Proceedings of Symposia in Pure Mathematics Volume 78, 2008 Universal unfoldings of Laurent polynomials and tt* structures Claude Sabbah ABSTRACT. This article surveys the relations between harmonic Higgs bundles and Saito structures which lead to tt* geometry on Frobenius manifolds. We give the main lines of the proof of the existence of a canonical tt* structure on the base space of the universal unfolding of convenient and nondegenerate Laurent polynomials. CONTENTS Introduction 2 1. Harmonic Frobenius manifolds 3 l.a. Harmonic Higgs bundles 3 l.b. Saito structures on holomorphic bundles 3 I.e. Harmonic Higgs bundles with supplementary structures 3 l.d. Saito structures with supplementary structures 6 I.e. Harmonic Frobenius manifolds 6 l.f. Examples 7 2. Integrable variations of twistor structures 10 2.a. Families of vector bundles on the Riemann sphere 11 2.b. Variation of twistor structures 12 2.c. Supplementary structures on a variation of twistor structure 14 2.d. A criterion for a Saito structure to be harmonic 17 3. Twistor structures through Fourier-Laplace transform 17 3.a. Laplace transform 18 3.b. Fourier transform of a sesquilinear pairing 18 3.c. Application to variations of Hodge structures 20 4. The harmonic Frobenius manifold attached to a Laurent polynomial 22 4.a. The Frobenius manifold attached to a Laurent polynomial 23 4.b. Twistor structures associated to tame functions 24 4.c. The canonical harmonic Frobenius manifold structure 26 References 27 2000 Mathematics Subject Classification. 53D45, 32S40, 14C30, 32L25, 34M55. Key words and phrases. Higgs bundle, Saito structure, tt*-structure, Fourier-Laplace trans- form, Frobenius manifold, Laurent polynomial. ©2008 American Mathematical Society
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