Showing papers on "Toric variety published in 2012"

Algebraic Geometry: A First Course

Abstract: 1: Affine and Projective Varieties. 2: Regular Functions and Maps. 3: Cones, Projections, and More About Products. 4: Families and Parameter Spaces. 5: Ideals of Varieties, Irreducible Decomposition. 6: Grassmannians and Related Varieties. 7: Rational Functions and Rational Maps. 8: More Examples. 9: Determinantal Varieties. 10: Algebraic Groups. 11: Definitions of Dimension and Elementary Examples. 12: More Dimension Computations. 13: Hilbert Functions and Polynomials. 14: Smoothness and Tangent Spaces. 15: Gauss Maps, Tangential and Dual Varieties. 16: Tangent Spaces to Grassmannians. 17: Further Topics Involving Smoothness and Tangent Spaces. 18: Degree. 19: Further Examples and Applications of Degree. 20: Singular Points and Tangent Cones. 21: Parameter Spaces and Moduli Spaces. 22: Quadrics.

D-branes at del Pezzo singularities: global embedding and moduli stabilisation

Abstract: In the context of type IIB string theory we combine moduli stabilisation and model building on branes at del Pezzo singularities in a fully consistent global compactification. By means of toric geometry, we classify all the Calabi-Yau manifolds with 3 < h 1,1 < 6 which admit two identical del Pezzo singularities mapped into each other under the orientifold involution. This effective singularity hosts the visible sector containing the Standard Model while the Kahler moduli are stabilised via a combination of D-terms, perturbative and non-perturbative effects supported on hidden sectors. We present concrete models where the visible sector, containing the Standard Model, gauge and matter content, is built via fractional D3-branes at del Pezzo singularities and all the Kahler moduli are fixed providing an explicit realisation of both KKLT and LARGE volume scenarios, the latter with D-term uplifting to de Sitter minima. We perform the consistency checks for global embedding such as tadpole, K-theory charges and Freed-Witten anomaly cancellation. We briefly discuss phenomenological and cosmological implications of our models.

Moduli stabilisation for chiral global models

Abstract: We combine moduli stabilisation and (chiral) model building in a fully con-sistent global set-up in Type IIB/F-theory. We consider compactifications on Calabi-Yau orientifolds which admit an explicit description in terms of toric geometry. We build globally consistent compactifications with tadpole and Freed-Witten anomaly cancellation by choosing appropriate brane set-ups and world-volume fluxes which also give rise to SU(5) or MSSM-like chiral models. We fix all the Kahler moduli within the Kahler cone and the regime of validity of the 4D effective field theory. This is achieved in a way compatible with the local presence of chirality. The hidden sector generating the non-perturbative effects is placed on a del Pezzo divisor that does not have any chiral intersection with any other brane. In general, the vanishing D-term condition implies the shrinking of the rigid divisor supporting the visible sector. However, we avoid this problem by generating r < n D-term conditions on a set of n intersecting divisors. The remaining (n − r) flat directions are fixed by perturbative corrections to the Kahler potential. We illustrate our general claims in an explicit example. We consider a K3-fibred Calabi-Yau with four Kahler moduli, that is a hypersurface in a toric ambient space and admits a ‘simple’ F-theory up-lift. We present explicit choices of brane set-ups and fluxes which lead to three different phenomenological scenarios: the first with GUT-scale strings and TeV-scale SUSY by fine-tuning the background fluxes; the second with an exponentially large value of the volume and TeV-scale SUSY without fine-tuning the background fluxes; and the third with a very anisotropic configuration that leads to TeV-scale strings and two micron-sized extra dimensions. The K3 fibration structure of the Calabi-Yau three-fold is also particularly suitable for cosmological purposes.

Log-concavity of characteristic polynomials and the Bergman fan of matroids

Abstract: In a recent paper, the first author proved the log-concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a long-standing conjecture of Read in graph theory. We extend the proof to all realizable matroids, making progress towards a more general conjecture of Rota–Heron–Welsh. Our proof follows from an identification of the coefficients of the reduced characteristic polynomial as answers to particular intersection problems on a toric variety. The log-concavity then follows from an inequality of Hodge type.

