Notes on direct images of pluricanonical bundles (Q6055057)

Given a surjective morphism \(f\colon X\to Y\) of smooth projective varieties, \(\dim Y = n\), the author proves a case of the relative Fujita conjecture; namely if \(\mathcal{L}\) is an ample line bundle on \(Y\) and \(j\) is the smallest integer such that \(\mathcal{L}^{\otimes j}\) is a base point free, then the sheaf \(f_*\omega_X^{\otimes m}\otimes \mathcal{L}^{\ell}\) is globally generated whenever \(\ell \geq m(jn+1)\). The result improves the bound proposed in [\textit{M. Popa} and \textit{C. Schnell}, Algebra Number Theory 8, No. 9, 2273--2295 (2014; Zbl 1319.14022)], which is \(\ell\geq m(jn+j)\) in this set-up. In fact, a re-run of the proof of Popa and Schnell's method reveals the stronger bound presented in this paper. For low dimensional \(Y\) (\(n\leq 4\)), if one is only concerned about the global generation at a general point, this bound is weaker than that in [\textit{Y. Dutta}, Ann. Inst. Fourier 70, No. 4, 1545--1561 (2020; Zbl 1472.14007)] which is \(\ell\geq m(n+1)\). More generally, for high values of \(m\) stronger bounds for general global generation were proven in [\textit{Y. Deng}, Int. Math. Res. Not. 2021, No. 23, 17611--17633 (2021; Zbl 1481.14012); \textit{Y. Dutta} and \textit{T. Murayama}, Algebra Number Theory 13, No. 2, 425--454 (2019; Zbl 1423.14048); \textit{M. Iwai}, Math. Z. 294, No. 1--2, 201--208 (2020; Zbl 1434.32031)] The author also proves the statement for log canonical pair \((X,\Delta)\) and \(Y\) any projective variety. Most importantly, over an algebraically closed field in positive characteristic \(p\), the author proves a similar statement for F-pure pairs \((X,\Delta)\) and with generically ample relative (twisted)-canonical bundle. The bound in this case is the same, however the statement holds only for pluricanonical bundle of certain minimum twist. The main input in this case is the Lemma 3.4 where the author shows a global generation statement for the \(p^e\)-th Frobenius image \(F^e_{Y*}(\mathcal{E}\otimes \mathcal{L}^{\otimes e})\) for any coherent sheaf \(\mathcal{E}\) and some appropriate bound \(a_e\).