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We provide a relation between the geometric framework for q-Painlevé equations and cluster Poisson varieties by using toric models of rational surfaces associated with q-Painlevé equations. We introduce the notion of seeds of q-Painlevé type by the negative semi-definiteness of symmetric bilinear forms associated with seeds, and classify the mutation equivalence classes of these seeds. This classification coincides with the classification of q-Painlevé equations given by Sakai. We realize q-Painlevé systems as automorphisms on cluster Poisson varieties associated with seeds of q-Painlevé type. [ABSTRACT FROM AUTHOR]

Using the theory of Klyachko filtrations for reflexive sheaves on toric varieties, we give a description of toric foliations and their singularities in terms of combinatorial data. We extend Spicer's results about co-rank one toric foliations to those of any ranks on Q -factorial toric varieties and prove that the toric foliated Minimal Model Program exists. [ABSTRACT FROM AUTHOR]

We develop an algebro-geometric formulation for neural networks in machine learning using the moduli space of framed quiver representations. We find natural Hermitian metrics on the universal bundles over the moduli whose expressions are independent of dimension vector, and show that their Ricci curvatures give a Kähler metric on the moduli. Moreover, we use toric moment maps to construct activation functions and prove the universal approximation theorem for the softmax function (also known as Boltzmann distribution) using toric geometry of the complex projective space. [ABSTRACT FROM AUTHOR]

In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond under mirror symmetry to the eight families of smooth del Pezzo surfaces with very ample anticanonical bundle. [ABSTRACT FROM AUTHOR]

We construct new supersymmetric AdS 2 × M 4 solutions of D = 6 gauged supergravity, where M 4 are certain four-dimensional orbifolds. After uplifting to massive type IIA supergravity these correspond to the near-horizon limit of a system of N D4-branes and N f D8-branes wrapped on M 4 . In one class of solutions, is a spindle fibered over a smooth Riemann surface of genus g > 1 , while in another class is a spindle fibered over another spindle. Both classes can be thought of as orbifold generalizations of Hirzebruch surfaces and, in the second case, we describe the solutions in terms of toric geometry. We show that the entropy associated with these solutions is reproduced by extremizing an entropy function obtained by gluing gravitational blocks, using a general recipe for orbifolds that we propose. We also discuss how our prescription can be used to define an off-shell central charge whose extremization reproduces the gravitational central charge of analogous AdS 3 × M 4 solutions of D = 7 gauged supergravity, arising from wrapping M5-branes on M 4 . [ABSTRACT FROM AUTHOR]

Call a Laurent polynomial W 'complete' if its Newton polytope is full-dimensional with zero in its interior. Suppose W is a Laurent polynomial with coefficients in the positive part of the field of (generalised) Puiseaux series. Here a Puiseaux or generalised Puiseux series (with exponents in R) is called 'positive' if the coefficient of its leading term is in R > 0. We show that W has a unique positive critical point p crit , i.e. all of whose coordinates are positive, if and only if W is complete. For any complete, positive Laurent polynomial W in r variables we also obtain from its positive critical point p crit a canonically associated 'tropical critical point' d crit ∈ R r by considering the valuations of the coordinates of p crit. Moreover we give a finite recursive construction of d crit in terms of a generalisation of the Newton polytope that we call the 'Newton datum' of W. We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also show that our theorem carries over to the case where the exponents of monomials appearing in W are not integral but in R , even though W is then no longer Laurent. Finally, we describe applications to both algebraic and symplectic toric geometry inspired by mirror symmetry. On the one hand, in the algebraic context of a complete toric variety X Σ we apply our results to obtain for any divisor class [ D ] satisfying a certain integrality property, a canonical choice of torus-invariant representative. This generalises the standard toric boundary divisor of X Σ to divisor classes other than the anti-canonical class. On the other hand, our result generalises a result of [11] and relates to the construction of canonical non-displaceable Lagrangian tori for toric symplectic orbifolds using [13,37]. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the structure of the saturation of ideals generated by sparse homogeneous polynomials over a projective toric variety X with respect to the irrelevant ideal of X. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on X. In particular, we prove that toric Sylvester forms yield bases of some graded components of I sat / I , where I denotes an ideal generated by n + 1 generic forms, n is the dimension of X and I sat is the saturation of I with respect to the irrelevant ideal of the Cox ring of X. Then, to illustrate the relevance of toric Sylvester forms we provide three consequences in elimination theory over smooth toric varieties: (1) we introduce a new family of elimination matrices that can be used to solve sparse polynomial systems by means of linear algebra methods, including overdetermined polynomial systems; (2) by incorporating toric Sylvester forms to the classical Koszul complex associated to a polynomial system, we obtain new expressions of the sparse resultant as a determinant of a complex; (3) we explore the computation of the toric residue of the product of two forms. [ABSTRACT FROM AUTHOR]

