The homological index and the De Rham complex on singular varieties

Journal of Singularities
volume 9 (2014), 1-26

Received 15 May 2012. Received in revised form 15 May 2014.

DOI: 10.5427/jsing.2014.9a

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Abstract:

We discuss several methods of computation of the homological index originated in a paper by X. Gómez-Mont for vector fields given on singular complex varieties. Our approach takes into account basic properties of holomorphic and regular meromorphic differential forms and is applicable in different settings depending on concrete types of varieties. Among other things, we describe how to compute the index in the case of Cohen-Macaulay curves, graded normal surfaces and complete intersections by elementary calculations. For quasihomogeneous complete intersections with isolated singularities, an explicit formula for the index is obtained; it is a direct consequence of earlier results of the author. Indeed, in this case the computation of the homological index is reduced to the use of Newton's binomial formula only.

Keywords:

holomorphic differential forms; contracted De Rham complex; regular meromorphic forms; torsion and cotorsion; generating functions; graded complete intersections; Lebelt resolutions

Mathematical Subject Classification:

32S25, 14F10, 14F40, 58K45, 58K70

Author(s) information:

A.G. Aleksandrov
Institute for Control Sciences
Russian Academy of Sciences
Profsojuznaja str. 65, GSP-7, B-342,
Moscow, 117997, Russian Federation