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

Abstract

Let $X$ be a compact complex manifold, consider a small deformation $\varphi :\mathcal{X}\to B$ 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.