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Hodge theoretic aspects of mirror symmetry

2008; American Mathematical Society; Linguagem: Inglês

10.1090/pspum/078/2483750

2324-707X

Ludmil Katzarkov, Maxim Kontsevich, Tony Pantev,

Advanced Topics in Algebra

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 the new Hodge theoretic invariants with mirror symmetry and derive non-commutative analogues of some of the more interesting consequences of Hodge theory such as unobstructedness and the construction of canonical coordinates on moduli.