Deformations of the tangent bundle of a projective hypersurface

TL;DR

This paper computes the dimension of the space of infinitesimal deformations of the tangent bundle of certain hypersurfaces, showing all are unobstructed, and proves the moduli space component containing the tangent bundle is rational.

Contribution

It explicitly determines the deformation space dimension of the tangent bundle for hypersurfaces and proves the associated moduli component is rational, with all deformations unobstructed.

Findings

Dimension of deformation space: ${n+d-1 race d} (d-1)$

All infinitesimal deformations are unobstructed

The moduli component is a rational variety

Abstract

For a nonsingular hypersurface $X \subset \mathbb{P}^n, n \geq 4,$ of degree $d \geq 2$, we show that the space $H^1(X, \End(T_X))$ of infinitesimal deformations of the tangent bundle $T_X$ has dimension ${n+d-1 \choose d} (d-1)$ and all infinitesimal deformations are unobstructed even though $H^2(X, \End(T_X))$ can be nonzero. Furthermore, we prove that the irreducible component of the moduli space of stable bundles containing the tangent bundle is a rational variety, by constructing an explicit birational model.