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Showing papers on "Toric variety published in 2010"


Journal ArticleDOI
TL;DR: In this paper, the authors construct global F-theory GUT models on del Pezzo surfaces in compact Calabi-Yau fourfolds realized as complete intersections of two hypersurface constraints, which allow for the phenomenologically relevant Yukawa couplings and GUT breaking to the MSSM via hypercharge flux while preventing dimension-4 proton decay.

271 citations


Journal ArticleDOI
TL;DR: In this article, a complete classification up to diffeomorphism of closed almost toric four-manifolds is presented, and a key step in the proof is a geometric classification of the singular affine structures that can occur on the base of a closed almost-toric fourmanifold.
Abstract: Almost toric manifolds form a class of singular Lagrangian fibered symplectic manifolds that is a natural generalization of toric manifolds. Notable examples include the K3 surface, the phase space of the spherical pendulum and rational balls useful for symplectic surgeries. The main result of the paper is a complete classification up to diffeomorphism of closed almost toric four-manifolds. A key step in the proof is a geometric classification of the singular affine structures that can occur on the base of a closed almost toric four-manifold.

114 citations


Journal ArticleDOI
TL;DR: In this paper, the authors construct a modular compactification of spaces of maps from nonsingular curves to smooth projective toric varieties, which generalize Givental's compactifications, when the complex structure of the curve is allowed to vary and markings are included.

104 citations


Journal ArticleDOI
TL;DR: In this paper, a class of global F-theory GUT models were constructed using toric geometry and a split spectral cover to generate chiral matter on the 10 curves in order to get more degrees of freedom in phenomenology.
Abstract: Making use of toric geometry we construct a class of global F-theory GUT models. The base manifolds are blowups of Fano threefolds and the Calabi-Yau fourfold is a complete intersection of two hypersurfaces. We identify possible GUT divisors and construct SO(10) models on them using the spectral cover construction. We use a split spectral cover to generate chiral matter on the 10 curves in order to get more degrees of freedom in phenomenology. We use abelian flux to break SO(10) to SU(5) ×U(1) which is interpreted as a flipped SU(5) model. With the GUT Higgses in the SU(5) × U(1) model it is possible to further break the gauge symmetry to the Standard Model. We present several phenomenologically attractive examples in detail.

100 citations


Posted Content
TL;DR: In this paper, the Okounkov body of a divisor D on a projective variety X admits a flat degeneration to the corresponding toric variety, which is functorial in an appropriate sense.
Abstract: Let \Delta be the Okounkov body of a divisor D on a projective variety X. We describe a geometric criterion for \Delta to be a lattice polytope, and show that in this situation X admits a flat degeneration to the corresponding toric variety. This degeneration is functorial in an appropriate sense.

87 citations


Posted Content
TL;DR: In this paper, a natural isomorphism between the Frobenius manifold structures of the quantum cohomology of the toric manifold and Saito's theory of singularities of the potential function constructed in \cite{fooo09} via the Lagrangian Floer cohomologies deformed by ambient cycles is constructed.
Abstract: 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) quantum cohomology of the toric manifold and of Saito's theory of singularities of the potential function constructed in \cite{fooo09} via the Floer cohomology deformed by ambient cycles. Our proof of the isomorphism involves the open-closed Gromov-Witten theory of one-loop.

84 citations


Journal ArticleDOI
TL;DR: In this article, a simple class of Calabi-Yau 3-folds arising from toric geometry and vector bundles on these manifolds is studied, and the authors show that anomaly-free positive monads exist on only 11 of these toric 3folds with a total number of bundles of about 2000.
Abstract: We systematically approach the construction of heterotic E 8 × E 8 Calabi-Yau models, based on compact Calabi-Yau three-folds arising from toric geometry and vector bundles on these manifolds. We focus on a simple class of 101 such three-folds with smooth ambient spaces, on which we perform an exhaustive scan and find all positive monad bundles with SU(N), N = 3; 4; 5 structure groups, subject to the heterotic anomaly cancellation constraint. We find that anomaly-free positive monads exist on only 11 of these toric three-folds with a total number of bundles of about 2000. Only 21 of these models, all of them on three-folds realizable as hypersurfaces in products of projective spaces, allow for three families of quarks and leptons. We also perform a preliminary scan over the much larger class of semi-positive monads which leads to about 44000 bundles with 280 of them satisfying the three-family constraint. These 280 models provide a starting point for heterotic model building based on toric three-folds.

80 citations


Journal ArticleDOI
TL;DR: In this article, a homogeneous locally nil-potent derivation on the normal affine domain is proposed, where ∂ generates a k+-action on X that is normalized by the torus action.
Abstract: Let X = Spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus \( \mathbb{T} \) of dimension n. Let also ∂ be a homogeneous locally nilpotent derivation on the normal affine \( {\mathbb{Z}^n} \)-graded domain A, so that ∂ generates a k+-action on X that is normalized by the \( \mathbb{T} \)-action.

