Real K3 surfaces without real points, equivariant determinant of the Laplacian, and the Borcherds Phi-function

摘要

We consider an equivariant analogue of a conjecture of Borcherds. Let (Y, sigma) be a real K3 surface without real points. We shall prove that the equivariant determinant of the Laplacian of (Y, sigma) with respect to a sigma-invariant Ricci-flat Kahler metric is expressed as the norm of the Borcherds phi-function at the "period point". Here the period of (Y, sigma) is not the one in algebraic geometry.