Journal of Differential Geometry

$G$-invariant holomorphic Morse inequalities

Martin Puchol

Abstract

Consider an action of a connected compact Lie group on a compact complex manifold $M$, and two equivariant vector bundles $L$ and $E$ on $M$, with $L$ of rank $1$. The purpose of this paper is to establish holomorphic Morse inequalities à la Demailly for the invariant part of the Dolbeault cohomology of tensor powers of $L$ twisted by $E$. To do so, we define a moment map $\mu$ by the Kostant formula and we define the reduction of $M$ under a natural hypothesis on $\mu^{-1} (0)$. Our inequalities are given in term of the curvature of the bundle induced by $L$ on this reduction.

Article information

Source
J. Differential Geom., Volume 106, Number 3 (2017), 507-558.

Dates
First available in Project Euclid: 15 July 2017

https://projecteuclid.org/euclid.jdg/1500084025

Digital Object Identifier
doi:10.4310/jdg/1500084025

Mathematical Reviews number (MathSciNet)
MR3680556

Zentralblatt MATH identifier
06846959

Citation

Puchol, Martin. $G$-invariant holomorphic Morse inequalities. J. Differential Geom. 106 (2017), no. 3, 507--558. doi:10.4310/jdg/1500084025. https://projecteuclid.org/euclid.jdg/1500084025