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$$tt^*$$ geometry, Frobenius manifolds, their connections, and the construction for singularities. (English) Zbl 1040.53095
This paper constructs and studies a natural generalization of variations of Hodge structures called $$tt^*$$-geometry, as defined by S. Cecotti and C. Vafa [Topological-antitopological fusion, Nucl. Phys. B 367, 359–461 (1991), Commun. Math. Phys. 158, 569–644 (1993; Zbl 0787.58049)] together with Frobenius manifold structures on the base space of a semi-universal unfolding of a hypersurface singularity (not necessarily quasi-homogeneous) in a unified way. Both, Frobenius structures and $$tt^*$$-structures are first generalized and studied on abstract holomorphic vector bundles. These are related by a careful analysis of associated flat connections with poles and flat pairings on the pull-back bundle $$\pi^*K$$, where $$\pi$$ is the projection $$\pi:{P}^1 \times M \to M$$. Bundles with such a (generalized) $$tt^*$$-structure are special cases of C. T. Simpson’s “harmonic bundles” [J. Am. Math. Soc. 3, 713–770 (1990; Zbl 0713.58012)].
In the second half of the paper these investigations are specialized to the case of the tangent bundle $$TM$$. Using oscillating integrals the author then reconstructs the Saito Frobenius manifold structure on the semi-universal unfolding of a hypersurface singularity, previously done using the Gauss-Manin connection [C. Hertling, Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts Math. 151, Cambridge University Press (2002; Zbl 1023.14018)], and he shows that it has a canonical compatible $$tt^*$$-geometry – or rather, CDV-structure (Cecotti-Dubrovin-Vafa) – outside of a real analytic subvariety $$R\subset M$$, invariant under the flow of $$E-\overline{E}$$, where $$E$$ is the Euler vector field of the Frobenius manifold.
Finally, the author shows that starting at suitable semi-simple or nilpotent points and going sufficiently far along the flow of the real vector field $$E+\overline{E}$$ one no longer meets this bad locus $$R$$; the Hermitian form $$h:=g(\cdot,\kappa \cdot)$$ is positive definite, where $$g$$ is the metric for the Frobenius manifold, and $$\kappa$$ is the anti-involution of the $$tt^*$$-structure; and the endomorphism $$\mathcal Q$$ given by the $$tt^*$$-structure has eigenvalues tending toward a known set.

##### MSC:
 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects) 32S30 Deformations of complex singularities; vanishing cycles 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 53C28 Twistor methods in differential geometry
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##### References:
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