Pseudoeffective thresholds and cohomology of twisted symmetric tensor fields on irreducible Hermitian symmetric spaces (Q6630739)

After Mori's solution to Hartshorne conjecture, the classification of Fano manifolds with certain positivity conditions has attracted lots of attention. In the past few years, several works imply that the bigness of tangent bundles is a rather restrictive property. A vector bundle \(E\) on a smooth projective variety \(X\) is called big, if the tautological line bundle \(\mathcal{O}_{\mathbb{P}(E)}(1)\) is big where \(\mathbb{P}(E)\) is the Grothendieck projectivization of the vector bundle \(E\), or equivalently, \(H^0(X,\text{Sym}^rE\otimes L^{-1})\neq 0\) holds for some positive integer \(r\) and some ample line bundle \(L\) on \(X\).\N\NIn the paper under review, the author introduces the notion of pseudoeffective thresholds to measure the bigness of tangent bundles. More precisely, given a smooth projective variety \(X\), the tautological class \(\xi=c_1(\mathcal{O}_{\mathbb{P}(T_X)}(1))\) of the Grothendieck projectivization \(\mathbb{P}(T_X)\) of the tangent bundle, and an ample divisor \(A\) on \(X\), the author defines the pseudoeffective threshold of \(T_X\) with respect to \(A\) as\N\[\N\Delta(X,A)=\sup\{\varepsilon\in\mathbb{R}~|~\xi-\varepsilon\pi^*A~\text{is effective}\},\N\]\Nwhere \(\pi\colon\mathbb{P}(T_X)\to X\) is the natural projection. Therefore, \(T_X\) is big if and only if \(\Delta(X,A)>0\). Some useful properties of the pseudoeffective threshold and related conjectures characterizing projective spaces and hyperquadrics are discussed in Section 3.\N\NOn the other hand, the author calculates the pseudoeffective thresholds for irreducible Hermitian symmetric spaces of compact type (IHSS for short) (see Corollary 1.4). To obtain this, the author uses the irreducible decomposition of \(\text{Sym}^rT_X\) to establish a result on the cohomology twisted symmetric tensor fields on IHSS (see Theorem 1.3); this also gives a cohomological characterization of the rank of Hermitian symmetric space of compact type (see Corollary 1.5).