The Newton--Okounkov body of a big divisor D on a smooth surface is a numerical invariant in the form of a convex polygon. We study the geometric significance of the shape of Newton--Okounkov polygons of ample divisors, showing that they share several important properties of Newton polygons on toric surfaces. In concrete terms, sides of the polygon are associated to some particular irreducible curves, and their lengths are determined by the intersection numbers of these curves with D. As a consequence of our description, we determine the numbers k such that D admits some k-gon as a Newton--Okounkov body, elucidating the relationship of these numbers with the Picard number of the surface, which was first hinted at by work of Küronya, Lozovanu and Maclean. [ABSTRACT FROM AUTHOR]

In previous work, Darvas, George and Smith obtained inequalities between the large scale asymptotic of the J functional with respect to the d1 metric on the space of toric Kähler metrics/rays. In this work we prove sharpness of these inequalities on all toric Kähler manifolds, and study the extremizing potentials/rays. On general Kähler manifolds we show that existence of radial extremizers is equivalent with the existence of plurisupported currents, as introduced and studied by McCleerey. [ABSTRACT FROM AUTHOR]

This paper defines quantum toric varieties associated with an arbitrary fan in a finitely generated subgroup of some R d. This is a generalization of the results of the article [14] of Katzarkov, Lupercio, Meersseman and Verjovsky. [ABSTRACT FROM AUTHOR]

The SYZ conjecture suggests a folklore that "Lagrangian multi-sections are mirror to holomorphic vector bundles". In this paper, we prove this folklore for Lagrangian multi-sections inside the cotangent bundle of a vector space, which are equivariantly mirror to complete toric varieties by the work of Fang-Liu-Treumann-Zaslow. We also introduce the Lagrangian realization problem , which asks whether one can construct an unobstructed Lagrangian multi-section with asymptotic conditions prescribed by a tropical Lagrangian multi-section. We solve the realization problem for tropical Lagrangian multi-sections over a complete 2-dimensional fan that satisfy the so-called N -generic condition. As an application, we show that every rank 2 toric vector bundle on the projective plane is mirror to a Lagrangian multi-section. [ABSTRACT FROM AUTHOR]

We develop a general theory for the existence of extremal Kähler metrics of Poincaré type in the sense of Auvray (J Reine Angew Math 722:1–64, 2017), defined on the complement of a torus invariant divisor of a smooth compact toric variety. In the case when the divisor is smooth, we obtain a list of necessary conditions which must be satisfied for such a metric to exist. Using the explicit methods of Apostolov et al. (Ann Sci Ecole Norm Supp (4) 48:1075–1112, 2015; J Reine Angew Math 721:109–147, 2016, https://doi.org/10.1515/crelle-2014-0060) together with the computational approach of Sektnan (N Y J Math 24:317–354, 2018), we show that on a Hirzebruch complex surface the necessary conditions are also sufficient. In particular, on such a complex surface the complement of the infinity section admits an extremal Kähler metric of Poincaré type, whereas the complement of a fibre fixed by the torus action admits a complete ambitoric extremal Kähler metric which is not of Poincaré type. [ABSTRACT FROM AUTHOR]

In this paper, we present explicit toric construction of moduli space of quasi maps from ℙ 1 with two marked points to ℙ 1 × ℙ 1 , which was first proposed by Jinzenji and prove that it is a compact orbifold. We also determine its Chow ring and compute its Poincaré polynomial for some lower degree cases. [ABSTRACT FROM AUTHOR]

Felsenstein's classical model for Gaussian distributions on a phylogenetic tree is shown to be a toric variety in the space of concentration matrices. We present an exact semialgebraic characterization of this model, and we demonstrate how the toric structure leads to exact methods for maximum likelihood estimation. Our results also give new insights into the geometry of ultrametric matrices. [ABSTRACT FROM AUTHOR]

Given a quasi-compact and quasi-separated (qcqs) scheme X of finite type over a Noetherian ring R having an ample family of line bundles, we construct a closed embedding of X into a smooth (qcqs) scheme over R having an ample family of line bundles. Such a smooth scheme arises as an open subscheme of the multihomogeneous Proj of a Z n -graded ring. [ABSTRACT FROM AUTHOR]

Entropic regularization is a method for large-scale linear programming. Geometrically, one traces intersections of the feasible polytope with scaled toric varieties, starting at the Birch point. We compare this to log-barrier methods, with reciprocal linear spaces, starting at the analytic center. We revisit entropic regularization for unbalanced optimal transport, and we develop the use of optimal conic couplings. We compute the degree of the associated toric variety, and we explore algorithms like iterative scaling. [ABSTRACT FROM AUTHOR]

We show that compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds. [ABSTRACT FROM AUTHOR]

