Transcendental base-point freeness and minimal models for projective varieties

Algebraic Geometry Seminar

Organizers：

Caucher Birkar，Jia Jia 贾甲

Speaker：

Omprokash Das

Tata Institute of Fundamental Research

Time：

Fri., 15:30-16:30, Feb. 21, 2025

Venue：

B725, Shuangqing Complex Building A

Online：

Zoom Meeting ID: 262 865 5007

Passcode: YMSC

Title

Transcendental base-point freeness

and minimal models for projective varieties

The minimal model program for generalized pairs has become one of the fundamental tools for classifying higher dimensional algebraic varieties since its inception due to Birkar and Zhang. In this talk I will introduce an analytic version of generalized pairs, namely a triplet (X, B, T), where X is an analytic variety, B a boundary divisor and T is a bi-degree (1,1) current. The current T is the analog of a b-divisor which appears in the generalized pairs of Birkar and Zhang. We will then see that many expected results of MMP, e.g. the results parallel to BCHM still hold in this generality. Finally, as an application of this kind of MMP we will show that the transcendental base-point free theorem holds for projective varieties, which says that if X is a projective manifold and \alpha is a (1,1) Bott-Chern cohomology class on X such that \alpha-K_X is nef and big (in the analytic sense), then there is a projective morphism f:X \to Y to a normal compact Kahler variety Y and a Kahler form \omega_Y such that \alpha=f^*\omega_Y.