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Showing papers on "Toric variety published in 2019"


Journal ArticleDOI
TL;DR: In this article, the authors consider a class of supersymmetric AdS3 × Y7 solutions of type IIB supergravity with five-form flux only in the case that Y7 is toric, and show how the offshell central charge of the dual field theory can be obtained from the toric data.
Abstract: We consider d = 3, $$ \mathcal{N}=2 $$ gauge theories arising on membranes sitting at the apex of an arbitrary toric Calabi-Yau 4-fold cone singularity that are then further compactified on a Riemann surface, Σg , with a topological twist that preserves two supersymmetries. If the theories flow to a superconformal quantum mechanics in the infrared, then they have a D = 11 supergravity dual of the form AdS2 × Y9, with electric four-form flux and where Y9 is topologically a fibration of a Sasakian Y7 over Σg . These D = 11 solutions are also expected to arise as the near horizon limit of magnetically charged black holes in AdS4 × Y7, with a Sasaki-Einstein metric on Y7. We show that an off-shell entropy function for the dual AdS2 solutions may be computed using the toric data and Kahler class parameters of the Calabi-Yau 4-fold, that are encoded in a master volume, as well as a set of integers that determine the fibration of Y7 over Σg and a Kahler class parameter for Σg . We also discuss the class of supersymmetric AdS3 × Y7 solutions of type IIB supergravity with five-form flux only in the case that Y7 is toric, and show how the off-shell central charge of the dual field theory can be obtained from the toric data. We illustrate with several examples, finding agreement both with explicit supergravity solutions as well as with some known field theory results concerning ℐ-extremization.

51 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of computing an off-shell entropy function for the dual AdS$_2$ solutions of the Calabi-Yau 4-fold, which are then further compactified on a Riemann surface.
Abstract: We consider $d=3$, $\mathcal{N}=2$ gauge theories arising on membranes sitting at the apex of an arbitrary toric Calabi-Yau 4-fold cone singularity that are then further compactified on a Riemann surface, $\Sigma_g$, with a topological twist that preserves two supersymmetries. If the theories flow to a superconformal quantum mechanics in the infrared, then they have a $D=11$ supergravity dual of the form AdS$_2\times Y_9$, with electric four-form flux and where $Y_9$ is topologically a fibration of a Sasakian $Y_7$ over $\Sigma_g$. These $D=11$ solutions are also expected to arise as the near horizon limit of magnetically charged black holes in AdS$_4\times Y_7$, with a Sasaki-Einstein metric on $Y_7$. We show that an off-shell entropy function for the dual AdS$_2$ solutions may be computed using the toric data and Kahler class parameters of the Calabi-Yau 4-fold, that are encoded in a master volume, as well as a set of integers that determine the fibration of $Y_7$ over $\Sigma_g$ and a Kahler class parameter for $\Sigma_g$. We also discuss the class of supersymmetric AdS$_3\times Y_7$ solutions of type IIB supergravity with five-form flux only in the case that $Y_7$ is toric, and show how the off-shell central charge of the dual field theory can be obtained from the toric data. We illustrate with several examples, finding agreement both with explicit supergravity solutions as well as with some known field theory results concerning ${\cal I}$-extremization.

46 citations


Book ChapterDOI
TL;DR: In this paper, the maximal cut of a Feynman integral is a GKZ hypergeometric series and the minimal differential operator acting on it is a trilogarithm.
Abstract: This expository text is about using toric geometry and mirror symmetry for evaluating Feynman integrals We show that the maximal cut of a Feynman integral is a GKZ hypergeometric series We explain how this allows to determine the minimal differential operator acting on the Feynman integrals We illustrate the method on sunset integrals in two dimensions at various loop orders The graph polynomials of the multi-loop sunset Feynman graphs lead to reflexive polytopes containing the origin and the associated variety are ambient spaces for Calabi-Yau hypersurfaces Therefore the sunset family is a natural home for mirror symmetry techniques We review the evaluation of the two-loop sunset integral as an elliptic dilogarithm and as a trilogarithm The equivalence between these two expressions is a consequence of (1) the local mirror symmetry for the non-compact Calabi-Yau three-fold obtained as the anti-canonical hypersurface of the del Pezzo surface of degree 6 defined by the sunset graph polynomial and (2) that the sunset Feynman integral is expressed in terms of the local Gromov-Witten prepotential of this del Pezzo surface

