Natural Lie Algebra bundles on rank two s-K\"ahler manifolds, abelian varieties and moduli of curves

TL;DR

This paper constructs natural Lie algebra bundles on rank two s-K"ahler manifolds, including abelian varieties, with flat connections in rigid cases, and explores examples like complex tori and moduli spaces.

Contribution

It introduces natural Lie algebra bundles on rank two s-K"ahler manifolds and analyzes their flat connections and actions on cohomology, expanding understanding of geometric structures.

Findings

Bundles have fibres isomorphic to so(s+1,s+1), su(s+1,s+1), and sl(2s+2,R)

In rigid cases, bundles admit natural flat connections

Examples include complex tori, abelian varieties, and moduli of elliptic curves

Abstract

We prove that one can obtain natural bundles of Lie algebras on rank two s-K\"ahler manifolds, whose fibres are isomorphic to so(s+1,s+1), su(s+1,s+1) and sl(2s + 2,\R). In the most rigid case (which includes complex tori and abelian varieties) these bundles have natural flat connections, whose flat global sections act naturally on cohomology. We also present several natural examples of manifolds which can be equipped with an s-K\"ahler structure with various levels of rigidity: complex tori and abelian varieties, cotangent bundles of smooth manifolds and moduli of pointed elliptic curves.