Showing papers on "Toric variety published in 2017"

Skeletons of stable maps I: rational curves in toric varieties

Abstract: We study the Berkovich analytification of the space of genus $0$ logarithmic stable maps to a toric variety $X$ and present applications to both algebraic and tropical geometry. On algebraic side, insights from tropical geometry give two new geometric descriptions of this space of maps -- (1) as an explicit toroidal modification of $\overline M_{0,n}\times X$ and (2) as a tropical compactification in a toric variety. On the combinatorial side, we prove that the tropicalization of the space of genus $0$ logarithmic stable maps coincides with the space of tropical stable maps, giving a large new collection of examples of faithful tropicalizations for moduli. Moreover, we identify the optimal settings in which the tropicalization of the moduli space of maps is faithful. The Nishinou--Siebert correspondence theorem is shown to be a consequence of this geometric connection between the algebraic and tropical moduli spaces.

Tropicalization is a non-Archimedean analytic stack quotient

Abstract: For a complex toric variety $X$ the logarithmic absolute value induces a natural retraction of $X$ onto the set of its non-negative points and this retraction can be identified with a quotient of $X(\mathbb{C})$ by its big real torus. We prove an analogous result in the non-Archimedean world: The Kajiwara-Payne tropicalization map is a non-Archimedean analytic stack quotient of $X^{an}$ by its big affinoid torus. Along the way, we provide foundations for a geometric theory of non-Archimedean analytic stacks, particularly focussing on analytic groupoids and their quotients, the process of analytification, and the underlying topological spaces of analytic stacks.

The Toric SO(10) F-Theory Landscape

Abstract: Supergravity theories in more than four dimensions with grand unified gauge symmetries are an important intermediate step towards the ultraviolet completion of the Standard Model in string theory. Using toric geometry, we classify and analyze six-dimensional F-theory vacua with gauge group SO(10) taking into account Mordell-Weil U(1) and discrete gauge factors. We determine the full matter spectrum of these models, including charged and neutral SO(10) singlets. Based solely on the geometry, we compute all matter multiplicities and confirm the cancellation of gauge and gravitational anomalies independent of the base space. Particular emphasis is put on symmetry enhancements at the loci of matter fields and to the frequent appearance of superconformal points. They are linked to non-toric Kahler deformations which contribute to the counting of degrees of freedom. We compute the anomaly coefficients for these theories as well by using a base-independent blow-up procedure and superconformal matter transitions. Finally, we identify six-dimensional supergravity models which can yield the Standard Model with high-scale supersymmetry by further compactification to four dimensions in an Abelian flux background.

The Hodge Numbers of Divisors of Calabi-Yau Threefold Hypersurfaces

Abstract: We prove a formula for the Hodge numbers of square-free divisors of Calabi-Yau threefold hypersurfaces in toric varieties. Euclidean branes wrapping divisors affect the vacuum structure of Calabi-Yau compactifications of type IIB string theory, M-theory, and F-theory. Determining the nonperturbative couplings due to Euclidean branes on a divisor $D$ requires counting fermion zero modes, which depend on the Hodge numbers $h^i({\cal{O}}_D)$. Suppose that $X$ is a smooth Calabi-Yau threefold hypersurface in a toric variety $V$, and let $D$ be the restriction to $X$ of a square-free divisor of $V$. We give a formula for $h^i({\cal{O}}_D)$ in terms of combinatorial data. Moreover, we construct a CW complex $\mathscr{P}_D$ such that $h^i({\cal{O}}_D)=h_i(\mathscr{P}_D)$. We describe an efficient algorithm that makes possible for the first time the computation of sheaf cohomology for such divisors at large $h^{1,1}$. As an illustration we compute the Hodge numbers of a class of divisors in a threefold with $h^{1,1}=491$. Our results are a step toward a systematic computation of Euclidean brane superpotentials in Calabi-Yau hypersurfaces.

The toric SO(10) F-theory landscape

Abstract: Supergravity theories in more than four dimensions with grand unified gauge symmetries are an important intermediate step towards the ultraviolet completion of the Standard Model in string theory. Using toric geometry, we classify and analyze six-dimensional F-theory vacua with gauge group SO(10) taking into account Mordell-Weil U(1) and discrete gauge factors. We determine the full matter spectrum of these models, including charged and neutral SO(10) singlets. Based solely on the geometry, we compute all matter multiplicities and confirm the cancellation of gauge and gravitational anomalies independent of the base space. Particular emphasis is put on symmetry enhancements at the loci of matter fields and to the frequent appearance of superconformal points. They are linked to non-toric Kahler deformations which contribute to the counting of degrees of freedom. We compute the anomaly coefficients for these theories as well by using a base-independent blow-up procedure and superconformal matter transitions. Finally, we identify six-dimensional supergravity models which can yield the Standard Model with high-scale supersymmetry by further compactification to four dimensions in an Abelian flux background.

