On the density of strebel differentials

Abstract

w 1. Statement of the Main Result Let X be a compact Riemann surface of genus g > 2 and denote by O the sheaf of germs of holomorphic differential 1-forms on X. Let q~H~ ~| be a nonzero holomorphic quadratic form on X, i.e. a form which can be expressed in a local coordinate z as f(z)dz 2, with f holomorphic. Denote by Xq the normalization of the complex curve defined in the total space of the cotangent bundle of X by the equation y2 =q(x), where x~X and yeT*. The surface Xq is the "Riemann surface of q~", i.e. under the projection 7r: (x,y)~-~x, the surface ~'q is a double cover of X ramified at the odd zeroes of q, and there is a canonical form __ 2 % ~ H~ O) such that 7r* q - ~oq.