39 citations


Posted Content
TL;DR: In this article, it was shown that a Cardy-like limit of the superconformal index of 4d ρ = 4$ SYM accounts for the entropy function whose Legendre transform corresponds to the entropy of the holographic dual AdS$_5$ rotating black hole.
Abstract: It has recently been claimed that a Cardy-like limit of the superconformal index of 4d $\mathcal{N}=4$ SYM accounts for the entropy function, whose Legendre transform corresponds to the entropy of the holographic dual AdS$_5$ rotating black hole. Here we study this Cardy-like limit for $\mathcal{N}=1$ toric quiver gauge theories, observing that the corresponding entropy function can be interpreted in terms of the toric data. Furthermore, for some families of models, we compute the Legendre transform of the entropy function, comparing with similar results recently discussed in the literature.

32 citations


Journal ArticleDOI
TL;DR: In this article, a logarithmically nonsingular moduli space of genus $1$ curves mapping to any toric variety is constructed, which is a birational modification of the principal component of the Abramovich-Chen-Gross--Gross-Siebert space.
Abstract: This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of singular curves of genus $1$. This volume focuses on logarithmic Gromov--Witten theory and tropical geometry. We construct a logarithmically nonsingular moduli space of genus $1$ curves mapping to any toric variety. The space is a birational modification of the principal component of the Abramovich--Chen--Gross--Siebert space of logarithmic stable maps and produces an enumerative genus $1$ curve counting theory. We describe the non-archimedean analytic skeleton of this moduli space and, as a consequence, obtain a full resolution to the tropical realizability problem in genus $1$.

32 citations


Journal ArticleDOI
TL;DR: The maximum likelihood (ML) degree of toric varieties, known as discrete exponential models in statistics, is studied, showing that the ML degree is equal to the degree of the toric variety for generic scalings, while it drops if and only if the scaling vector is in the locus of the principal A-determinant.

27 citations


Journal ArticleDOI
TL;DR: In this paper, the authors carried out the SYZ program for local Calabi-Yau manifolds of type A by developing an equivariant SYZ theory for the toric Calabi Yau manifold of infinite-type.

24 citations


Posted Content
TL;DR: In this article, the apolarity lemma is used to study the border rank of polynomials and tensors, analogous to the aponymy lemma for the case of the toric projective projective variety.
Abstract: We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a systematic approach. We also define a border rank version of the variety of sums of powers and analyse its usefulness in studying tensors and polynomials with large symmetries. In particular, it can be applied to provide lower bounds for the border rank of some very interesting tensors, such as the matrix multiplication tensor. We work in a general setting, where the base variety is not necessarily a Segre or Veronese variety, but an arbitrary smooth toric projective variety. A critical ingredient of our work is an irreducible component of a multigraded Hilbert scheme related to the toric variety in question.

21 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that if a toric variety X is smooth in codimension 2 then one can find a subgroup of SAut(X) generated by a finite number of one-parameter unipotent subgroups which has the same transitivity property.

19 citations


Journal ArticleDOI
TL;DR: In this paper, the authors systematically produce algebraic algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties, which admit an explicit treatment in terms of Toric geometry and graded ring theory.
Abstract: We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring theory. Our approach extends existing constructions of rational varieties with torus action of complexity one and delivers all Mori dream spaces with torus action. We exhibit the example class of ‘general arrangement varieties’ and obtain classification results in the case of complexity two and Picard number at most two, extending former work in complexity one.

19 citations


Journal ArticleDOI
TL;DR: In this article, a monodromy action on the monomially admissible Fukaya-Seidel categories of these Laurent polynomials was shown, as the arguments of their coefficients vary that corresponds under homological mirror symmetry to tensoring by a line bundle naturally associated to monomials whose coefficients are rotated.

Posted Content
TL;DR: A survey of polyhedral products can be found in this article, where the authors provide a brief historical overview of the development of this subject, summarize many of the main results and describe applications.
Abstract: A polyhedral product is a natural subspace of a Cartesian product that is specified by a simplicial complex. The modern formalism arose as a generalization of the spaces known as moment-angle complexes which were developed within the nascent subject of toric topology. This field, which began as a topological approach to toric geometry and aspects of symplectic geometry, has expanded rapidly in recent years. The investigation of polyhedral products and their homotopy theoretic properties has developed to the point where they are studied in various fields of mathematics far removed from their origin. In this survey, we provide a brief historical overview of the development of this subject, summarize many of the main results and describe applications.

