Recent developments in the minimal model program for Kähler varieties

Abstract

There are two main obstacles in extending the Minimal Model Program (MMP) for Kahler varieties: (i) given a compact Kahler manifold X, if the canonical bundle K_X is not nef, then does there exist a rational curve C\subset of X such that K_X\cdot C<0? (ii) If R is a K_X-negative extrema ray, can we contract it? When X is a projective variety, the positive answer to the first question comes from Mori’s famous Bend-Break theorem which involves reduction modulo prime characteristic, and thus not available for non-algebraic varieties. The second question also have positive answer in the projective case due to the Base-point free theorem, which says that if D is a nef supporting divisor of the extremal R and D-K_X is nef and big, then R can be contracted. However, a famous theorem of Moishezon says that if a compact Kahler manifold poses a big line bundle, then it is projective. Despite these tow major obstacles, in 2015–16, in a series of three papers, Campana, Höring and Peternell showed that if X is a compact Kahler 3-fold with terminal singularities, then MMP holds for X. They used deformation of curves on 3-folds to deal with Problem (i) and an ad-hoc contraction theorem of Fujiki for Problem (ii). Recently in a series of articles, jointly with Wenhao Ou, Christopher Hacon, Mihai Paun and Jose Yanez, we extended these results to dlt pairs in dimension 3 and some partial cases in dimension 4. Some of our methods are very different than those of Campana, Höring and Peternell and more suited for inductive arguments, which is desirable in higher dimensions. In this talk I survey the main results and techniques of our work.

About the Speaker

Omprokash Das

University of Utah

My area of interest is Algebraic Geometry. I am mainly interested in the Birational Geometry of algebraic varieties, namely, the Minimal Model Program and the singularities of the MMP in characteristic 0 and p.

Personal Homepage：

https://www.math.utah.edu/~das/om.html