The cohomological support locus of pluri-canonical sheaves and the Iitaka fibration (Q2922853)

Let \(X\) be a smooth projective variety of non negative Kodaira dimension, \(A\) an abelian variety and \(F\) a sheaf on \(X\). The cohomological support of \(F\) with respect to a map \(a: X \to A\) is \(V^i(F,a)\), the set of divisors \(\alpha \in \mathrm{Pic}^0(A)\) for which the \(i\)-cohomology \(h^i(F \otimes a^*\alpha) \neq 0\). In the particular case in which \(F=\omega_X\) is the canonical sheaf and \(a\) the Albanese map the \(V^i(\omega_X, a)\)'s are of interest in relation with pluricanonical maps of \(X\) (in particular to prove results of birationality of these maps). The main result of the paper (see Theorem 1.2) is on the cohomological support of some pluricanonical sheaves \(\omega_X^m\). Under some particular assumptions (see AS(1,2)), the subgroup of \(\mathrm{Pic}^0(X)\) generated by the translates through the origin of the components of \(V^0(\omega_X^m,a)\) (\(a\) the Albanese map) is \(I^*\mathrm{Pic}^0(S)\) (\(I:X \to S\) the Iitaka fibration). Moreover \(q(S)\) can be computed from \(q(X)\) and dimension and Kodaira dimension of \(X\) and of the general fiber \(F\) of the Stein factorization of the Albanese map. Applications to pluricanonical maps (of \(F\)) and some considerations on the assumptions AS(1,2) are also provided.