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Géométrie tt* et applications pluriharmoniques

Abstract : In this work we introduce the real differential geometric notion of a tt*-bundle (E,D,S), a metric tt*-bundle (E,D,S,g) and a symplectic tt*-bundle (E,D,S,w) on an abstract vector bundle E over an almost complex manifold (M,J). With notion we construct, generalizing Dubrovin,a correspondence between metric tt*-bundles over complex manifolds (M,J) and admissible} pluriharmonic maps from (M,J) into the pseudo-Riemannian symmetric space GL(r,R)/O(p,q) where (p,q) is the signature of the metric g. Moreover, we show a igidity result for tt*-bundles over compact Kähler manifolds and we obtain as application a special case of Lu's theorem.
In addition we study solutions of tt*-bundles (TM,D,S) on the tangent bundle TM of (M,J) and characterize an interesting class of these solutions which contains special complex manifolds and flat nearly Kähler manifolds. We analyze which elements of this class admit metric or symplectic tt*-bundles. Further we consider solutions coming from varitations of Hodge structures (VHS) and harmonic bundles. Applying our correspondence to harmonic bundles we generalize a correspondence given by Simpson. Analyzing the associated pluriharmonic maps we obtain roughly speaking for special Kähler manifolds the dual Gauß map and for VHS of odd weight the period map. In the case of non-integrable complex structures, we need to generalize the notions of pluriharmonic maps and some results.
Apart from the rigidity result we generalize all above results to para-complex geometry.
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https://tel.archives-ouvertes.fr/tel-01746630
Contributor : Lars Schäfer <>
Submitted on : Tuesday, April 24, 2007 - 6:51:32 PM
Last modification on : Friday, February 26, 2021 - 3:22:02 AM
Long-term archiving on: : Wednesday, April 7, 2010 - 12:28:48 AM

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Lars Schäfer. Géométrie tt* et applications pluriharmoniques. Mathématiques [math]. Université Henri Poincaré - Nancy 1, 2006. Français. ⟨NNT : 2006NAN10041⟩. ⟨tel-01746630v2⟩

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