The Minimal Model Program aims to classify all algebraic varieties X (up to birational equivalence) via three building blocks: Fano varieties, Calabi-Yau varieties and canonically polarized varieties. Given a fibration from X to Z, for inductive reasons, we want to induce canonically certain structures on both X and Z associated to the fibration. It turns out that the correct structures are foliations and generalized pairs. The latter were formally introduced by Birkar and Zhang. I will talk about some recent progress in the Minimal Model Program for both generalized pairs and foliations. This talk is based on a series of joint works with G.Chen, J. Han, J. Liu and N.Tsakanikas.

]]>In 1953, Kodaira proved what is now called the Kodaira vanishing theorem, which states that if L is an ample divisor on a complex projective manifold X, then

H^i(X,-L) = 0 for all i < dim(X). Since then, Kodaira's theorem and its generalizations for complex projective varieties – in particular, the Kawamata–Viehweg vanishing theorem and its relative version due to Kawamata–Matsuda–Matsuki – have become indispensable tools in algebraic geometry over fields of characteristic zero, particularly in birational geometry and the minimal model program. However, while the goal in birational geometry is to study birational equivalences between projective varieties, recent progress in the minimal model program has shown that it would be very useful to have a version of the Kawamata–Viehweg vanishing theorem that holds for schemes and other geometric objects that are not necessarily projective varieties. In this talk, I will discuss my version of the Kawamata–Viehweg vanishing theorem for projective morphisms of schemes of equal characteristic zero. This result implies analogous vanishing theorems for projective morphisms of algebraic spaces, formal schemes, and both complex and non-Archimedean analytic spaces. My vanishing theorem is optimal given known counterexamples to vanishing theorems in positive and mixed characteristic and has many applications to both algebraic geometry and commutative algebra. I will discuss some of these applications, in particular my joint work with Shiji Lyu on the minimal model program with scaling for projective morphisms of excellent schemes, algebraic spaces, formal schemes, and both complex and non-Archimedean analytic spaces.

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