Integration of simplicial forms and Deligne cohomology

We present two approaches to constructing an integration map for smooth Deligne cohomology. The first is defined in the simplicial model, where a class in Deligne cohomology is represented by a simplicial form, and the second in a related but more combinatorial model.Comment: 28 pages, section on products adde

Cohomology of affine Artin groups and applications

The result of this paper is the determination of the cohomology of Artin groups of type A_n, B_n and \tilde{A}_{n} with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type B_n with coefficients over the module \Q[q^{\pm 1},t^{\pm 1}]. Here the first (n-1) standard generators of the group act by (-q)-multiplication, while the last one acts by (-t)-multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type \tilde{A}_{n} as well as the cohomology of the classical braid group {Br}_{n} with coefficients in the n-dimensional representation presented in \cite{tong}. The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be K(\pi,1) spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.Comment: 21 pages, 4 figure

Cohomology of Line Bundles: Applications

Massless modes of both heterotic and Type II string compactifications on compact manifolds are determined by vector bundle valued cohomology classes. Various applications of our recent algorithm for the computation of line bundle valued cohomology classes over toric varieties are presented. For the heterotic string, the prime examples are so-called monad constructions on Calabi-Yau manifolds. In the context of Type II orientifolds, one often needs to compute equivariant cohomology for line bundles, necessitating us to generalize our algorithm to this case. Moreover, we exemplify that the different terms in Batyrev's formula and its generalizations can be given a one-to-one cohomological interpretation. This paper is considered the third in the row of arXiv:1003.5217 and arXiv:1006.2392.Comment: 56 pages, 8 tables, cohomCalg incl. Koszul extension available at http://wwwth.mppmu.mpg.de/members/blumenha/cohomcalg

Floer cohomology of torus fibers and real lagrangians in Fano toric manifolds

In this article, we consider the Floer cohomology (with $\Z_2$ coefficients) between torus fibers and the real Lagrangian in Fano toric manifolds. We first investigate the conditions under which the Floer cohomology is defined, and then develop a combinatorial description of the Floer complex based on the polytope of the toric manifold. We show that if the Floer cohomology is defined, and the Floer cohomology of the torus fiber is non-zero, then the Floer cohomology of the pair is non-zero. We use this result to develop some applications to non-displaceability and the minimum number of intersection points under Hamiltonian isotopy.Comment: v2: Modified exposition and new corollary adde