Hermitian Lie algebroids over analytic spaces

TL;DR

This paper develops the theory of Hermitian metrics on holomorphic Lie algebroids over complex analytic spaces, examining their geometric structures, cohomologies, and characteristic foliations to advance complex differential geometry.

Contribution

It introduces Hermitian metrics on holomorphic Lie algebroids and extends equivariant de Rham cohomology to this setting, linking geometric and cohomological aspects.

Findings

Defined Hermitian metrics on holomorphic Lie algebroids

Analyzed characteristic foliations and their inner products

Extended equivariant de Rham cohomology to Hermitian Lie algebroids

Abstract

We explore complex Riemannian geometry and Hermitian metrics on complex algebraic varieties and analytic spaces, respectively. In particular, we introduce Hermitian metrics on holomorphic Lie algebroids and examine the associated characteristic foliation with its canonically induced inner product. Furthermore, we study hypercohomologies related to the leaf space, leaves, and certain invariant subspaces arising from the characteristic foliation of a holomorphic Lie algebroid over a Hermitian manifold. Finally, we extends the concept of equivariant de Rham cohomology to the setting of Hermitian Lie algebroids.