Showing papers on "Toric variety published in 2020"

Virtual resolutions for a product of projective spaces

Abstract: Syzygies capture intricate geometric properties of a subvariety in projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety, minimal free resolutions over the Cox ring are too long and contain many geometrically superfluous summands. In this paper, we construct some much shorter free complexes that better encode the geometry.

Minimal surfaces and weak gravity

Abstract: We show that the Weak Gravity Conjecture (WGC) implies a nontrivial upper bound on the volumes of the minimal-volume cycles in certain homology classes that admit no calibrated representatives. In compactification of type IIB string theory on an orientifold X of a Calabi-Yau threefold, we consider a homology class [Σ] ∈ H4(X, ℝ) represented by a union Σ∪ of holomorphic and antiholomorphic cycles. The instanton form of the WGC applied to the axion charge [Σ] implies an upper bound on the action of a non-BPS Euclidean D3-brane wrapping the minimal-volume representative Σmin of [Σ]. We give an explicit example of an orientifold X of a hypersurface in a toric variety, and a hyperplane H ⊂ H4(X, ℝ), such that for any [Σ] ∈ H that satisfies the WGC, the minimal volume obeys Vol (Σmin) ≪ Vol(Σ∪): the holomorphic and antiholomorphic components recombine to form a much smaller cycle. In particular, the sub-Lattice WGC applied to X implies large recombination, no matter how sparse the sublattice. Non-BPS instantons wrapping Σmin are then more important than would be predicted from a study of BPS instantons wrapping the separate components of Σ∪. Our analysis hinges on a novel computation of effective divisors in X that are not inherited from effective divisors of the toric variety.

The nonequivariant coherent-constructible correspondence for toric stacks

Abstract: The nonequivariant coherent-constructible correspondence is a microlocal-geometric interpretation of homological mirror symmetry for toric varieties conjectured by Fang, Liu, Treumann, and Zaslow. We prove a generalization of this conjecture for a class of toric stacks which includes any toric variety and toric orbifold. Our proof is based on gluing descriptions of ∞-categories of both sides.

The Higher Dimensional Tropical Vertex

Abstract: We study log Calabi-Yau varieties obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary. Mirrors to such varieties are constructed by Gross-Siebert from a canonical scattering diagram built by using punctured log Gromov-Witten invariants of Abramovich-Chen-Gross-Siebert. We show that there is a piecewise linear isomorphism between the canonical scattering diagram and a scattering diagram defined algortihmically, following a higher dimensional generalisation of the Kontsevich-Soibelman construction. We deduce that the punctured log Gromov-Witten invariants of the log Calabi-Yau variety can be captured from this algorithmic construction. As a particular example, we compute these invariants for a non-toric blow-up of the three dimensional projective space along two lines. This generalizes previous results of Gross-Pandharipande-Siebert on "The Tropical Vertex" to higher dimensions.

Calabi-Yau Spaces in the String Landscape

Abstract: Calabi-Yau spaces, or Kahler spaces admitting zero Ricci curvature, have played a pivotal role in theoretical physics and pure mathematics for the last half-century. In physics, they constituted the first and natural solution to compactification of superstring theory to our 4-dimensional universe, primarily due to one of their equivalent definitions being the admittance of covariantly constant spinors. Since the mid-1980s, physicists and mathematicians have joined forces in creating explicit examples of Calabi-Yau spaces, compiling databases of formidable size, including the complete intersecion (CICY) dataset, the weighted hypersurfaces dataset, the elliptic-fibration dataset, the Kreuzer-Skarke toric hypersurface dataset, generalized CICYs etc., totaling at least on the order of 10^10 manifolds. These all contribute to the vast string landscape, the multitude of possible vacuum solutions to string compactification. More recently, this collaboration has been enriched by computer science and data science, the former, in bench-marking the complexity of the algorithms in computing geometric quantities and the latter, in applying techniques such as machine-learning in extracting unexpected information. These endeavours, inspired by the physics of the string landscape, have rendered the investigation of Calabi-Yau spaces one of the most exciting and inter-disciplinary fields. Invited contribution to the Oxford Research Encyclopedia of Physics, B.~Foster Ed., OUP, 2020

Levi–Kähler 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.

Toric Eigenvalue Methods for Solving Sparse Polynomial Systems

Abstract: We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact toric variety $X$. Our starting point is a homogeneous ideal $I$ in the Cox ring of $X$, which gives a global description of this subscheme. It was recently shown that eigenvalue methods for solving this problem lead to robust numerical algorithms for solving (nearly) degenerate sparse polynomial systems. In this work, we give a first description of this strategy for non-reduced, zero-dimensional subschemes of $X$. That is, we allow isolated points with arbitrary multiplicities. Additionally, we investigate the regularity of $I$ to provide the first universal complexity bounds for the approach, as well as sharper bounds for weighted homogeneous, multihomogeneous and unmixed sparse systems, among others. We disprove a recent conjecture regarding the regularity and prove an alternative version. Our contributions are illustrated by several examples.

