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A unified calculation for Gromov norm of Kähler calss of bounded symmetric domains

Yuan Liu

Abstract

We provide a unified way to calculate the Gromov norm of the Kähler class of all (compact manifolds uniformized by) bounded symmetric domains. This was done for three classical domains by Domin and Toledo and for the general case by Clerc and Ørsted. Here, the calculation is much simplified by a combination of the ideas in Domin-Toledo and a work of Toledo, with the help of the Polydisc Theorem. The equality is achieved if and only if the triangle is ideal with three vertices on the Shilov boundary.

A unified calculation for Gromov norm of Kähler calss of bounded symmetric domains

Abstract

We provide a unified way to calculate the Gromov norm of the Kähler class of all (compact manifolds uniformized by) bounded symmetric domains. This was done for three classical domains by Domin and Toledo and for the general case by Clerc and Ørsted. Here, the calculation is much simplified by a combination of the ideas in Domin-Toledo and a work of Toledo, with the help of the Polydisc Theorem. The equality is achieved if and only if the triangle is ideal with three vertices on the Shilov boundary.
Paper Structure (6 sections, 1 theorem, 18 equations)

This paper contains 6 sections, 1 theorem, 18 equations.

Table of Contents

  1. Introduction
  2. Reduction to the integral on triangles
  3. Estimation of the integral
  4. Background on the bounded symmetric domains
  5. Construction of special coordinates
  6. Estimation of the integral

Key Result

Theorem 3.1

With the notation as above, there exists a totally geodesic complex submanifold $D$ of $X$ such that $(D,g_0|_D)$ is isometric to a Poincaré polydisc $\Delta^r$ (equipped with product of the Poincaé metric) and

Theorems & Definitions (1)

  • Theorem 3.1: Polydisc Theorem