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Hodge theoretic aspects of mirror symmetry. (English) Zbl 1206.14009
Donagi, Ron Y. (ed.) et al., From Hodge theory to integrability and TQFT tt*-geometry. International workshop From TQFT to tt* and integrability, Augsburg, Germany, May 25–29, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4430-4/hbk). Proceedings of Symposia in Pure Mathematics 78, 87-174 (2008).
The paper under review is the first in a series aiming to develop a general procedure associating a cohomological field theory to any Calabi–Yau \(A_\infty\)-category satisfying certain technical conditions.
In the first part the notion of a non-commutative (nc) Hodge structure is introduced and developed. The authors in particular explain how nc-Hodge theory fits within the setup of categorical non-commutative geometry. Here, a non-commutative space is a \(\mathbb{C}\)-linear dg-category satisfying certain properties. For example, the nc-space associated to a complex variety is simply the derived category of coherent sheaves endowed with a dg-enhancement.
The second part explains how symplectic and complex geometry give rise to nc-Hodge structures and how these structures can be viewed as interesting invariants in, for example, Gromov–Witten theory.
In the last part variations of nc-Hodge structures are considered. In particular, the authors obtain unobstructedness results, generalized pre–Frobenius structures and some interesting geometric properties of period domains for nc-Hodge structures.
For the entire collection see [Zbl 1148.14002].

14A22 Noncommutative algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
19D55 \(K\)-theory and homology; cyclic homology and cohomology
34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
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