Commutative algebraic monoid structures on affine spaces

Abstract: We study commutative associative polynomial operations 𝔸n × 𝔸n → 𝔸n with unit on the affine space 𝔸n over an algebraically closed field of characteristic zero. A classification of such operations i...

Immaculate line bundles on toric varieties

Abstract: We call a sheaf on an algebraic variety immaculate if it lacks any cohomology including the zero-th one, that is, if the derived version of the global section functor vanishes. Such sheaves are the basic tools when building exceptional sequences, investigating the diagonal property, or the toric Frobenius morphism. In the present paper we focus on line bundles on toric varieties. First, we present a possibility of understanding their cohomology in terms of their (generalized) momentum polytopes. Then we present a method to exhibit the entire locus of immaculate divisors within the class group. This will be applied to the cases of smooth toric varieties of Picard rank two and three and to those being given by splitting fans. The locus of immaculate line bundles contains several linear strata of varying dimensions. We introduce a notion of relative immaculacy with respect to certain contraction morphisms. This notion will be stronger than plain immaculacy and provides an explanation of some of these linear strata.

On toric geometry and K-stability of Fano varieties

Abstract: We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano 3-fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3-fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3-folds by building degenerations to K-polystable toric Fano 3-folds.

Varieties of Tropical Ideals are Balanced

Abstract: Tropical ideals, introduced in arXiv:1609.03838, define subschemes of tropical toric varieties. We prove that the top-dimensional parts of their varieties are balanced polyhedral complexes of the same dimension as the ideal. This means that every subscheme of a tropical toric variety defined by a tropical ideal has an associated class in the Chow ring of the toric variety. A key tool in the proof is that specialization of variables in a tropical ideal yields another tropical ideal; this plays the role of hyperplane sections in the theory. We also show that elimination theory (projection of varieties) works for tropical ideals as in the classical case. The matroid condition that defines tropical ideals is crucial for these results.

ADE surfaces and their moduli

Abstract: We define a class of surfaces corresponding to the ADE root lattices and construct compactifications of their moduli spaces as quotients of projective varieties for Coxeter fans, generalizing Losev-Manin spaces of curves. We exhibit modular families over these moduli spaces, which extend to families of stable pairs over the compactifications. One simple application is a geometric compactification of the moduli of rational elliptic surfaces that is a finite quotient of a projective toric variety.

Tannakian classification of equivariant principal bundles on toric varieties

Abstract: Let X be a complete toric variety equipped with the action of a torus T, and G a reductive algebraic group, defined over an algebraically closed field K. We introduce the notion of a compatible ∑-filtered algebra associated to X, generalizing the notion of a compatible ∑-filtered vector space due to Klyachko, where ∑ denotes the fan of X. We combine Klyachko's classification of T-equivariant vector bundles on X with Nori's Tannakian approach to principal G-bundles, to give an equivalence of categories between T-equivariant principal G-bundles on X and certain compatible ∑-filtered algebras associated to X, when the characteristic of K is 0.

Numerical homotopies from Khovanskii bases

Abstract: We present numerical homotopy continuation algorithms for solving systems of equations on a variety in the presence of a finite Khovanskii basis. These take advantage of Anderson's flat degeneration to a toric variety. When Anderson's degeneration embeds into projective space, our algorithm is a special case of a general toric two-step homotopy algorithm. When Anderson's degeneration is embedded in a weighted projective space, we explain how to lift to a projective space and construct an appropriate modification of the toric homotopy. Our algorithms are illustrated on several examples using Macaulay2.

Cracked polytopes and Fano toric complete intersections

Abstract: We introduce the notion of cracked polytope, and – making use of joint work with Coates and Kasprzyk—construct the associated toric variety X as a subvariety of a smooth toric variety Y under certain conditions. Restricting to the case in which this subvariety is a complete intersection, we present a sufficient condition for a smoothing of X to exist inside Y. We exhibit a relative anti-canonical divisor for this smoothing of X, and show that the general member is simple normal crossings.

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.

Gelfand–tsetlin degenerations of representations and flag varieties

Abstract: Our main goal is to show that the Gelfand–Tsetlin toric degeneration of the type A flag variety can be obtained within a degenerate representation-theoretic framework similar to the theory of PBW degenerations. In fact, we provide such frameworks for all Grobner degenerations intermediate between the flag variety and the GT toric variety. These degenerations are shown to induce filtrations on the irreducible representations and the associated graded spaces are acted upon by a certain associative algebra. To achieve our goal, we construct embeddings of our Grobner degenerations into the projectivizations of said associated graded spaces in terms of this action. We also obtain an explicit description of the maximal cone in the tropical flag variety that parametrizes the Grobner degenerations we consider. In an addendum we propose an alternative solution to the problem which relies on filtrations and gradings by non-abelian ordered semigroups.