Moduli of stable maps in genus one and logarithmic geometry II

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 $1$.

Lagrangian fibers of Gelfand-Cetlin systems

Abstract: A Gelfand-Cetlin system is a completely integrable system defined on a partial flag manifold whose image is a rational convex polytope called a Gelfand-Cetlin polytope. Motivated by the study of Nishinou-Nohara-Ueda on the Floer theory of Gelfand-Cetlin systems, we provide a detailed description of topology of Gelfand-Cetlin fibers. In particular, we prove that any fiber over an interior point of a k-dimensional face of the Gelfand-Cetlin polytope is an isotropic submanifold and is diffeomorphic to $(S^1)^k \times N$ for some smooth manifold $N$. We also prove that such $N$'s are exactly the vanishing cycles shrinking to points in the associated toric variety via the toric degeneration. We also devise an algorithm of reading off Lagrangian fibers from the combinatorics of the ladder diagram.

On the integral cohomology ring of toric orbifolds and singular toric varieties

Abstract: We examine the integral cohomology rings of certain families of 2n–dimensional orbifolds X that are equipped with a well-behaved action of the n–dimensional real torus. These orbifolds arise from two distinct but closely related combinatorial sources, namely from characteristic pairs (Q,λ), where Q is a simple convex n–polytope and λ a labeling of its facets, and from n–dimensional fans Σ. In the literature, they are referred as toric orbifolds and singular toric varieties, respectively. Our first main result provides combinatorial conditions on (Q,λ) or on Σ which ensure that the integral cohomology groups H∗(X) of the associated orbifolds are concentrated in even degrees. Our second main result assumes these conditions to be true, and expresses the graded ring H∗(X) as a quotient of an algebra of polynomials that satisfy an integrality condition arising from the underlying combinatorial data. Also, we compute several examples.

Toric vector bundles and parliaments of polytopes

Abstract: We introduce a collection of convex polytopes associated to a torus-equivariant vector bundle on a smooth complete toric variety. We show that the lattice points in these polytopes correspond to ge ...

Characteristic classes of affine varieties and Plücker formulas for affine morphisms

Abstract: An enumerative problem on a variety V is usually solved by reduction to intersection theory in the cohomology of a compactification of V . However, if the problem is invariant under a “nice” group action on V (so that V is spherical), then many authors suggested a better home for intersection theory: the direct limit of the cohomology rings of all equivariant compactifications of V . We call this limit the affine cohomology of V and construct affine characteristic classes of subvarieties of a complex torus, taking values in the affine cohomology of the torus. This allows us to make the first steps in computing affine Thom polynomials. Classical Thom polynomials count how many fibers of a generic proper map of a smooth variety have a prescribed collection of singularities, and our affine version addresses the same question for generic polynomial maps of affine algebraic varieites. This notion is also motivated by developing an intersection-theoretic approach to tropical correspondence theorems: they can be reduced to the computation of affine Thom polynomials, because the fundamental class of a variety in the affine cohomology is encoded by the tropical fan of this variety. The first concrete answer that we obtain is the affine version of what were, historically speaking, the first three Thom poylnomials – the Plucker formulas for the degree and the number of cusps and nodes of a projectively dual curve. This, in particular, classifies toric varieties, whose projective dual is a hypersurface, computes the tropical fan of the variety of double tangent hyperplanes to a toric variety, and describes the Newton polytope of the hypersurface of non-Morse polynomials of a given degree. We also make a conjecture on the general form of affine Thom polynomials – a key ingredient is the n-ary fan, generalizing the secondary polytope.

Multiplication structure of the cohomology ring of real toric spaces

Abstract: A real toric space is a topological space which admits a well-behaved $\mathbb{Z}_2^k$-action. Real moment-angle complexes and real toric varieties are typical examples of real toric spaces. A real toric space is determined by a pair of a simplicial complex $K$ and a characteristic matrix $\Lambda$. In this paper, we provide an explicit $R$-cohomology ring formula of a real toric space in terms of $K$ and $\Lambda$, where $R$ is a commutative ring with unity in which $2$ is a unit. Interestingly, it has a natural $(\mathbb{Z} \oplus \operatorname*{row} \Lambda)$-grading. As corollaries, we compute the cohomology rings of (generalized) real Bott manifolds in terms of binary matroids, and we also provide a criterion for real toric spaces to be cohomology symplectic.

