Split distributions on Grassmann manifolds and smooth quadric hypersurfaces

TL;DR

The paper investigates holomorphic distributions on Grassmann manifolds and smooth quadric hypersurfaces, establishing when their tangent and conormal sheaves split into line bundles and how this relates to the geometry of their singular loci. By leveraging Ottaviani’s splitting criterion and Bott–Weil–Bott vanishing, it proves that splitting of $T_{\mathscr{F}}$ enforces arithmetically Cohen–Macaulay or Buchsbaum properties for the singular locus $Z$ on $Gr(k,n)$ and $Q_n$, and conversely under suitable codimension and cohomology hypotheses, $Z$ dictates splitting of the distributions. The work extends known results from $\mathbb{P}^n$ and Fano threefolds to homogeneous varieties, providing explicit criteria for both tangent and conormal sheaves, including dimension-specific refinements and connections between the geometry of $Z$ and the ambient sheaves. These results deepen the understanding of how distribution geometry interacts with the ambient algebraic structure of Grassmannians and quadrics, with potential implications for classifications of foliations on these spaces.

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.

Theorems & Definitions (56)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4