Local Vanishing and Hodge Filtration for Rational Singularities

Abstract: Given an n-dimensional variety Z with rational singularities, we conjecture that for a resolution of singularities whose reduced exceptional divisor E has simple normal crossings, the (n-1)-th higher direct image of the sheaf of differential forms with log poles along E vanishes. We prove this when Z has isolated singularities and when it is a toric variety. We deduce that for a divisor D with isolated rational singularities on a smooth complex n-dimensional variety X, the generation level of Saito's Hodge filtration on the localization of the structure sheaf along D is at most n-3.

A comparison of positivity in complex and tropical toric geometry

Abstract: Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the cone of invariant closed positive currents on the complex toric variety with closed positive currents on the tropicalization. In a subsequent paper, this correspondence will be used to develop a Bedford-Taylor theory of plurisubharmonic functions on the tropicalization.

On uniqueness of additive actions on complete toric varieties

Abstract: By an additive action on an algebraic variety $X$ we mean a regular effective action $\mathbb{G}_a^n\times X\to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. In this paper, we give a uniqueness criterion for additive action on a complete toric variety.

The bipermutahedron

Abstract: The harmonic polytope and the bipermutahedron are two related polytopes which arose in Ardila, Denham, and Huh's work on the Lagrangian geometry of matroids. We study the bipermutahedron. We show that its faces are in bijection with the vertex-labeled and edge-labeled multigraphs with no isolated vertices; the generating function for its f-vector is a simple evaluation of the three variable Rogers--Ramanujan function. We show that the h-polynomial of the bipermutahedral fan is the biEulerian polynomial, which counts bipermutations according to their number of descents. We construct a unimodular triangulation of the product of n triangles that is combinatorially equivalent to (the triple cone over) the nth bipermutahedral fan. Ehrhart theory then gives us a formula for the biEulerian polynomial, which we use to show that this polynomial is real-rooted and that the h-vector of the bipermutahedral fan is log-concave and unimodal. We describe all the deformations of the bipermutahedron; that is, the ample cone of the bipermutahedral toric variety. We prove that among all polytopes in this family, the bipermutahedron has the largest possible symmetry group. Finally, we show that the Minkowski quotient of the bipermutahedron and the harmonic polytope equals 2.

Automorphisms of nonnormal toric varieties

Abstract: In this paper we prove criteria for a nonnormal toric variety to be flexible, to be rigid and to be almost rigid. For rigid and almost rigid toric varieties we describe the automorphism group explicitly.

Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks

Abstract: We introduce a global Landau-Ginzburg model which is mirror to several toric Deligne-Mumford stacks and describe the change of the Gromov-Witten theories under discrepant transformations. We prove a formal decomposition of the quantum cohomology D-modules (and of the all-genus Gromov-Witten potentials) under a discrepant toric wall-crossing. In the case of weighted blowups of weak-Fano compact toric stacks along toric centres, we show that an analytic lift of the formal decomposition corresponds, via the $\widehat{\Gamma}$-integral structure, to an Orlov-type semiorthogonal decomposition of topological $K$-groups. We state a conjectural functoriality of Gromov-Witten theories under discrepant transformations in terms of a Riemann-Hilbert problem.

Topology of smoothings of non-isolated singularities of complex surfaces

Abstract: We prove that the boundaries of the Milnor fibers of smoothings of non-isolated reduced complex surface singularities are graph manifolds. Moreover, we give a method, inspired by previous work of Nemethi and Szilard, to compute associated plumbing graphs.

Rigidity of rationally connected smooth projective varieties from dynamical viewpoints

Abstract: Let $X$ be a rationally connected smooth projective variety of dimension $n$. We show that $X$ is a toric variety if and only if $X$ admits an int-amplified endomorphism with totally invariant ramification divisor. We also show that $X\cong (\mathbb{P}^1)^{\times n}$ if and only if $X$ admits a surjective endomorphism $f$ such that the eigenvalues of $f^*|_{\text{N}^1(X)}$ (without counting multiplicities) are $n$ distinct real numbers greater than $1$.

Quantum (Non-commutative) Toric Geometry: Foundations

Abstract: In this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we define their category and show that it is equivalent to a category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantum Toric Varieties. Unlike classical toric varieties, quantum toric varieties admit moduli and we define their moduli spaces, prove that these spaces are orbifolds and, in favorable cases, up to homotopy, they admit a complex structure.