K\"ahler-Ricci flow on rational homogeneous varieties

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.