K\"ahler-Ricci flow on rational homogeneous varieties

TL;DR

This paper explicitly describes solutions to the Kähler-Ricci flow on rational homogeneous varieties, linking geometric evolution with algebraic invariants and providing bounds for geometric quantities.

Contribution

It offers an explicit description of homogeneous Kähler-Ricci flow solutions using representation theory, connecting geometric invariants with algebraic and symplectic embedding properties.

Findings

Explicit solutions for homogeneous Kähler-Ricci flow

Bounds for curvature, volume, diameter, and Laplacian eigenvalues

Relations between flow invariants and algebraic geometric constants

Abstract

In this work, we study the K\"{a}hler-Ricci flow on rational homogeneous varieties exploring the interplay between projective algebraic geometry and representation theory which underlies the classical Borel-Weil theorem. By using elements of representation theory of semisimple Lie groups and Lie algebras, we give an explicit description for all solutions of the homogeneous K\"{a}hler-Ricci flow on rational homogeneous varieties. This description enables us to compute explicitly the maximal existence time for any homogeneous solution and obtain explicit upper and lower bounds for several geometric quantities along the flow, including curvatures, volume, diameter, and the first non-zero eigenvalue of the Laplacian. As an application of our main result, we investigate the relationship between numerical invariants associated to ample divisors and numerical invariants arising from solutions of the homogeneous K\"{a}hler-Ricci flow. In the particular setting of full flag varieties, we prove that the numerical invariants obtained from solutions of the homogeneous K\"{a}hler-Ricci flow can be related to certain well-known invariants which appear in some different contexts, including the global Seshadri constant of ample line bundles, the maximum possible radius of embeddings of symplectic and K\"{a}hler balls, and the log canonical threshold of ample $\mathbb{Q}$-divisors. From this, we obtain constraints in terms of the scalar and Ricci curvatures, respectively, for symplectic embeddings of open Euclidean balls, and for the triviality of analytic multiplier ideal sheaves defined by ample $\mathbb{Q}$-divisors.