69 citations


Posted Content
TL;DR: The theory of local models of Shimura varieties has been surveyed in this paper, where the authors give an overview of the results on their geometry and combinatorics obtained in the last 15 years.
Abstract: We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and combinatorics obtained in the last 15 years. We also exhibit their connections to other classes of algebraic varieties such as nilpotent orbit closures, affine Schubert varieties, quiver Grassmannians and wonderful completions of symmetric spaces.

58 citations


Journal ArticleDOI
TL;DR: In this paper, the Chow and Hilbert quotient of a projective scheme X by the action of an algebraic torus T in an auxiliary toric variety is given, and geometric invariant theory descriptions of these canonical quotients are provided.
Abstract: We give explicit equations for the Chow and Hilbert quotients of a projective scheme X by the action of an algebraic torus T in an auxiliary toric variety. As a consequence we provide geometric invariant theory descriptions of these canonical quotients, and obtain other GIT quotients of X by variation of GIT quotient. We apply these results to find equations for the moduli space (M) over bar (0,n) of stable genus-zero n-pointed curves as a subvariety of a smooth toric variety defined via tropical methods.

55 citations


Journal ArticleDOI
TL;DR: In this paper, a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrally-constructible sheaves in a vector space was proved.
Abstract: We prove a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrally-constructible sheaves on a vector space At the level of K-theory, the theorem recovers Morelli's description of the K-theory of a smooth projective toric variety Specifically, let $X$ be a proper toric variety of dimension $n$ and let $M_\bR = \mathrm{Lie}(T_\bR^\vee)\cong \bR^n$ be the Lie algebra of the compact dual (real) torus $T_\bR^\vee\cong U(1)^n$ Then there is a corresponding conical Lagrangian $\Lambda \subset T^*M_\bR$ and an equivalence of triangulated dg categories $\Perf_T(X) \cong \Sh_{cc}(M_\bR;\Lambda),$ where $\Perf_T(X)$ is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on $X$ and $\Sh_{cc}(M_\bR;\Lambda)$ is the triangulated dg category of complex of sheaves on $M_\bR$ with compactly supported, constructible cohomology whose singular support lies in $\Lambda$ This equivalence is monoidal---it intertwines the tensor product of coherent sheaves on $X$ with the convolution product of constructible sheaves on $M_\bR$

Journal ArticleDOI
TL;DR: In this article, it was shown that the Betti realization of a smooth proper and exact mapping of log analytic spaces is a topological fibration, whose fibers are orientable manifolds (possibly with boundary).
Abstract: We show that the introduction of polar coordinates in toric geometry smoothes a wide class of equivariant mappings, rendering them locally trivial in the topological category. As a consequence, we show that the Betti realization of a smooth proper and exact mapping of log analytic spaces is a topological fibration, whose fibers are orientable manifolds (possibly with boundary). This turns out to be true even for certain noncoherent log structures, including some families familiar from mirror symmetry. The moment mapping plays a key role in our proof. 14D06, 14M25, 14F45, 32S30; 53D20, 14T05

Journal ArticleDOI
TL;DR: In this article, it was shown that the pushforward of the Lebesgue measure on the Okounkov body is equal to a Duistermaat-Heckman measure of a certain deformation of the manifold.
Abstract: We associate to a test configuration of an ample line bundle a filtration of the section ring of the line bundle. Using the recent work of Boucksom-Chen we get a concave function on the Okounkov body whose law with respect to Lebesgue measure determines the asymptotic distribution of the weights of the test configuration. We show that this is a generalization of a well-known result in toric geometry. As an application, we prove that the pushforward of the Lebesgue measure on the Okounkov body is equal to a Duistermaat-Heckman measure of a certain deformation of the manifold. Via the Duisteraat-Heckman formula, we get as a corollary that in the special case of an effective $\mathbb{C^{\times}}$-action on the manifold lifting to the line bundle, the pushforward of the Lebesgue measure on the Okounkov body is piecewise polynomial.