Journal ArticleDOI
TL;DR: The notion of monocritical toric metrized divisors was introduced in this article, where it was shown that for any generic D-small sequence of algebraic points of X and every place v of K, the sequence of their Galois orbits on the analytic space X an converges to a measure.
Abstract: We study the distribution of Galois orbits of points of small height on proper toric varieties, and its application to the Bogomolov problem. We introduce the notion of monocritical toric metrized divisor. We prove that a toric metrized divisor D on a proper toric variety X over a global eld K is monocritical if and only if for every generic D-small sequence of algebraic points of X and every place v of K, the sequence of their Galois orbits on the analytic space X an converges to a measure. When this is the case, the limit measure is a translate of the natural measure on the compact torus sitting in the principal orbit of X. The key ingredient is the study of the v-adic modulus distribution of Ga- lois orbits of generic D-small sequences of algebraic points. In particular, we characterize all their cluster measures. We generalize the Bogomolov problem by asking when a subvariety of the principal orbit of a proper toric variety that has the same essential minimum than the ambient variety, must be a translate of a subtorus. We prove that the generalized Bogomolov problem has a positive answer for monocritical toric metrized divisors, and we give several examples of toric metrized divisors for which the Bogomolov problem has a negative answer.

Journal ArticleDOI
TL;DR: In this paper, a holomorphic function W known as the Floer potential is defined for a weakly unobstructed Lagrangian torus in a symplectic manifold X, and a canonical A ∞ -functor from the Fukaya category of X to the category of matrix factorizations of W is constructed.

Journal ArticleDOI
TL;DR: In this article, the cost of solving systems of sparse polynomial equations by homotopy continuation is investigated, and a toric Newton operator is defined on that toric variety.
Abstract: This paper investigates the cost of solving systems of sparse polynomial equations by homotopy continuation. First, a space of systems of n-variate polynomial equations is specified through n monomial bases. The natural locus for the roots of those systems is known to be a certain toric variety. This variety is a compactification of $$(\mathbb {C}\setminus \{0\})^n$$ , dependent on the monomial bases. A toric Newton operator is defined on that toric variety. Smale’s alpha theory is generalized to provide criteria of quadratic convergence. Two condition numbers are defined, and a higher derivative estimate is obtained in this setting. The Newton operator and related condition numbers turn out to be invariant through a group action related to the momentum map. A homotopy algorithm is given and is proved to terminate after a number of Newton steps which is linear on the condition length of the lifted homotopy path. This generalizes a result from Shub (Found Comput Math 9(2):171–178, 2009. https://doi.org/10.1007/s10208-007-9017-6 ).

Journal ArticleDOI
TL;DR: In this article, it was shown that the semipositive envelope is a continuous semipoSitive metric on L-an and that the non-archimedean Monge-Ampere equation has a solution.
Abstract: Let L be an ample line bundle on a smooth projective variety X over a non-archimedean field K. For a continuous metric on L-an, we show in the following two cases that the semipositive envelope is a continuous semipositive metric on L-an and that the non-archimedean Monge-Ampere equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that X is a surface defined geometrically over the function field of a curve over a perfect field k of positive characteristic. The second case holds in higher dimensions if we assume resolution of singularities over k. The proof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplier ideals by test ideals. Finally, the appendix by Burgos and Sombra provides an example of a semipositive metric whose retraction is not semipositive. The example is based on the construction of a toric variety which has two SNC-models which induce the same skeleton but different retraction maps.

Posted Content
TL;DR: The log-local principle of van Garrel-Graber-Ruddat conjectures that the genus 0 log Gromov-Witten theory of maximal tangency of $(X,D)$ is equivalent to the genus ε-local GA of the projective toric complex variety as mentioned in this paper.
Abstract: Let $X$ be a smooth projective complex variety and let $D=D_1+\cdots+D_l$ be a reduced normal crossing divisor on $X$ with each component $D_j$ smooth, irreducible, and nef. The log-local principle of van Garrel-Graber-Ruddat conjectures that the genus 0 log Gromov-Witten theory of maximal tangency of $(X,D)$ is equivalent to the genus 0 local Gromov-Witten theory of $X$ twisted by $\bigoplus_{j=1}^l\mathcal{O}(-D_j)$. We prove that an extension of the log-local principle holds for $X$ a (not necessarily smooth) $\mathbb{Q}$-factorial projective toric variety, $D$ the toric boundary, and descendent point insertions.


