property of being defined for holomorphic vector fields on arbitrary complex analytic isolated singularity germs (V, 0). When the germ (V, 0) is a complete intersection, the homological index coincides with the GSV–index, by [17, 68].

If we now let V be a compact complex variety with isolated singularities, one has a well–defined notion of the total homological index for holomorphic vector fields on V with isolated singularities, defined in the usual way. This total index ought to be independent of the choice of vector field, being therefore an invariant of V . This is the case if V is a local complete intersection, and the corresponding global invariant is the 0–degree Fulton–Johnson class of V , as we will see in Chap. 11.

It would be interesting to know what the homological index measures for singular germs and varieties which are not local ICIS. This is related with extending the notion of Milnor number to isolated singularity germs which are not complete intersections (see Chap. 9).