Expected Centre of Mass of the Random Kodaira Embedding

Let X ⊂ P N - 1 be a smooth projective variety. To each g ∈ S L ( N , C ) which induces the embedding g · X ⊂ P N - 1 given by the ambient linear action we can associate a matrix μ ¯ X ( g ) called the centre of mass, which depends nonlinearly on g . With respect to the probability measure on S L ( 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.