Journal ArticleDOI
TL;DR: In this article, it was shown that a crepant resolution π : Y → X admits a complete Ricci-flat Kahler metric asymptotic to the cone metric in every Kahler class.
Abstract: We prove that a crepant resolution π : Y → X of a Ricci-flat Kahler cone X admits a complete Ricci-flat Kahler metric asymptotic to the cone metric in every Kahler class in $${H^2_c(Y,\mathbb{R})}$$ . A Kahler cone $${(X,\bar{g})}$$ is a metric cone over a Sasaki manifold (S, g), i.e. $${X=C(S):=S\times\mathbb{R}_{ >0 }}$$ with $${\bar{g}=dr^2 +r^2 g}$$ , and $${(X,\bar{g})}$$ is Ricci-flat precisely when (S, g) Einstein of positive scalar curvature. This result contains as a subset the existence of ALE Ricci-flat Kahler metrics on crepant resolutions $${\pi:Y\rightarrow X=\mathbb{C}^n /\Gamma}$$ , with $${\Gamma\subset SL(n,\mathbb{C})}$$ , due to P. Kronheimer (n = 2) and D. Joyce (n > 2). We then consider the case when X = C(S) is toric. It is a result of A. Futaki, H. Ono, and G. Wang that any Gorenstein toric Kahler cone admits a Ricci-flat Kahler cone metric. It follows that if a toric Kahler cone X = C(S) admits a crepant resolution π : Y → X, then Y admits a T n -invariant Ricci-flat Kahler metric asymptotic to the cone metric $${(X,\bar{g})}$$ in every Kahler class in $${H^2_c(Y,\mathbb{R})}$$ . A crepant resolution, in this context, is a simplicial fan refining the convex polyhedral cone defining X. We then list some examples which are easy to construct using toric geometry.

Posted Content
TL;DR: A survey of a series of papers [FOOO3, FOOO4,FOOO5] in which they developed the method of calculation of Floer cohomology of Lagrangian torus orbits in compact toric manifolds can be found in this article.
Abstract: This article is a survey of a series of papers [FOOO3,FOOO4,FOOO5] in which we developed the method of calculation of Floer cohomology of Lagrangian torus orbits in compact toric manifolds, and its applications to symplectic topology and to mirror symmetry. In this article we summarize the main ingredients of calculation and illustrate them by examples. The second half of the survey is devoted to discussion of the most recent result from [FOOO5] (arXiv:1009.1648) where the mirror symmetry between the two Frobenius manifolds arising from the big quantum cohomology and from the K. Saito theory of singularities was established.

Journal ArticleDOI
TL;DR: A combinatorial proof of White's conjecture that the toric ideal associated with the basis of a matroid is generated by quadrics corresponding to symmetric exchanges is presented by using a lemma proposed by Blasiak.
Abstract: White conjectured that the toric ideal associated with the basis of a matroid is generated by quadrics corresponding to symmetric exchanges. We present a combinatorial proof of White's conjecture for matroids of rank 3 by using a lemma proposed by Blasiak.

Journal ArticleDOI
TL;DR: In this paper, the authors studied compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5.
Abstract: We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using existence result of [25], we show that a co-oriented compact toric contact 5-manifold whose moment cone has 4 facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on $S^2\times S^3$ admitting two non isometric and non transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

Journal ArticleDOI
TL;DR: In this article, the authors show that three-dimensional smooth, compact toric varieties (SCTV) can admit G-structures rendering them suitable for internal manifolds in supersymmetric flux compactifications.
Abstract: Three-dimensional smooth, compact toric varieties (SCTV), when viewed as real six-dimensional manifolds, can admit G-structures rendering them suitable for internal manifolds in supersymmetric flux compactifications. We develop techniques which allow us to systematically construct G-structures on SCTV and read off their torsion classes. We illustrate our methods with explicit examples, one of which consists of an infinite class of toric CP^1 bundles. We give a self-contained review of the relevant concepts from toric geometry, in particular the subject of the classification of SCTV in dimensions less or equal to 3. Our results open up the possibility for a systematic construction and study of supersymmetric flux vacua based on SCTV.

Posted Content
TL;DR: In this article, the authors consider the problem of finding a toric variety on which a rational monomial map is algebraically stable and give conditions for when such a variety exists.
Abstract: Given a rational monomial map, we consider the question of finding a toric variety on which it is algebraically stable. We give conditions for when such variety does or does not exist. We also obtain several precise estimates of the degree sequences of monomial maps on $¶^n$. Finally, we characterize polynomial maps which are algebraically stable on $(¶^1)^n$.

Posted Content
TL;DR: In this paper, the authors established asymptotic formulas for the number of integral points of bounded height on toric varieties, where the integral points are bounded by a fixed number of vertices.
Abstract: We establish asymptotic formulas for the number of integral points of bounded height on toric varieties.

Posted Content
Jan-Li Lin1
TL;DR: In this paper, the pullback maps on cohomology groups for equivariant rational maps (i.e., monomial maps) on toric varieties were derived based on intersection theory.
Abstract: We study the pullback maps on cohomology groups for equivariant rational maps (i.e., monomial maps) on toric varieties. Our method is based on the intersection theory on toric varieties. We use the method to determine the dynamical degrees of monomial maps and compute the degrees of the Cremona involution.

