On pseudoeffective thresholds and cohomology of twisted symmetric tensor fields on irreducible Hermitian symmetric spaces

Let $X$ be an irreducible Hermitian symmetric space of compact type (IHSS for short). In this paper, we give the irreducible decomposition of $Sy{m}^r{T}_X$. As a by-product, we give a cohomological characterization of the rank of $X$. Moreover, we introduce pseudoeffective thresholds to measure the bigness of tangent bundles of smooth complex projective varieties precisely and calculate them for irreducible Hermitian symmetric spaces of compact type.