Behrouz Taji (UNSW)

PDL C-38

Title pre-talk: Hodge theoretic tools in birational geometry

Abstract pre-talk: I will discuss a few applications of some basic tools in the theory of variation of Hodge structures (VHS) to the birational geometry of families of varieties.

Keywords pre-talk: projective families of varieties, weak semipositivity, VHS, Shafarevich-type problems, minimal model program

Abstract pre-talk: I will discuss a few applications of some basic tools in the theory of variation of Hodge structures (VHS) to the birational geometry of families of varieties.

Keywords pre-talk: projective families of varieties, weak semipositivity, VHS, Shafarevich-type problems, minimal model program

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Abstract main talk: It was discovered by Parshin that finiteness conjectures for the number of rational points of smooth projective curves of high genus (Mordell Conjecture) reduces to finiteness problems for families of such curves over a fixed base scheme of dimension one (Shafarevich Conjecture). The latter was resolved by Faltings,

Arakelov and others through two major breakthroughs: (1) Deligne-Mumford's results on the moduli of stable curves and (2) a certain numerical inequality for smooth families of projective curves over one dimensional base which only depends on a natural invariant of the fibers. (2) is nowadays referred to as an Arakelov inequality.

In this talk, after covering some of the historical background, I will discuss our recent generalization of Arakelov inequality to families of canonically polarized manifolds over arbitrary base schemes, which affirmatively answers a question of Viehweg. This is based on joint work (https://arxiv.org/abs/2205.761 ) with Sándor Kovács.

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