Projective Contact Manifolds

TL;DR

This paper characterizes certain projective contact manifolds, showing they are often projectivized tangent bundles, and explores properties of their canonical bundles and nef subsheaves.

Contribution

It proves that projective contact manifolds with specific Betti number and canonical bundle conditions are projectivized tangent bundles, advancing understanding of their structure.

Findings

Projective contact manifolds with b_2 ≥ 2 and non-nef canonical bundle are P(T_Y).

Canonical bundle of such manifolds is never nef unless K_X^2=0 and K_X is not trivial.

Studied nef rank 1 subsheaves in cotangent bundles proportional to the canonical bundle.

Abstract

We prove that a projective contact manifold X with second Betti number at least 2 whose canonical bundle K_X is not nef, is always the projectivised tangent bundle P(T_Y) of a projective manifold Y. It is expected that the canonical bundle of a projective contact manifold is never nef; we prove this unless possibly K_X^2 = 0 and K_X is not numerically trivial. Moreover we study more generally nef subsheaves of rank 1 in the cotangent bundle which are proportional to the canonical bundle.