Grothendieck-riemann-roch and the Moduli of Enriques Surfaces

A (complex) Enriques surface is a projective smooth connected algebraic surface Y over C with H(Y,OY ) = H (Y,OY ) = (0), (Ω 2 Y ) ⊗ 2 ≃ OY but Ω 2 Y 6≃ OY ([CF]). In this note we give a short proof of the fact that the coarse moduli space of complex Enriques surfaces is quasi-affine. This was first shown by Borcherds [B] using the denominator function of a generalized Kac-Moody superalgebra (the “Φ-function”). From this, it follows (see [BKPS]) that any complete family of complex Enriques surfaces is isotrivial. Our observation here is that the ingredient of Borcherds’ theory of infinite products can be completely removed from the proof and can be replaced by a simple use of the Grothendieck-RiemannRoch theorem.