Expected Centre of Mass of the Random Kodaira Embedding

Abstract

AbstractLet$$X \subset \mathbb {P}^{N-1}$$X⊂PN-1be a smooth projective variety. To each$$g \in SL (N , \mathbb {C})$$g∈SL(N,C)which induces the embedding$$g \cdot X \subset \mathbb {P}^{N-1}$$g·X⊂PN-1given by the ambient linear action we can associate a matrix$$\bar{\mu }_X (g)$$μ¯X(g)called the centre of mass, which depends nonlinearly ong. With respect to the probability measure on$$SL (N , \mathbb {C})$$SL(N,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.