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2 FORMAL FORMALITY OF THE HYPERCOMMUTATIVE ALGEBRAS OF LOW DIMENSIONAL CALABIYAU VARIETIES
"... ar ..."
Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties
 J. Alg. Geom
, 1994
"... We consider families F(∆) consisting of complex (n − 1)dimensional projective algebraic compactifications of ∆regular affine hypersurfaces Zf defined by Laurent polynomials f with a fixed ndimensional Newton polyhedron ∆ in ndimensional algebraic torus T = (C ∗ ) n. If the family F(∆) defined by ..."
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Cited by 467 (20 self)
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by a Newton polyhedron ∆ consists of (n − 1)dimensional CalabiYau varieties then the dual, or polar, polyhedron ∆ ∗ in the dual space defines another family F( ∆ ∗ ) of CalabiYau varieties, so that we obtain the remarkable duality between two different families of CalabiYau varieties. It is shown
CalabiYau algebras
"... Abstract. We introduce some new algebraic structures arising naturally in the geometry of CY manifolds and mirror symmetry. We give a universal construction of CY algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative ..."
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Cited by 156 (1 self)
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potential. Representation varieties of the CY algebra are intimately related to the set of critical points, and to the sheaf of vanishing cycles of the potential. Numerical invariants, like ranks of cyclic homology groups, are expected to be given by ‘matrix integrals ’ over representation varieties. We
Modularity of CalabiYau Varieties
 GLOBAL ASPECTS OF COMPLEX GEOMETRY
, 2006
"... In this paper we discuss recent progress on the modularity of CalabiYau varieties. We focus mostly on the case of surfaces and threefolds. We will also discuss some progress on the structure of the Lfunction in connection with mirror symmetry. Finally, we address some questions and open problems. ..."
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In this paper we discuss recent progress on the modularity of CalabiYau varieties. We focus mostly on the case of surfaces and threefolds. We will also discuss some progress on the structure of the Lfunction in connection with mirror symmetry. Finally, we address some questions and open problems.
ARITHMETIC OF CALABI–YAU VARIETIES
, 2004
"... ... of Göttingen. We address the modularity questions of Calabi–Yau varieties of dimension ≤ 3 defined over Q. The uptodate reference on the modularity of Calabi–Yau varieties is Yui [Yu03]. ..."
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Cited by 2 (0 self)
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... of Göttingen. We address the modularity questions of Calabi–Yau varieties of dimension ≤ 3 defined over Q. The uptodate reference on the modularity of Calabi–Yau varieties is Yui [Yu03].
QUOTIENTS OF CALABIYAU VARIETIES
, 2007
"... Let X be a CalabiYau variety over C, that is, a projective variety with canonical singularities whose canonical class is numericaly trivial. Let G be a finite group acting on X and consider the quotient variety X/G. The aim of this paper is to determine the place of X/G in the birational classifica ..."
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Let X be a CalabiYau variety over C, that is, a projective variety with canonical singularities whose canonical class is numericaly trivial. Let G be a finite group acting on X and consider the quotient variety X/G. The aim of this paper is to determine the place of X/G in the birational
CFT’s from CalabiYau Fourfolds
 Nucl. Phys. B584
"... We consider F/M/Type IIA theory compactified to four, three, or two dimensions on a CalabiYau fourfold, and study the behavior near an isolated singularity in the presence of appropriate fluxes and branes. We analyze the vacuum and soliton structure of these models, and show that near an isolated ..."
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Cited by 277 (14 self)
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We consider F/M/Type IIA theory compactified to four, three, or two dimensions on a CalabiYau fourfold, and study the behavior near an isolated singularity in the presence of appropriate fluxes and branes. We analyze the vacuum and soliton structure of these models, and show that near an isolated
Generalized CalabiYau manifolds
 Q. J. Math
"... A geometrical structure on evendimensional manifolds is defined which generalizes the notion of a CalabiYau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphisms and closed 2forms. In the special case o ..."
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Cited by 330 (3 self)
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A geometrical structure on evendimensional manifolds is defined which generalizes the notion of a CalabiYau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphisms and closed 2forms. In the special case
Superconformal field theory on threebranes at a CalabiYau singularity
 Nucl. Phys. B
, 1998
"... Just as parallel threebranes on a smooth manifold are related to string theory on AdS5 × S 5, parallel threebranes near a conical singularity are related to string theory on AdS5 × X5, for a suitable X5. For the example of the conifold singularity, for which X5 = (SU(2) × SU(2))/U(1), we argue that ..."
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Cited by 690 (37 self)
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Just as parallel threebranes on a smooth manifold are related to string theory on AdS5 × S 5, parallel threebranes near a conical singularity are related to string theory on AdS5 × X5, for a suitable X5. For the example of the conifold singularity, for which X5 = (SU(2) × SU(2))/U(1), we argue that string theory on AdS5 × X5 can be described by a certain N = 1 supersymmetric gauge theory which we describe in detail.
AbelJacobi maps for hypersurfaces and non commutative CalabiYau’s
 Comm. Cont. Math
"... Abstract. It is well known that the Fano scheme of lines on a cubic 4fold is a symplectic variety. We generalize this fact by constructing a closed (2n − 4)form on the Fano scheme of lines on a (2n − 2)dimensional hypersurface Yn of degree n. We provide several definitions of this form — via the ..."
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the Abel–Jacobi map, via Hochschild homology, and via the linkage class — and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Yn we show that the Fano scheme is birational to a certain moduli space of sheaves of a (2n−4)dimensional Calabi–Yau variety X arising naturally
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