Tropical compactification and the Gromov–Witten theory of \mathbb {P}^1

Abstract: We use tropical and non-archimedean geometry to study the moduli space of genus 0, stable maps to $$\mathbb {P}^1$$ relative to two points. This space is exhibited as a tropical compactification in a toric variety. Moreover, the fan of this toric variety may be interpreted as a moduli space for tropical relative stable maps with the same discrete data. As a consequence, we confirm an expectation of Bertram and the first two authors, that the tropical Hurwitz cycles are tropicalizations of classical Hurwitz cycles. As a second application, we obtain a full descendant correspondence for genus 0 relative invariants of $$\mathbb {P}^1$$ .

Beyond Aztec Castles: Toric Cascades in the dP 3 Quiver

Abstract: Given one of an infinite class of supersymmetric quiver gauge theories, string theorists can associate a corresponding toric variety (which is a Calabi–Yau 3-fold) as well as an associated combinatorial model known as a brane tiling. In combinatorial language, a brane tiling is a bipartite graph on a torus and its perfect matchings are of interest to both combinatorialists and physicists alike. A cluster algebra may also be associated to such quivers and in this paper we study the generators of this algebra, known as cluster variables, for the quiver associated to the cone over the del Pezzo surface d P 3. In particular, mutation sequences involving mutations exclusively at vertices with two in-coming arrows and two out-going arrows are referred to as toric cascades in the string theory literature. Such toric cascades give rise to interesting discrete integrable systems on the level of cluster variable dynamics. We provide an explicit algebraic formula for all cluster variables that are reachable by toric cascades as well as a combinatorial interpretation involving perfect matchings of subgraphs of the d P 3 brane tiling for these formulas in most cases.

Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals (with an Appendix by Jos\'e Ignacio Burgos Gil and Mart\'in Sombra)

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, we show in the following two cases that the semipositive envelope is a continuous semipositive metric on L 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.

Non-commutative crepant resolutions for some toric singularities II

Abstract: Using the theory of dimer models Broomhead proved that every 3-dimensional Gorenstein affine toric variety Spec R admits a toric non-commutative crepant resolution (NCCR). We give an alternative proof of this result by constructing a tilting bundle on a (stacky) crepant resolution of Spec R using standard toric methods. Our proof does not use dimer models.

Duality for toric Landau–Ginzburg models

Abstract: We introduce a duality construction for toric Landau-Ginzburg models, applicable to complete intersections in toric varieties via the sigma model / Landau-Ginzburg model correspondence. This construction is shown to reconstruct those of Batyrev-Borisov, Berglund-H"ubsch, Givental, and Hori-Vafa. It can be done in more general situations, and provides partial resolutions when the above constructions give a singular mirror. An extended example is given: the Landau-Ginzburg models dual to elliptic curves in (P^1)^2 .

Quantum d-modules for toric nef complete intersections

Abstract: Let X be a smooth projective toric variety with k ample line bundles. Let Z be the zero locus of k generic sections. It is well known that the ambient quantum 𝒟-module of Z is cyclic i.e. is defined by an ideal of differential operators. In this paper, we give an explicit construction of this ideal as a quotient ideal of a GKZ system associated to the toric data of X and the line bundles. This description can be seen as a “left cancellation procedure”. We consider some examples where this description enables us to compute generators of this ideal, and thus to give a presentation of the ambient quantum 𝒟-module.

Liftability of the Frobenius morphism and images of toric varieties

Abstract: We formulate a conjecture characterizing smooth projective varieties in positive characteristic whose Frobenius morphism can be lifted modulo $p^2$ - we expect that such varieties, after a finite etale cover, admit a Zariski-locally trivial fibration with toric fibers over an ordinary abelian variety. We prove that this assertion implies a conjecture of Occhetta and Wiśniewski, which states that a smooth image of a projective toric variety is a toric variety. In order to deal with an important special case, we develop a logarithmic variant of the characterization of ordinary varieties with trivial tangent bundle due to Mehta and Srinivas. Furthermore, we verify our conjecture for surfaces, Fano threefolds, and homogeneous spaces (answering a question posed by Buch-Thomsen-Lauritzen-Mehta). Our proofs are based on a comprehensive theory of Frobenius liftings together with a variety of other techniques including deformation theory of rational curves and Frobenius splittings.

