Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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Hermitian Curvature Flow on complex homogeneous manifolds

Yury Ustinovskiy

created by daniele on 22 May 2018
modified on 03 Jul 2018

19 jul 2018 -- 14:30

Aula Tricerri, DiMaI, Firenze

Abstract.

We present a version of the Hermitian curvature flow (HCF) introduced in arXiv:1604.04813. This flow is proved to preserve many complex-analytic curvature (semi)positivity conditions. In the talk, we focus on complex homogeneous manifolds equipped with submersion metrics, since those satisfy dual-Nakano semipositivity. We prove that the finite-dimensional space of submersion metrics is invariant under the HCF and write down the corresponding ODE on the space of Hermitian forms on the underlying Lie algebra. Using these computations we construct HCF-Einstein metrics on G-homogeneous manifolds, where G is a complexification of a compact simple Lie group. We conjecture that under the HCF any induced metric on such a manifold pinches towards the HCF-Einstein metric. For a nilpotent or solvable complex Lie group equipped with an induced metric we investigate the blow-up behavior of the HCF.

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