Claus Hertling

Nilpotent orbits of a generalization of Hodge structures

Abstract We study a generalization of Hodge structures which first appeared in the work of Cecotti and Vafa. It consists of twistors, that is, holomorphic vector bundles on ℙ1, with additional structure, a flat connection on ℂ*, a real subbundle and a pairing. We call these objects TERP-structures. We generalize to TERP-structures a correspondence of Cattani, Kaplan and Schmid between nilpotent orbits of Hodge structures and polarized mixed Hodge structures. The proofs use work of Simpson and Mochizuki on variations of twistor structures and a control of the Stokes structures of the poles at zero and infinity. The results are applied to TERP-structures which arise via oscillating integrals from holomorphic functions with isolated singularities.

Examples of non-commutative Hodge structures

We show that, under a condition called minimality, if the Stokes matrix of a connection with a pole of order two and no ramification gives rise, when added to its adjoint, to a positive semi- definite Hermitian form, then the associated integrable twistor structure (or TERP structure, or non- commutative Hodge structure) is pure and polarized.

SEMISTABLE BUNDLES ON CURVES AND REDUCIBLE REPRESENTATIONS OF THE FUNDAMENTAL GROUP

This note is an attempt to generalize Bolibruchs theorem from the projective line to curves of higher genus. We show that an irreducible representation of the fundamental group of an open in a curve of higher genus has always a representation as a regular system of differential equations on a semistable bundle of degree 0. Vice-versa, we show that given such a bundle and 3 points on the curve, one can construct an irreducible representation of the curve minus the 3 points such that an associated regular system of differential equations lives on this bundle.

Unfoldings of meromorphic connections and a construction of Frobenius manifolds

Let M be a complex manifold. A structure of a Frobenius manifold on M defined by B. Dubrovin consists of several pieces of data of which the most important are: (a) the choice of a flat structure on M represented by a subsheaf of flat vector fields T M f of the sheaf of holomorphic vector fields T M; (b) a commutative and associative O M-bilinear multiplication o on T M.

An update on semisimple quantum cohomology and F-manifolds

AbstractIn the first section of this note, we show that Theorem 1.8.1 of Bayer-Manin can be strengthened in the following way: If the even quantum cohomology of a projective algebraic manifold V is generically semisimple, then V has no odd cohomology and is of Hodge-Tate type. In particular, this answers a question discussed by G. Ciolli. In the second section, we prove that an analytic (or formal ) supermanifold M with a given supercommutative associative

Bernstein Polynomial and Tjurina Number

\mathcal{O}_M

Brieskorn Lattices and Torelli Type Theorems for Cubics in ℙ3 and for Brieskorn-Pham Singularities with Coprime Exponents

-bilinear multiplication on its tangent sheaf