A $$\mathbb Q$$ Q -factorial complete toric variety is a quotient of a poly weighted space

Abstract: We prove that every \(\mathbb {Q}\)-factorial complete toric variety is a finite abelian quotient of a poly weighted space (PWS), as defined in our previous work (Rossi and Terracini in Linear Algebra Appl 495:256–288, 2016. doi:10.1016/j.laa.2016.01.039). This generalizes the Batyrev–Cox and Conrads description of a \(\mathbb {Q}\)-factorial complete toric variety of Picard number 1, as a finite quotient of a weighted projective space (WPS) (Duke Math J 75:293–338, 1994, Lemma 2.11) and (Manuscr Math 107:215–227, 2002, Prop. 4.7), to every possible Picard number, by replacing the covering WPS with a PWS. By Buczynska’s results (2008), we get a universal picture of coverings in codimension 1 for every \(\mathbb {Q}\)-factorial complete toric variety, as topological counterpart of the \(\mathbb {Z}\)-linear universal property of the double Gale dual of a fan matrix. As a consequence, we describe the bases of the subgroup of Cartier divisors inside the free group of Weil divisors and the bases of the Picard subgroup inside the class group, respectively, generalizing to every \(\mathbb {Q}\)-factorial complete toric variety the description given in Rossi and Terracini (2016, Thm. 2.9) for a PWS.

Notes on toric varieties from Mori theoretic viewpoint, II

Abstract: We give new estimates of lengths of extremal rays of birational type for toric varieties. We can see that our new estimates are the best by constructing some examples explicitly. As applications, we discuss the nefness and pseudo-effectivity of adjoint bundles of projective toric varieties. We also treat some generalizations of Fujita's freeness and very ampleness for toric varieties.

On toric locally conformally Kähler manifolds

Abstract: We study compact toric strict locally conformally Kahler manifolds. We show that the Kodaira dimension of the underlying complex manifold is $$-\infty $$ , and that the only compact complex surfaces admitting toric strict locally conformally Kahler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.

Cohomology of toric origami manifolds with acyclic proper faces

Abstract: Toric origami manifolds are generalizations of symplectic toric manifolds, where the origami symplectic form, in contrast to the usual sym- plectic form, is allowed to degenerate in a good controllable way. It is widely known that symplectic toric manifolds are encoded by Delzant polytopes. The cohomology and equivariant cohomology rings of a symplectic toric manifold can be described in terms of the corresponding polytope. One can obtain a similar description for the cohomology of a toric origami manifold M in terms of the orbit space M=T when M is orientable and the orbit space M=T is con- tractible; this was done by Holm and Pires in (9). Generally, the orbit space of a toric origami manifold need not be contractible. In this paper we study the topology of orientable toric origami manifolds for the wider class of examples: we require that any proper face of the orbit space is acyclic, while the orbit space itself may be arbitrary. Furthermore, we give a general description of the equivariant cohomology ring of torus manifolds with locally standard torus action in the case when proper faces of the orbit space are acyclic and the free part of the action is a trivial torus bundle.

Toric contact geometry in arbitrary codimension

Abstract: We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.

Shift operators and toric mirror theorem

Abstract: We give a new proof of Givental's mirror theorem for toric manifolds using shift operators of equivariant parameters. The proof is almost tautological: it gives an A-model construction of the I-function and the mirror map. It also works for non-compact or non-semipositive toric manifolds.

Arithmetic purity of strong approximation for homogeneous spaces

Abstract: We prove that any open subset $U$ of a semi-simple simply connected quasi-split linear algebraic group $G$ with ${codim} (G\setminus U, G)\geq 2$ over a number field satisfies strong approximation by establishing a fibration of $G$ over a toric variety. We also prove a similar result of strong approximation with Brauer-Manin obstruction for a partial equivariant smooth compactification of a homogeneous space where all invertible functions are constant and the semi-simple part of the linear algebraic group is quasi-split. Some semi-abelian varieties of any given dimension where the complements of a rational point do not satisfy strong approximation with Brauer-Manin obstruction are given.

Levi-Kahler reduction of CR structures, products of spheres, and toric geometry

Abstract: We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi-Kahler quotient, for constructing Kahler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi-Kahler quotients of toric CR manifolds, and in particular, products of odd dimensional spheres. We obtain explicit descriptions and characterizations of such quotients, and find Levi-Kahler quotients of products of 3-spheres which are extremal in a weighted sense introduced by G. Maschler and the first author.

Pseudograph and its associated real toric manifold

Abstract: Given a simple graph $G$, the graph associahedron $P_G$ is a convex polytope whose facets correspond to the connected induced subgraphs of $G$. Graph associahedra have been studied widely and are found in a broad range of subjects. Recently, S. Choi and H. Park computed the rational Betti numbers of the real toric variety corresponding to a graph associahedron under the canonical Delzant realization. In this paper, we focus on a pseudograph associahedron which was introduced by Carr, Devadoss and Forcey, and then discuss how to compute the Poincare polynomial of the real toric variety corresponding to a pseudograph associahedron under the canonical Delzant realization.