Expected centre of mass of the random Kodaira embedding

Abstract

Let $X\subset {\mathbb{P}}^{N-1}$ be a smooth projective variety. To each $g\in SL(N,\mathbb{C})$ which induces the embedding $g\cdot X\subset {\mathbb{P}}^{N-1}$ given by the ambient linear action we can associate a matrix ${\bar{\mu }}_X(g)$ called the centre of mass, which depends nonlinearly on $g$. With respect to the probability measure on $SL(N,\mathbb{C})$ induced by the Haar measure and the Gaussian unitary ensemble, we prove that the expectation of the centre of mass is a constant multiple of the identity matrix for any smooth projective variety.

v2: 18 pages, minor corrections. v3: 14 pages, corrected proof, final version

References in corpus (4)

• Large deviations for Gibbs measures with singular Hamiltonians and emergence of Kahler-Einstein metrics
• Stability and integration over Bergman metrics
• Mapping properties of the Hilbert and Fubini--Study maps in Kähler geometry
• Entropy of Bergman measures of a toric Kaehler manifold