A guide to tropicalizations

Abstract: Tropicalizations form a bridge between algebraic and convex geometry. We generalize basic results from tropical geometry which are well-known for special ground elds to arbitrary non-archimedean valued elds. To achieve this, we develop a theory of toric schemes over valuation rings of rank 1. As a basic tool, we use techniques from non-archimedean analysis. MSC2010: 14T05, 14M25, 32P05 X) of polyhedra in R n . This process is called tropicaliza- tion and it can be used to transform a problem from algebraic geometry into a corresponding problem in convex geometry which is usually easier. If the toric co- ordinates are well suited to the problem, it is sometimes possible to use a solution of the convex problem to solve the original algebraic problem. Another strategy is to vary the ambient torus to compensate the loss of information due to the tropi- calization process. Tropicalization originates from a paper of Bergman (Berg) on logarithmic limit sets. The convex structure of the tropical variety Trop(X) was worked out by Bieri{Groves (BG) with applications to geometric group theory in mind. Sturmfels (Stu) pointed out that Trop(X) is a subcomplex of the Grobner complex. In fact, the polyhedral complex Trop(X) has some natural weights satisfying a balancing condition which appears rst in Speyer's thesis (Spe). This relies on the description

Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties

Abstract: We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampere equation in R^n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P. Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties (X,D), saying that (X,D) admits a (singular) Kahler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and extend to the log Fano setting the seminal result of Zhou-Wang concerning the case when X is smooth and D is trivial. Li's toric formula for the greatest lower bound on the Ricci curvature is also generalized. More generally, we obtain Kahler-Ricci solitons on any log Fano variety and show that they appear as the large time limit of the Kahler-Ricci flow. Furthermore, using duality, we also confirm a conjecture of Donaldson concerning solutions to Abreu's boundary value problem on the convex body P. in the case of a given canonical measure on the boundary of P.

Flag varieties, toric varieties, and suspensions: Three instances of infinite transitivity

Abstract: We say that a group acts infinitely transitively on a set if for every the induced diagonal action of is transitive on the cartesian th power with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of normal affine cones over flag varieties, the second of nondegenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups. Bibliography: 42 titles.

Tropical intersection theory from toric varieties

Abstract: We apply ideas from intersection theory on toric varieties to tropical intersection theory. We introduce mixed Minkowski weights on toric varieties which interpolate between equivariant and ordinary Chow cohomology classes on compact toric varieties. These objects fit into the framework of tropical intersection theory developed by Allermann and Rau. Standard facts about intersection theory on toric varieties are applied to show that the definitions of tropical intersection product on tropical cycles in $${\mathbb{R}^n}$$ given by Allermann–Rau and Mikhalkin are equivalent. We introduce an induced tropical intersection theory on subvarieties on a toric variety. This gives a conceptual proof that the intersection of tropical ψ-classes on $${\overline{\mathcal{M}}_{0,n}}$$ used by Kerber and Markwig computes classical intersection numbers.

The Geometry of T-Varieties

Abstract: This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established the- ory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others.

Test configurations and Okounkov bodies

Abstract: We associate to a test configuration for a polarized variety a filtration of the section ring of the line bundle. Using the recent work of Boucksom and 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 C-x-action on the manifold lifting to the line bundle, the pushforward of the Lebesgue measure on the Okounkov body is piecewise polynomial.

Picard-Fuchs equations for relative periods and Abel-Jacobi map for Calabi-Yau hypersurfaces

Abstract: We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a Calabi-Yau hypersurface in toric variety, and deriving a general formula for the $d$-exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for the Abel-Jacobi map, which has played an important role in the recent study of D-branes (by Morrison and Walcher). Using the variation formalism, we prove that the relative periods of toric B-branes on Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics literature (by Alim, Hecht, Mayr, and Mertens), and discuss the relations between several works in the recent study of open string mirror symmetry. We also give the general solutions to the enhanced hypergeometric system.

