Variations of (para-)Hodge structures and their period maps in tt*-geometry

Lars Schäfer

Abstract: We introduce the notion of variations of Hodge structures (VHS) in para-complex geometry and define the associated period map. Moreover, we construct VHS from special (pa\-ra-)complex and (para-)Kähler manifolds and prove that they provide solutions of (metric) $tt^*$-bundles (cf. [H] for the complex case). In the case of odd weight we relate the period map to the (para-)pluriharmonic maps associated to $tt^*$-bundles (cf. [S2], [S3]).

\item[H] C. Hertling: tt*-geometry, Frobenius manifolds, their connections, and the construction for singularities. J. Reine Angew. Math. {\bf 555} (2003), 77--161. \item[S2] L. Schäfer, tt*-bundles and pluriharmonic maps. Ann. Global Anal. Geom. {\bf 28}(3) (2005), 285--300. \item[S3] L. Schäfer, $tt^*$-bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps. Diff. Geom. and Appl. {\bf 24}(1) (2006), 60--89.

Keywords: tt*-bundles, (para-)pluriharmonic maps, special (para-)complex and special (para-)Kähler manifolds, variations of (para-)Hodge structure, period maps

Classification (MSC2000): 53C43, 58E20, 32G20

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