Recent advances in Hodge theory, the theory of singular hermitian metrics and moduli theory of higher dimensional varieties have led to major breakthroughs in solving long-standing problmes in complex algebraic geometry, in particular birational geometry.

In Hodge theory, Saito’s theory of Hodge modules have proved to be a specially relevant framework for the study of hyperbolicity properties of the base spaces of families of smooth varieties, admitting a (relative) good minimal model. In particular, it has been shown that such base spaces are always of log-general type, whenever the family has maximal variation, proving a deep conjecture of Viehweg (Viehweg, Zuo, Kebekus, Kovács, Campana, Paun, Popa, Schnell and others).

Furthermore, these generalized Hodge theoretic notions have resulted in new (and more general) proofs of vanishing results in birational geometry (Popa, Kovács, Mustata, Wu, Arapura and others).

In a different but closely related direction, a new, deeper understanding of singular metrics on higher rank, singular sheaves with “good" curvature properties has emerged (Berndtsson, Paun, Takayama, Cao and others). Major applications to Iitaka’s conjecture for Kodaira dimension of algebraic fibre spaces have (subsequently) followed (Pǎun, Cao, Hacon, Popa, Schnell and others).

In moduli theory, the construction of moduli spaces of higher dimensional varieties is a fundamental problem. Here, as the result of Kollár and Kovács show, construction of “reasonable" moduli functors require close analysis of Hodge theoretic aspects of singularities of stable varieties.

The aim of this week is to further investigate these emerging methods with a view towards applications in birational geometry and moduli spaces. Here are the outlines of the main topics that will be covered in this week.

• Hodge theory. This includes an introduction to the theory of Hodge modules. Investigation of new applications of this theory in the study of Viehweg’s hyperbolicity conjecture and its various generalizations.

• Moduli of higher dimensional varieties. This consists of studying the Hodge theoretic aspects of degeneration of singularities of stable varieties and their role in construction of moduli functors for higher dimensional varieties.

• Interactions between Analytic and Hodge theories. The main aim here is to study how the two areas complement each other specially within the context of moduli problems and the Iitaka conjecture.

​Junior women mathematicians and those from underrepresented minority groups will be given priority for financial help.