Posted Content
TL;DR: In this article, it was shown that there is a fully faithful embedding of the perfect derived category of a proper toric variety into the derived categories of constructible sheaves on a compact torus.
Abstract: We prove the following result of Bondal's: that there is a fully faithful embedding $\kappa$ of the perfect derived category of a proper toric variety into the derived category of constructible sheaves on a compact torus. We compare this result to a torus-equivariant version considered in joint work with Fang, Liu, and Zaslow. There we showed that in the torus-equivariant version the image of the embedding is cut out by microlocal conditions. To establish a similar characterization of the image of $\kappa$ is an open problem.

Journal ArticleDOI
08 Jul 2010
TL;DR: In this paper, a moduli space for stable spherical varieties over projective spaces is introduced, which includes the spherical varieties under a reductive group $G$ and their flat equivariant degenerations.
Abstract: We introduce a notion of stable spherical variety which includes the spherical varieties under a reductive group $G$ and their flat equivariant degenerations. Given any projective space $\bP$ where $G$ acts linearly, we construct a moduli space for stable spherical varieties over $\bP$, that is, pairs $(X,f)$, where $X$ is a stable spherical variety and $f : X \to \bP$ is a finite equivariant morphism. This space is projective, and its irreducible components are rational. It generalizes the moduli space of pairs $(X,D)$, where $X$ is a stable toric variety and $D$ is an effective ample Cartier divisor on $X$ which contains no orbit. The equivariant automorphism group of $\bP$ acts on our moduli space; the spherical varieties over $\bP$ and their stable limits form only finitely many orbits. A variant of this moduli space gives another view to the compactifications of quotients of thin Schubert cells constructed by Kapranov and Lafforgue.

Journal ArticleDOI
TL;DR: In this paper, a class of global F-theory GUT models were constructed using toric geometry and a split spectral cover to generate chiral matter on the 10 curves in order to get more degrees of freedom in phenomenology.
Abstract: Making use of toric geometry we construct a class of global F-theory GUT models. The base manifolds are blowups of Fano threefolds and the Calabi-Yau fourfold is a complete intersection of two hypersurfaces. We identify possible GUT divisors and construct SO(10) models on them using the spectral cover construction. We use a split spectral cover to generate chiral matter on the 10 curves in order to get more degrees of freedom in phenomenology. We use abelian flux to break SO(10) to SU(5)\times U(1) which is interpreted as a flipped SU(5) model. With the GUT Higgses in the SU(5)\times U(1) model it is possible to further break the gauge symmetry to the Standard Model. We present several phenomenologically attractive examples in detail.

Journal ArticleDOI
TL;DR: In this article, the authors use the Grossberg-Karshon degeneration of Bott-Samelson varieties to toric varieties and the description of cohomology of line bundles on toric lines bundles to deduce vanishing results for the cohomologies of lines bundles on Bott-samelson varieties.

Journal ArticleDOI
TL;DR: A deterministic algorithm is obtained that can be seen as a sparse version of an algorithm of Lecerf, with a polynomial complexity in the volume of the Newton polytope.

Journal ArticleDOI
TL;DR: In this article, it was shown that the usual sufficient criterion for a generic hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to hypersurfaces in complete simplicial toric varieties.
Abstract: We show that the usual sufficient criterion for a generic hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to hypersurfaces in complete simplicial toric varieties. This sufficient condition is always satisfied by generic K3 surfaces embedded in Fano toric 3-folds.

Journal ArticleDOI
TL;DR: In this article, it was shown that 1-pointed closed Gromov-Witten invariants for semi-Fano toric manifolds are equal to the mirror superpotential for the Hirzebruch surface.
Abstract: We prove that open Gromov-Witten invariants for semi-Fano toric manifolds of the form $X=\mathbb{P}(K_Y\oplus\mathcal{O}_Y)$, where $Y$ is a toric Fano manifold, are equal to certain 1-pointed closed Gromov-Witten invariants of $X$. As applications, we compute the mirror superpotentials for these manifolds. In particular, this gives a simple proof for the formula of the mirror superpotential for the Hirzebruch surface $\mathbb{F}_2$.

Posted Content
TL;DR: In this paper, the authors studied toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group.
Abstract: We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify k-forms of projective spaces when K/k is cyclic, and we also study k-forms of surfaces.

Posted Content
TL;DR: In this article, the authors constructed two polyhedral lower bounds and one polyhedral upper bound for the nef cone of a projective variety Y using an embedding of Y into a toric variety.
Abstract: The nef cone of a projective variety Y is an important and often elusive invariant. In this paper we construct two polyhedral lower bounds and one polyhedral upper bound for the nef cone of Y using an embedding of Y into a toric variety. The lower bounds generalize the combinatorial description of the nef cone of a Mori dream space, while the upper bound generalizes the F-conjecture for the nef cone of the moduli space \bar{M}_{0,n} to a wide class of varieties.