The homological index and the De Rham complex on singular varieties (Q2879059)

Gomez-Mont introduced the concept of the homological index of a vector field on a reduced pure-dimensional complex analytic space [\textit{X. Gómez-Mont}, J. Algebr. Geom. 7, No. 4, 731--752 (1998; Zbl 0956.32029)]. He considers the alternating sum of dimensions of the homology groups of the truncated de Rham complex of holomorphic differential forms whose differential is defined by the contraction of differential forms along the vector field, i.e., the Euler-Poincaré characteristic of the contracted de Rham complex. Under certain finiteness assumptions the homological index is equal to the classical local topological index up to a constant not depending on the vector field.NEWLINENEWLINESeveral methods of computation of the homological index for some types of singular varieties (Cohen-Macaulay curves, graded normal surfaces, complete intersections) are discussed. The homology of the contracted de Rham complex can be computed with the use of meromorphic differential forms. In the case of quasihomogeneous complete intersections with isolated singularities an explicit formula for the homological index is given.