Surfaces of general type and sl_2-triples

Abstract

The sl_2-triples play a fundamental role for the structure theory of Lie algebras, and representation theory in general. Here we investigate sl_2-triples of global vector fields on schemes X in positive characteristics p>0, and develop a general theory for actions of the corresponding height-one group scheme G=SL_2[F]. Sending a point to the Lie algebra of its stabilizer defines rational maps to various Grassmann varieties. For surfaces of general type, this yields fibrations in curves of genus g at least 2 over the projective line. Using properties of the corresponding moduli stack M_g, we prove that there are no smooth surfaces of general type with an sl_2-triple. On the other hand, employing Lefschetz pencils and Frobenius pullbacks we show that canonical surfaces of general type with such triples exist in abundance. In this connection, we classify the rational double points where the tangent sheaf is free or the evaluation pairing with K\"ahler differentials is surjetive, including characteristic two.