Expected centre of mass of the random Kodaira embedding

TL;DR

The paper proves that the expected centre of mass matrix, associated with the random Kodaira embedding of a smooth projective variety under certain probability measures, is proportional to the identity matrix.

Contribution

It establishes that the average centre of mass matrix is a scalar multiple of the identity for any smooth projective variety under Haar and Gaussian measures.

Findings

Expected centre of mass is proportional to the identity matrix.

The result holds for any smooth projective variety.

The proof involves probabilistic measures on the special linear group.

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.