Birational Geometry, Kähler–Einstein Metrics and Degenerations: Moscow, Shanghai and Pohang, April–November 2019

Vol. 409. Cham : Springer, 2023.

Editor-in-chief: I. Cheltsov, X. Chen, Katzarkov L. V., J. Park

This book collects the proceedings of a series of conferences dedicated to birational geometry of Fano varieties held in Moscow, Shanghai and Pohang

The conferences were focused on the following two related problems:

•  existence of Kähler–Einstein metrics on Fano varieties

•  degenerations of Fano varieties

on which two famous conjectures were recently proved. The first is the famous Borisov–Alexeev–Borisov Conjecture on the boundedness of Fano varieties, proved by Caucher Birkar (for which he was awarded the Fields medal in 2018), and the second one is the (arguably even more famous) Tian–Yau–Donaldson Conjecture on the existence of Kähler–Einstein metrics on (smooth) Fano varieties and K-stability, which was proved by Xiuxiong Chen, Sir Simon Donaldson and Song Sun. The solutions for these longstanding conjectures have opened new directions in birational and Kähler geometries. These research directions generated new interesting mathematical problems, attracting the attention of mathematicians worldwide.

These conferences brought together top researchers in both fields (birational geometry and complex geometry) to solve some of these problems and understand the relations between them. The result of this activity is collected in this book, which contains contributions by sixty nine mathematicians, who contributed forty three research and survey papers to this volume. Many of them were participants of the Moscow–Shanghai–Pohang conferences, while the others helped to expand the research breadth of the volume—the diversity of their contributions reflects the vitality of modern Algebraic Geometry.

Chapters

Interpretations of Spectra

Katzarkov L. V., Lee K. S., Svoboda J. et al., , in: Birational Geometry, Kähler–Einstein Metrics and Degenerations: Moscow, Shanghai and Pohang, April–November 2019Vol. 409.: Cham: Springer, 2023. P. 371–407.

The studies of homological mirror symmetry as correspondence of Lefshetz pencils was initiated as part of the general theory of categorical linear systems. In this paper, we look at the monodromy of these linear systems via a new notion of noncommutative spectrum. ...

Added: May 25, 2023

Keywords: Minimal model program, Birational rigidity, the Cremona group, Fano variety, Kahler-Einstein metrics, K-stability, rationally connected variety, Calabi problem, Sasaki-Einstein metric

Publication based on the results of: Spectral networks, spectra and F-bundles (2023)