Posted Content
TL;DR: In this paper, the role of tropical moduli spaces in logarithmic Gromov-Witten theory is discussed, and the authors use them to study the virtual class of curves in a product of pairs.
Abstract: We discuss the role of subdivisions of tropical moduli spaces in logarithmic Gromov-Witten theory, and use them to study the virtual class of curves in a product of pairs. Our main result is that the cycle-valued logarithmic Gromov-Witten theory of $X\times Y$ decomposes into a product of pieces coming from $X$ and $Y$, but this decomposition must be considered in a blowup of the moduli space of curves. This blowup is specified by tropical moduli data. As an application, we show that the cycle of curves in a toric variety with fixed contact orders is a product of virtual strict transforms of double ramification cycles. The formalism we outline offers a unified viewpoint on a number of recent results in logarithmic Gromov-Witten theory, including works of Herr, Holmes-Pixton-Schmitt, and Nabijou and the author.

Posted Content
TL;DR: In this article, the notion of a valuation into the semifield of piecewise linear functions is used to give a classification of torus equivariant flat families of finite type over a toric variety base.
Abstract: Using the notion of a valuation into the semifield of piecewise linear functions, we give a classification of torus equivariant flat families of finite type over a toric variety base, by certain piecewise linear maps between fans. As a consequence we derive a classification of toric vector bundles phrased in terms of tropicalized linear spaces. We use these tools to give a characterization of the Mori dream space property for a projectivized toric vector bundle.

Posted Content
TL;DR: In this paper, a theory of average sizes of kernels of generic matrices with support constraints defined in terms of graphs and hypergraphs is developed, and the authors apply this theory to study unipotent groups associated with graphs.
Abstract: We develop a theory of average sizes of kernels of generic matrices with support constraints defined in terms of graphs and hypergraphs. We apply this theory to study unipotent groups associated with graphs. In particular, we establish strong uniformity results pertaining to zeta functions enumerating conjugacy classes of these groups. We deduce that the numbers of conjugacy classes of $\mathbf{F}_q$-points of the groups under consideration depend polynomially on $q$. Our approach combines group theory, graph theory, toric geometry, and $p$-adic integration. Our uniformity results are in line with a conjecture of Higman on the numbers of conjugacy classes of unitriangular matrix groups. Our findings are, however, in stark contrast to related results by Belkale and Brosnan on the numbers of generic symmetric matrices of given rank associated with graphs.

Book ChapterDOI
TL;DR: In this paper, it is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group.
Abstract: It is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group. We prove partial results towards this conjecture. We also show that it would give a new proof of the K-polystability of constant scalar curvature polarised manifolds.

Posted Content
TL;DR: In this paper, the authors give a combinatorial criterion for the tangent bundle on a smooth toric variety to be stable with respect to a given polarisation in terms of the corresponding lattice polytope.
Abstract: We give a combinatorial criterion for the tangent bundle on a smooth toric variety to be stable with respect to a given polarisation in terms of the corresponding lattice polytope. Furthermore, we show that for a smooth toric surface and a smooth toric variety of Picard rank 2, there exists an ample line bundle with respect to which the tangent bundle is stable if and only if it is an iterated blow-up of projective space.

Journal ArticleDOI
28 Jan 2019
TL;DR: In this article, the authors give an interim report on some improvements and generalizations of the Abbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate ample hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in linear time in
Abstract: We give an interim report on some improvements and generalizations of the Abbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate ample hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in linear time in $p$. These are illustrated with a number of examples including K3 surfaces, Calabi-Yau threefolds, and a cubic fourfold. The latter example is a non-special cubic fourfold appearing in the Ranestad-Voisin coplanar divisor on moduli space; this verifies that the coplanar divisor is not a Noether-Lefschetz divisor in the sense of Hassett.

Journal ArticleDOI
TL;DR: In this paper, the Chow group of projective toric toric varieties has been used to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of a toric variety.
Abstract: Let $X \subset Y$ be closed (possibly singular) subschemes of a smooth projective toric variety $T$. We show how to compute the Segre class $s(X,Y)$ as a class in the Chow group of $T$. Building on this, we give effective methods to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of $T$. Our methods may be implemented without using Groebner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used.

