Cohomology of twisted symmetric powers of cotangent bundles of smooth complete intersections

TL;DR

This paper develops explicit resolutions for cohomology calculations of twisted symmetric powers of cotangent bundles in smooth complete intersections, enabling computational implementation and leading to new geometric insights.

Contribution

It introduces explicit resolutions of sheaves on projectivized tangent bundles, improving cohomology computations and applications in algebraic geometry.

Findings

Recovered known vanishing theorems and proved their optimality.

Improved results on symmetric algebra of global sections of cotangent bundles.

Provided explicit computational methods for cohomology and new examples of surfaces.

Abstract

In this paper, we provide two different resolutions of structural sheaves of projectivized tangent bundles of smooth complete intersections. These resolutions allow in particular to obtain convenient (and completely explicit) descriptions of cohomology of twisted symmetric powers of cotangent bundles of complete intersections, which are easily implemented on computer. We then provide several applications. First, we recover the known vanishing theorems on the subject, and show that they are optimal via some non-vanishing theorems. Then, we study the symmetric algebra of global sections of symmetric powers of $\Omega_X(1)$, where $X$ is a smooth complete intersection of codimension $c < N/2$, improving the known results in the literature. We also study partial ampleness of cotangent bundles of general hypersurfaces. Finally, we illustrate how the explicit descriptions of cohomology can be implemented on computer. In particular, this allows to exhibit new and simple examples of family of surfaces along which the canonically twisted pluri-genera do not remain constant.