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140
Generalized complex geometry
, 2007
"... Generalized complex geometry encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deformation theory, relation to Poisson geometry, and local structure theory. We also define and s ..."
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Cited by 295 (7 self)
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Generalized complex geometry encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deformation theory, relation to Poisson geometry, and local structure theory. We also define and study generalized complex branes, which interpolate between flat bundles on
Strong homotopy algebras of a Kähler manifold
 math.AG/9809172, Int. Math. Res. Notices
, 1999
"... It is shown that any compact Kähler manifold M gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the Dolbeault complex. In these algebras the product of two harmonic differ ..."
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Cited by 96 (9 self)
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It is shown that any compact Kähler manifold M gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the Dolbeault complex. In these algebras the product of two harmonic differential forms is again harmonic. If M happens to be a CalabiYau manifold, there exists a third strongly homotopy algebra closely related to the BarannikovKontsevich extended moduli space of complex structures. 1
Noncommutative curves and noncommutative surfaces
 Bulletin of the American Mathematical Society
"... Abstract. In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative grad ..."
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Cited by 91 (8 self)
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Abstract. In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing
Topological string amplitudes, complete intersection Calabi–Yau spaces and threshold corrections
, 2005
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Noncommutative differential calculus, homotopy . . .
, 2000
"... We define a notion of a strong homotopy BV algebra and apply it to deformation theory problems. Formality conjectures for Hochschild cochains are formulated. We prove several results supporting these conjectures. ..."
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Cited by 57 (1 self)
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We define a notion of a strong homotopy BV algebra and apply it to deformation theory problems. Formality conjectures for Hochschild cochains are formulated. We prove several results supporting these conjectures.
FourierMukai transforms for K3 and elliptic fibrations
"... Abstract. Given a nonsingular variety with a K3 fibration π: X → S we construct dual fibrations ˆπ: Y → S by replacing each fibre Xs of π by a twodimensional moduli space of stable sheaves on Xs. In certain cases we prove that the resulting scheme Y is a nonsingular variety and construct an equiv ..."
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Cited by 55 (5 self)
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Abstract. Given a nonsingular variety with a K3 fibration π: X → S we construct dual fibrations ˆπ: Y → S by replacing each fibre Xs of π by a twodimensional moduli space of stable sheaves on Xs. In certain cases we prove that the resulting scheme Y is a nonsingular variety and construct an equivalence of derived categories of coherent sheaves Φ: D(Y) → D(X). Our methods also apply to elliptic and abelian surface fibrations. As an application we show how the equivalences Φ identify certain moduli spaces of stable bundles on elliptic threefolds with Hilbert schemes of curves. 1.
Quantum periods  I. Semiinfinite variations of Hodge structures
, 2000
"... We introduce a generalization of variations of Hodge structures living over the moduli spaces of noncommutative deformations of complex manifolds. The Hodge structure associated with a point of such moduli space is an element of Sato type grassmanian of semiinnite subspaces in H (X; C )[[~ \Ga ..."
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Cited by 53 (1 self)
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We introduce a generalization of variations of Hodge structures living over the moduli spaces of noncommutative deformations of complex manifolds. The Hodge structure associated with a point of such moduli space is an element of Sato type grassmanian of semiinnite subspaces in H (X; C )[[~ \Gamma1 ; ~]]. Periods associated with such semiinnite Hodge structures serve in order to extend mirror symmetry relations in dimensions greater then three.
Topological Open pBranes
, 2000
"... By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3form Cfield leads to deformations of the algebras of multivectors on the Dirichletbrane worldvolume as 2algebras. This would shed some new light on geometry of Mtheory 5brane and associated decoupled ..."
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Cited by 45 (1 self)
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By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3form Cfield leads to deformations of the algebras of multivectors on the Dirichletbrane worldvolume as 2algebras. This would shed some new light on geometry of Mtheory 5brane and associated decoupled theories. We show that, in general, topological open pbrane has a structure of (p + 1)algebra in the bulk, while a structure of palgebra in the boundary. The bulk/boundary correspondences are exactly as of the generalized Deligne conjecture (a theorem of Kontsevich) in the algebraic world of palgebras. It also imply that the algebras of quantum observables of (p − 1)brane are “close to ” the algebras of its classical observables as palgebras. We interpret above as deformation quantization of (p − 1)brane, generalizing the p = 1 case. We argue that there is such quantization based on the direct relation between BV master equation and Ward identity of the bulk topological theory. The path integral of the theory will lead to the explicit formula. We also discuss some applications to