Journal ArticleDOI
TL;DR: In this article, the authors introduce symplectic reduction in the framework of nonrational toric geometry, where the action of a general, not necessarily closed, Lie subgroup of the torus is studied.

Journal ArticleDOI
TL;DR: Recently, Kuwagaki et al. as mentioned in this paperang-Liu-Treumann-Zaslow showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves for toric stacks.
Abstract: Given a smooth projective toric variety $$X_\Sigma $$ of complex dimension n, Fang–Liu–Treumann–Zaslow (Invent Math 186(1):79–114, 2011) showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves $$Coh(X_\Sigma )$$ into the dg derived category of constructible sheaves on a torus $$Sh(T^n, \Lambda _\Sigma )$$ . Recently, Kuwagaki (The nonequivariant coherent-constructible correspondence for toric stacks, 2016. arXiv:1610.03214 ) proved that the quasi-embedding is a quasi-equivalence, and generalized the result to toric stacks. Here we give a different proof in the smooth projective case, using non-characteristic deformation of sheaves to find twisted polytope sheaves that co-represent the stalk functors.

Journal ArticleDOI
TL;DR: In this article, it was shown that a normal projective variety X is a toric variety if and only if X is of Fano type and smooth in codimension 2 and if there is a reduced divisor D such that X admits a quasi-etale cover such that it lifts to Cartier's Cartier.
Abstract: Let X be a normal projective variety and $$f:X\rightarrow X$$ a non-isomorphic polarized endomorphism. We give two characterizations for X to be a toric variety. First we show that if X is $$\mathbb {Q}$$ -factorial and G-almost homogeneous for some linear algebraic group G such that f is G-equivariant, then X is a toric variety. Next we give a geometric characterization: if X is of Fano type and smooth in codimension 2 and if there is an $$f^{-1}$$ -invariant reduced divisor D such that $$f|_{X\backslash D}$$ is quasi-etale and $$K_X+D$$ is $$\mathbb {Q}$$ -Cartier, then X admits a quasi-etale cover $${\widetilde{X}}$$ such that $${\widetilde{X}}$$ is a toric variety and f lifts to $${\widetilde{X}}$$ . In particular, if X is further assumed to be smooth, then X is a toric variety.

Journal ArticleDOI
TL;DR: In this article, it was shown that for many Calabi-Yau threefolds with elliptic or genus one fibrations mirror symmetry factorizes between the fiber and the base of the fibration.
Abstract: We find that for many Calabi-Yau threefolds with elliptic or genus one fibrations mirror symmetry factorizes between the fiber and the base of the fibration. In the simplest examples, the generic CY elliptic fibration over any toric base surface B that supports an elliptic Calabi-Yau threefold has a mirror that is an elliptic fibration over a dual toric base surface $$ \tilde{B} $$ that is related through toric geometry to the line bundle −6KB. The Kreuzer-Skarke database includes all these examples and gives a wide range of other more complicated constructions where mirror symmetry also factorizes. Since recent evidence suggests that most Calabi-Yau threefolds are elliptic or genus one fibered, this points to a new way of understanding mirror symmetry that may apply to a large fraction of smooth Calabi-Yau threefolds. The factorization structure identified here can also apply for CalabiYau manifolds of higher dimension.

Posted Content
TL;DR: In this paper, Batyrev et al. gave a complete answer to the question of (semi)stability of tangent bundle of any nonsingular projective complex toric variety with Picard number 2 by using combinatorial crietrion of an equivariant sheaf.
Abstract: We give a complete answer to the question of (semi)stability of tangent bundle of any nonsingular projective complex toric variety with Picard number 2 by using combinatorial crietrion of (semi)stability of an equivariant sheaf. We also give a complete answer to the question of (semi)stability of tangent bundle of all toric Fano 4-folds with Picard number (\leq) 3 which are classified by Batyrev \cite{batyrev}. We have constructed a collection of equivariant indecomposable rank 2 vector bundles on Bott tower and pseudo-symmetric toric Fano varieties. Further in case of Bott tower, we have shown the existence of an equivariant stable rank 2 vector bundle with certain Chern classes with respect to a suitable polarization.