Space Of Holomorphic Differentials Research Articles

Cohomology classes of strata of differentials

We introduce a space of stable meromorphic differentials with poles of prescribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of multiplicities of zeros of the differential. The main goal of this paper is to compute the Poincar\'e-dual cohomology classes of all strata. We prove that all these classes are tautological and give an algorithm to compute them. In a second part of the paper we study the Picard group of the strata. We use the tools introduced in the first part to deduce several relations in these Picard groups.

Explicit Galois representations of automorphisms on holomorphic differentials in characteristic p

We determine the representation of the Galois group for the cyclotomic function fields in characteristic p>0 induced by the natural action on the space of holomorphic differentials via construction of an explicit canonical basis of differentials. This includes those cases which present wild ramification and finite automorphism groups with non-cyclic p-part, which have remained elusive. We also obtain information on the gap sequences of ramified primes and extend these results to rank one Drinfel'd modules.

On the galois module structure of semisimple holomorphic differentials

LetL/K be a finite Galoisp-extension of algebraic function fields of one variable over an algebraically closed fieldk of characteristicp, with Galois groupG=Gal(L/K). The space ΏLs(0) of semisimple holomorphic differentials ofL is thek-vector space of holomorphic differentials which are fixed by the Cartier operator. We obtain the isomorphism classes and multiplicities of the summands in a Krull-Schmidt decomposition of thek[G]-module ΏLs(0) into a direct sum of indecomposablek[G]-modules.

Picard-type theorems for holomorphic mappings

where c. are constants and f' :C § T(W) is the derivative of the curve f. J Proof. We start the proof of the theorem by recalling some facts from algebraic geometry (see [2] for details). Let W be a compact complex manifold, D be a divisor on W with normal intersection, m = dim H~ be the dimension of the space of holomorphic differentials, and m + k = dim H ~ D)) be the dimension of the space of D-logarithmically meromorphic 1-forms on W.