# On equivariant indices of 1-forms on varieties

## TL;DR

This paper compares two equivariant indices of 1-forms on complex varieties with group actions, showing they coincide on smooth varieties and proposing their difference as an equivariant Milnor number.

## Contribution

It establishes the equality of equivariant homological and radial indices on smooth G-varieties and introduces their difference as a new invariant, the equivariant Milnor number.

## Key findings

- Indices coincide on smooth G-varieties.
- Difference defines an equivariant Milnor number.
- Provides tools for studying singularities with symmetry.

## Abstract

For a G-invariant holomorphic 1-form with an isolated singular point on a germ of a complex-analytic G-variety with an isolated singular point (G is a finite group) one has notions of the equivariant homological index and of the (reduced) equivariant radial index as elements of the ring of complex representations of the group. We show that on a germ of a smooth complex-analytic G-variety these indices coincide. This permits to consider the difference between them as a version of the equivariant Milnor number of a germ a G-variety with an isolated singular point.