Morse theory and toric vector bundles

TL;DR

This paper explores the relationship between Morse theory and toric vector bundles, demonstrating how Morse-theoretic filtrations align with Klyachko filtrations and providing conditions for complexes of sheaves to correspond to vector bundles.

Contribution

It establishes a connection between Morse theory and the filtrations of toric vector bundles, offering new criteria for identifying vector bundles via microlocal conditions.

Findings

Morse-theoretic filtrations coincide with Klyachko filtrations for toric vector bundles.

Provided Morse-theoretic conditions for complexes of sheaves to correspond to vector bundles.

Extended the coherent-constructible correspondence to include filtrations and microlocal criteria.

Abstract

Morelli's computation of the K-theory of a toric variety X associates a polyhedrally constructible function on a real vector space to every equivariant vector bundle E on X. The coherent-constructible correspondence lifts Morelli's constructible function to a complex of constructible sheaves kappa(E). We show that certain filtrations of the cohomology of kappa(E) coming from Morse theory coincide with the Klyachko filtrations of the generic stalk of E. We give Morse-theoretic (i.e. microlocal) conditions for a complex of constructible sheaves to correspond to a vector bundle, and to a nef vector bundle.