Holomorphic differentials as functions of moduli

Holomorphic Differentials as Functions of Moduli

The purpose of this note is to strengthen the results of [3 ] and to indicate a very brief derivation of some theorems announced without proof in [ l ; 3 ] . We begin by indicating a correction to [3]. Let Si and S2 be Riemann surfaces, ƒ an orientation preserving (orientation reversing) homeomorphism of bounded eccentricity of Si onto 52 and [ƒ] the homotopy class of/; then (Si, [ƒ], S2) is ca...

Moduli of bounded holomorphic functions in the ball

We prove that there is a continuous non-negative function g on the unit sphere in C d, d ≥ 2, whose logarithm is integrable with respect to Lebesgue measure, and which vanishes at only one point, but such that no non-zero bounded analytic function m in the unit ball, with boundary values m⋆, has |m⋆| ≤ g almost everywhere. The proof analyzes the common range of co-analytic Toeplitz operators in...

Two-dimensional Yang-Mills theory and moduli spaces of holomorphic differentials

We describe and solve a new double scaling limit of large N Yang-Mills theory on a two-dimensional torus. We find the exact strong-coupling expansion in this limit and describe its relation to the conventional Gross-Taylor series. The limit retains only the chiral sector of the full gauge theory and the coefficients of the expansion determine the asymptotic Hurwitz numbers, in the limit of infi...

Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials

Symmetric Differentials of Rank 1 and Holomorphic Maps

Let X be a projective manifold of dimension n, a symmetric differential of degree m is a section of the m-th symmetric power of the sheaf of holomorphic 1-forms, SΩX . A symmetric differential w of degree m is of rank 1 if it can be locally written in the form w|U = fμ, where μ ∈ H(U,ΩX) and f ∈ O(U). Symmetric differentials of degree 1, i.e. holomorphic 1-forms, are trivially of rank 1. Holomo...