Algebraic Geometry Seminar

The theory of D-modules provides very powerful tools and solved many important problems. In this talk, I will introduce a natural way to generalize the D-module theory in logarithmic geometry. I will explain log Bernstein inequality and define log holonomic D-modules on smooth log schemes. Then I will explain log constructibility by using Kato-Nakayama spaces associated to log schemes. If time allowed, I will also explain how the theory is related to the classical b-function theory as well as log Riemann-Hilbert correspondence. This is based on an ongoing project with Andreas Hohl.

In this talk, I will report recent progress on the minimal model program for foliations, applying the theory of Birkar--Zhang’s generalized pairs. Particularly, I will discuss the ACC for lc thresholds and the canonical bundle formula for foliations. Part of this talk is based on a series of joint works with Omprokash Das, Yujie Luo, and Roktim Mascharak, Fanjun Meng, and Lingyao Xie.

We show that the Chern numbers of a smooth complex projective threefold are bounded by a constant which depends only on the topological type of the threefold, provided that the cubic form of the threefold has non-zero discriminant. This is a joint work with Paolo Cascini.

In this talk, I will discuss the behavior of positivity, hyperbolicity, and Kodaira dimension under smooth morphisms of complex quasi-projective manifolds. This includes a vast generalization of a classical result: a fibration from a projective surface of non-negative Kodaira dimension to a projective line has at least three singular fibers. Furthermore, I will explain a proof of Popa's conjecture on the superadditivity of the log Kodaira dimension over bases of dimension at most three. These theorems are applications of the main technical result, namely the logarithmic base change theorem.