Introduction to Equivariant Cohomology in Algebraic Geometry

Abstract: Introduced by Borel in the late 1950’s, equivariant cohomology encodes information about how the topology of a space interacts with a group action. Quite some time passed before algebraic geometers picked up on these ideas, but in the last twenty years, equivariant techniques have found many applications in enumerative geometry, Gromov-Witten theory, and the study of toric varieties and homogeneous spaces. In fact, many classical algebro-geometric notions, going back to the degeneracy locus formulas of Giambelli, are naturally statements about certain equivariant cohomology classes. These lectures survey some of the main features of equivariant cohomology at an introductory level. The first part is an overview, including basic definitions and examples. In the second lecture, I discuss one of the most useful aspects of the theory: the possibility of localizing at fixed points without losing information. The third lecture focuses on Grassmannians, and describes some recent “positivity” results about their equivariant cohomology rings.

On the unipotence of autoequivalences of toric complete intersection Calabi–Yau categories

Abstract: Consider the derived category of coherent sheaves, Db(X), on a compact Calabi–Yau complete intersection X in a toric variety. The scope of this work is to establish the (quasi-)unipotence of a class of elements in the group of autoequivalences, Aut(Db(X)). This is achieved by associating singularity categories, modelled by matrix factorizations, to the toric data. Each of these triangulated categories is equivalent to the derived category of coherent sheaves on X. The idea is then that, although the singularity categories share the group of autoequivalences, on each category there are elements in Aut(Db(X)), whose (quasi-)unipotence relations are easier to see than on the other categories.

Global embeddings for branes at toric singularities

Abstract: We describe how local toric singularities, including the Toric Lego construction, can be embedded in compact Calabi-Yau manifolds. We study in detail the addition of D- branes, including non-compact avor branes as typically used in semi-realistic model building. The global geometry provides constraints on allowable local models. As an illustration of our discussion we focus on D3 and D7-branes on (the partially resolved) (dP0) 3 singularity,

Categorified duality in Boij-S\"oderberg Theory and invariants of free complexes

Abstract: We present a robust categorical foundation for the duality theory introduced by Eisenbud and Schreyer to prove the Boij-Soderberg conjectures describing numerical invariants of syzygies. The new foundation allows us to extend the reach of the theory substantially. More explicitly, we construct a pairing between derived categories that simultaneously categorifies all the functionals used by Eisenbud and Schreyer. With this new tool, we describe the cone of Betti tables of finite, minimal free complexes having homology modules of specified dimensions over a polynomial ring, and we treat many examples beyond polynomial rings. We also construct an analogue of our pairing between derived categories on a toric variety, yielding toric/multigraded analogues of the Eisenbud-Schreyer functionals.

Remarks on Lagrangian intersections in toric manifolds

Abstract: I will consider two natural Lagrangian intersection problems in the context of symplectic toric manifolds: displaceability of torus orbits and of a torus orbit with the real part of the toric manifold. The remarks address the fact that one can use simple cartesian product and symplectic reduction considerations to go from basic examples to much more sophisticated ones. I will show in particular how rigidity results for the above Lagrangian intersection problems in weighted projective spaces can be combined with these considerations to prove analogous results for all monotone toric symplectic manifolds. We also discuss non-monotone and/or non-Fano examples, including some with a continuum of non-displaceable torus orbits. This is joint work with Leonardo Macarini.

Disk counting on toric varieties via tropical curves

Abstract: In this paper, we define two numbers. One is defined by counting tropical curves with a stop, and the other is the number of holomorphic disks in toric varieties with Lagrangian boundary condition. Both of these curves should satisfy some incidence conditions. We show that these numbers coincide. These numbers can be considered as Gromov-Witten type invariants for holomorphic disks, and they have similarities as well as differences to the counting numbers of closed holomorphic curves. We study several aspects of them.

