The Jumping Phenomenon of the Dimensions of Cohomology Groups of Tangent Sheaf

TL;DR

This paper investigates how the dimensions of cohomology groups of the tangent sheaf on a compact complex manifold change under small deformations, providing formulas for the obstructions involved.

Contribution

It introduces a formula for the obstructions to deform classes in cohomology groups, explaining the jumping phenomenon of these dimensions during deformations.

Findings

Derived formulas for obstructions to deformation

Analyzed conditions for dimension jumps in cohomology groups

Provided insights into the deformation behavior of tangent sheaf cohomology

Abstract

Let $X$ be a compact complex manifold, consider a small deformation $\phi: \mathcal{X} \to B$ of $X$, the dimensions of the cohomology groups of tangent sheaf $H^q(X_t,\mathcal{T}_{X_t})$ may vary under this deformation. This paper will study such phenomenons by studying the obstructions to deform a class in $H^q(X,\mathcal{T}_X)$ with the parameter $t$ and get the formula for the obstructions.