Split distributions on Grassmann manifolds and smooth quadric hypersurfaces

TL;DR

This paper investigates the splitting behavior of tangent and conormal sheaves of holomorphic distributions on Grassmann manifolds and smooth quadric hypersurfaces, extending known results from Fano threefolds and projective spaces.

Contribution

It provides new criteria for when these sheaves split as sums of line bundles on these complex manifolds, generalizing previous work.

Findings

Conditions under which sheaves split as sums of line bundles.

Relationship between singular set properties and sheaf structure.

Extension of splitting results to Grassmannians and quadrics.

Abstract

This work is dedicated to studying holomorphic distributions on Grassmann manifolds and smooth quadric hypersurfaces. In special, we prove, under certain conditions, when the tangent and conormal sheaves of a distribution splits as a sum of line bundles on these manifolds, generalizing the previous works on Fano threefolds and $\mathbb{P}^{n}$. We analyze how the algebro-geometric properties of the singular set of singular holomorphic distributions relate to their associated sheaves.