Arithmetic positivity on toric varieties

Abstract: We continue with our study of the arithmetic geometry of toric varieties In this text, we study the positivity properties of metrized R-divisors in the toric setting For a toric metrized R-divisor, we give formulae for its arithmetic volume and its chi-arithmetic volume, and we characterize when it is arithmetically ample, nef, big or pseudo-effective, in terms of combinatorial data As an application, we prove a Dirichlet's unit theorem on toric varieties, we give a characterization for the existence of a Zariski decomposition of a toric metrized R-divisor, and we prove a toric arithmetic Fujita approximation theorem

Equivariant Chow cohomology of nonsimplicial toric varieties

Abstract: For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne shows that the equivariant Chow cohomology of X_P is the Sym(N)--algebra C^0(P) of integral piecewise polynomial functions on P. We use the Cartan-Eilenberg spectral sequence to analyze the associated reflexive sheaf on Proj(N), showing that the Chern classes depend on subtle geometry of P and giving criteria for the splitting of the sheaf as a sum of line bundles. For certain fans associated to the reflection arrangement A_n, we describe a connection between C^0(P) and logarithmic vector fields tangent to A_n.

Algebraic and combinatorial Brill-Noether theory

Abstract: The interplay between algebro-geometric and combinatorial Brill-Noether theory is studied The Brill-Noether variety of a graph shown to be non-empty if the Brill-Noether number is non-negative, as a consequence of the analogous fact for smooth projective curves Similarly, the existence of a graph for which the Brill-Noether variety is empty implies the emptiness of the corresponding Brill-Noether variety for a general curve The main tool is a refinement of Baker's Specialization Lemma

Mutations of Laurent Polynomials and Flat Families with Toric Fibers

Abstract: We give a general criterion for two toric varieties to appear as fibers in a flat family over P 1 . We apply this to show that certain birational transformations mapping

Invariants of toric seiberg duality

Abstract: Three-branes at a given toric Calabi–Yau singularity lead to different phases of the conformal field theory related by toric (Seiberg) duality. Using the dimer model/brane tiling description in terms of bipartite graphs on a torus, we find a new invariant under Seiberg duality, namely the Klein j-invariant of the complex structure parameter in the distinguished isoradial embedding of the dimer, determined by the physical R-charges. Additional number theoretic invariants are described in terms of the algebraic number field of the R-charges. We also give a new compact description of the a-maximization procedure by introducing a generalized incidence matrix.

Algebraic stability and degree growth of monomial maps

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.

Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces

Abstract: We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface $H$ in a toric variety $V$ we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of $V\times\mathbb{C}$ along $H\times 0$, under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to $H$. The main applications concern affine hypersurfaces of general type, for which our results provide a geometric basis for various mirror symmetry statements that appear in the recent literature. We also obtain analogous results for complete intersections.

Holomorphic orbidiscs and Lagrangian Floer cohomology of symplectic toric orbifolds

Abstract: We develop Floer theory of Lagrangian torus fibers in compact symplectic toric orbifolds. We first classify holomorphic orbi-discs with boundary on Lagrangian torus fibers. We show that there exists a class of basic discs such that we have one-to-one correspondences between a) smooth basic discs and facets of the moment polytope, and b) between basic orbi-discs and twisted sectors of the toric orbifold. We show that there is a smooth Lagrangian Floer theory of these torus fibers, which has a bulk-deformation by fundamental classes of twisted sectors of the toric orbifold. We show by several examples that such bulk-deformation can be used to illustrate the very rigid Hamiltonian geometry of orbifolds. We define its potential and bulk-deformed potential, and develop the notion of leading order potential. We study leading term equations analogous to the case of toric manifolds by Fukaya, Oh, Ohta and Ono.

The Universal Kummer Threefold

Abstract: The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in 7-dimensional projective space. We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the Gopel variety, and over the reflection representation of type E7. We develop classical themes such as theta functions and Coble's quartic hypersurface using current tools from combinatorics, geometry, and commutative algebra. Symbolic and numerical computations for genus 3 moduli spaces appear alongside toric and tropical methods.