Topic

# Algebraic torus

About: Algebraic torus is a research topic. Over the lifetime, 298 publications have been published within this topic receiving 4083 citations.

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TL;DR: In this paper, the generalized Jacobian of an algebraic curve with at most ordinary double points was systematically compactified in finitely many different ways and described their structure in terms of torus embeddings.

Abstract: The generalized Jacobian variety of an algebraic curve with at most ordinary double points is an extension of an abelian variety by an algebraic torus. Using the geometric invariant theory, we systematically compactify it in finitely many different ways and describe their structure in terms of torus embeddings. Our compactifications include all known good ones.

205 citations

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TL;DR: In this article, it was shown that ordered product factorizations in the tropical vertex group are equivalent to calculations of genus zero relative Gromov-Witten invariants of toric surfaces.

Abstract: Elements of the tropical vertex group are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove that ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus zero relative Gromov-Witten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold Gromov-Witten theory

201 citations

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TL;DR: In this article, a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties is provided.

Abstract: We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our approach extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.

182 citations

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TL;DR: In this article, the authors introduce the notion of a multi-fan, which is a generalization of a fan in the theory of toric variety in algebraic geometry and define topological invariants of a multfan.

Abstract: We introduce the notion of a multi-fan. It is a generalization of that of a fan in the theory of toric variety in algebraic geometry. Roughly speaking a toric variety is an algebraic variety with an action of algebraic torus of the same dimension as that of the variety, and a fan is a combinatorial object associated with the toric variety. Algebro-geometric properties of the toric variety can be described in terms of the associated fan. We develop a combinatorial theory of multi-fans and define ``topological invariants'' of a multi-fan. A smooth manifold with an action of a compact torus of half the dimension of the manifold and with some orientation data is called a torus manifold. We associate a multi-fan with a torus manifold, and apply the combinatorial theory to describe topological invariants of the torus manifold. A similar theory is also given for torus orbifolds. As a related subject a generalization of the Ehrhart polynomial concerning the number of lattice points in a convex polytope is discussed.

136 citations

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TL;DR: In this article, the authors give a new definition of vertex algebra which includes chiral algebras as a special case, but allows for fields which are neither meromorphic nor anti-meromorphic, and show that for algebraic tori the isomorphism of N=2 SCVA's implies the equivalence of the derived categories of coherent sheaves corresponding to the tori or their noncommutative generalizations.

Abstract: A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. We give a new definition of a vertex algebra which includes chiral algebras as a special case, but allows for fields which are neither meromorphic nor anti-meromorphic. To any complex torus equipped with a flat Kahler metric and a closed 2-form we associate an N=2 superconformal vertex algebra (N=2 SCVA) in the sense of our definition. We find a criterion for two different tori to produce isomorphic N=2 SCVA's. We show that for algebraic tori the isomorphism of N=2 SCVA's implies the equivalence of the derived categories of coherent sheaves corresponding to the tori or their noncommutative generalizations (Azumaya algebras over tori). We also find a criterion for two different tori to produce N=2 SCVA's related by a mirror morphism. If the 2-form is of type (1,1), this condition is identical to the one proposed by Golyshev, Lunts, and Orlov, who used an entirely different approach inspired by the Homological Mirror Symmetry Conjecture of Kontsevich. Our results suggest that Kontsevich's conjecture must be modified: coherent sheaves must be replaced with modules over Azumaya algebras, and the Fukaya category must be ``twisted'' by a closed 2-form. We also describe the implications of our results for BPS D-branes on Calabi-Yau manifolds.

130 citations