GSV-indices as residues

Journal of Singularities
volume 9 (2014), 206-218

Received 23 February 2012. Received in revised form 23 January 2013.

DOI: 10.5427/jsing.2014.9p

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Abstract:

We introduce a local invariant for a vector field v on a complete intersection V with an isolated singularity as the residue of the relevant Chern class of the ambient tangent bundle by a frame consisting of v and some natural meromorphic vector fields associated with defining functions of V. We then show that the residue coincides with the GSV-index as well as the virtual index of v so that it provides another interpretation of these indices. As an application, we give an algebraic formula for the GSV-indices of holomorphic vector fields on singular curves.

Author(s) information:

Tatsuo Suwa
Department of Mathematics
Hokkaido University
Sapporo 060-0810, Japan
email: tsuwa@sci.hokudai.ac.jp