| Description |
We prove that the zero-locus of any global holomorphic log-one-form on a projective log-smooth pair of log-general type must be non-empty. This result is a natural generalization of the work of M. Popa and C. Schnell in 2015, which states that, the zero-locus of any global holomorphic one-form on a smooth projective variety of general type must be non-empty. We first give a simplified proof of the result of Popa and Schnell. Instead of using generic vanishing of mixed Hodge modules on abelian varieties, we obtain a new proof using only Kodaira-Saito Vanishing. To prove our result about log-one forms, we apply Saito's mixed Hodge modules theory. We prove some logarithmic comparison theorems and we use a filtered log-$\mathscr{D}$-module to represent a mixed Hodge module, instead of using filtered $\mathscr{D}$-modules. The structure of the proof still follows the general outline of the work of Popa and Schnell. There are two important applications of our main result. One of them is that we get an affirmative answer (in a much more general setting) to a question posed by F. Catanese and M. Schneider. Another application is that we give an answer to the algebraic hyperbolicity part of Shafarevich's conjecture, with the generic fiber being Kawamata-log-terminal (klt) and of log-general type. |
