Hodge theoretic aspects of mirror symmetry

  title={Hodge theoretic aspects of mirror symmetry},
  author={Ludmil Katzarkov and Maxim Kontsevich and Tony Pantev},
  journal={arXiv: Algebraic Geometry},
We discuss the Hodge theory of algebraic non-commutative spaces and analyze how this theory interacts with the Calabi-Yau condition and with mirror symmetry. We develop an abstract theory of non-commutative Hodge structures, investigate existence and variations, and propose explicit construction and classification techniques. We study the main examples of non-commutative Hodge structures coming from a symplectic or a complex geometry possibly twisted by a potential. We study the interactions of… 

Figures from this paper

Mirror Symmetry, Singularity Theory and Non-commutative Hodge Structures
We review a version of the mirror correspondence for smooth toric varieties with a numerically effective anticanonical bundle. We give a precise description of the so-called B-model, which involves
Equivariant Hodge theory and noncommutative geometry
We develop a version of Hodge theory for a large class of smooth cohomologically proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative
Non-commutative Hodge structures: Towards matching categorical and geometric examples
The subject of the present work is the de Rham part of non-commutative Hodge structures on the periodic cyclic homology of differential graded algebras and categories. We discuss explicit formulas
Family Floer cohomology and mirror symmetry
Ideas of Fukaya and Kontsevich-Soibelman suggest that one can use Strominger-Yau-Zaslow's geometric approach to mirror symmetry as a torus duality to construct the mirror of a symplectic manifold
Lagrangian floer theory and mirror symmetry on compact toric manifolds
In this paper we study Lagrangian Floer theory on toric manifolds from the point of view of mirror symmetry. We construct a natural isomorphism between the Frobenius manifold structures of the (big)
Real and integral structures in quantum cohomology I: toric orbifolds
We study real and integral structures in the space of solutions to the quantum differential equations. First we show that, under mild conditions, any real structure in orbifold quantum cohomology
Formulae in noncommutative Hodge theory
  • Nick Sheridan
  • Mathematics
    Journal of Homotopy and Related Structures
  • 2019
We prove that the cyclic homology of a saturated $$A_\infty $$ A ∞ category admits the structure of a ‘polarized variation of Hodge structures’, building heavily on the work of many authors: the main
Categorical geometry : Hodge structures and K-theory
These Hodge structures on the cohomology of smooth complex projective varieties revealed themselves to be a powerful tool in the study of the geometry of such varieties. Among other things we can
Differential equations aspects of quantum cohomology
The quantum differential equations can be regarded as examples of equations with certain universal properties which are of wider interest beyond quantum cohomology itself. We present this point of


Non-Commutative Periods and Mirror Symmetry¶in Higher Dimensions
Abstract: We study an analog for higher-dimensional Calabi–Yau manifolds of the standard predictions of Mirror Symmetry. We introduce periods associated with “non-commutative” deformations of
Nilpotent orbits of a generalization of Hodge structures
Abstract We study a generalization of Hodge structures which first appeared in the work of Cecotti and Vafa. It consists of twistors, that is, holomorphic vector bundles on ℙ1, with additional
Homological mirror symmetry and torus fibrations
In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the Fukaya
Real and integral structures in quantum cohomology I: toric orbifolds
We study real and integral structures in the space of solutions to the quantum differential equations. First we show that, under mild conditions, any real structure in orbifold quantum cohomology
Quantum periods, I: semi-infinite variations of Hodge structures
We introduce a generalization of variations of Hodge structures living over moduli spaces of non-commutative deformations of complex mani- folds. Hodge structure associated with a point of such
tt* geometry and mixed Hodge structures
tt* geometry is a generalization of variation of Hodge structures (section 2). Also the nilpotent orbits of Schmid and the relation to polarized mixed Hodge structures generalize; part of this is
Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves
We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface Xk
Affine Structures and Non-Archimedean Analytic Spaces
In this paper we propose a way to construct an analytic space over a non-archimedean field, starting with a real manifold with an affine structure which has integral monodromy. Our construction is
Operads and Motives in Deformation Quantization
The algebraic world of associative algebras has many deep connections with the geometric world of two-dimensional surfaces. Recently, D. Tamarkin discovered that the operad of chains of the little