Abstract We carry out the SYZ program for the local Calabi–Yau manifolds of type A ˜ by developing an equivariant SYZ theory for the toric Calabi–Yau manifolds of infinite-type. Mirror geometry is shown to be expressed in terms of the Riemann theta functions and generating functions of open Gromov–Witten invariants, whose modular properties are found and studied in this article. Our work also provides a mathematical justification for a mirror symmetry assertion of the physicists Hollowood–Iqbal–Vafa (Hollowood et al., 2008). [ABSTRACT FROM AUTHOR]

On a given compact complex manifold or orbifold (M, J), we study the existence of Hermitian metrics Q g in the conformal classes of Kähler metrics on (M, J), such that the Ricci tensor of Q g is of type (1, 1) with respect to the complex structure, and the scalar curvature of Q g is constant. In real dimension 4, such Hermitian metrics provide a Riemannian counterpart of the Einstein-Maxwell equations in general relativity, and have been recently studied in [3, 34, 35, 33]. We show how the existence problem of such Hermitian metrics (which we call in any dimension conformally Kähler, Einstein-Maxwell metrics) fits into a formal momentum map interpretation, analogous to results by Donaldson and Fujiki [22, 25] in the constant scalar curvature Kähler case. This leads to a suitable notion of a Futaki invariant which provides an obstruction to the existence of conformally Kähler, Einstein-Maxwell metrics invariant under a certain group of automorphisms which are associated to a given Kähler class, a real holomorphic vector field on (M, J), and a positive normalization constant. Specializing to the toric case, we further define a suitable notion of K-polystability and show it provides a (stronger) necessary condition for the existence of toric, conformally Kähler, Einstein-Maxwell metrics. We use the methods of [4] to show that on a compact symplectic toric 4-orbifold with second Betti number equal to 2, K-polystability is also a sufficient condition for the existence of (toric) conformally Kähler, Einstein-Maxwell metrics, and the latter are explicitly described as ambitoric in the sense of [3]. As an application, we exhibit many new examples of conformally Kähler, Einstein-Maxwell metrics defined on compact 4-orbifolds, and obtain a uniqueness result for the construction in [34]. [ABSTRACT FROM AUTHOR]

Abstract We present a notion of a random toric surface modeled on a notion of a random graph. We then study some threshold phenomena related to the smoothness of the resulting surfaces. [ABSTRACT FROM AUTHOR]

Abstract Fixing a weakly unobstructed Lagrangian torus in a symplectic manifold X , we define a holomorphic function W known as the Floer potential. We construct a canonical A ∞ -functor from the Fukaya category of X to the category of matrix factorizations of W. It provides a unified way to construct matrix factorizations from Lagrangian Floer theory. The technique is applied to toric Fano manifolds to transform Lagrangian branes to matrix factorizations and prove homological mirror symmetry. Using the method, we also obtain an explicit expression of the matrix factorization mirror to the real locus of the complex projective space. [ABSTRACT FROM AUTHOR]

We introduce an invariant of a finite point configuration A⊂R1+n which we denote the cuspidal form of A. We use this invariant to extend Esterov's characterization of dual-defective point configurations to exponential sums; the dual variety associated with A has codimension at least 2 if and only if A does not contain any iterated circuit. [ABSTRACT FROM AUTHOR]

In this paper we study the smallest non-zero eigenvalue λ1 of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for λ1 in terms of moment polytope data. We show that this bound can only be attained for CPn endowed with the Fubini-Study metric and therefore CPn endowed with the Fubini-Study metric is spectrally determined among all toric Kähler metrics. We also study the equivariant counterpart of λ1 which we denote by λ1T. It is the smallest non-zero eigenvalue of the Laplacian restricted to torus-invariant functions. We prove that λ1T is not bounded among toric Kähler metrics thus generalizing a result of Abreu-Freitas on S2. In particular, λ1T and λ1 do not coincide in general. [ABSTRACT FROM AUTHOR]

Using the structure of the jet schemes of rational double point singularities, we construct “minimal embedded toric resolutions” of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every “minimal embedded toric resolution”. We prove that this correspondence is bijective except for the E8 singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities. [ABSTRACT FROM AUTHOR]

We investigate the relationship between stability and the existence of extremal Kähler metrics on certain toric surfaces. In particular, we consider how log stability depends on weights for toric surfaces whose moment polytope is a quadrilateral. For quadrilaterals, we give a computable criterion for stability with 0 weights along two of the edges of the quadrilateral. This in turn implies the existence of a definite log-stable region for quadrilaterals. This uses constructions due to Apostolov-Calderbank-Gauduchon and Legendre. [ABSTRACT FROM AUTHOR]

The author gives an explicit formula on the Ehrhart polynomial of a 3-dimensional simple integral convex polytope by using toric geometry. [ABSTRACT FROM AUTHOR]

Let be a smooth projective toric variety with ample line bundles. Let be the zero locus of generic sections. It is well known that the ambient quantum -module of 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 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 -module. [ABSTRACT FROM AUTHOR]

We study actions of diagonalizable groups on toroidal schemes (i.e. logarithmically regular logarithmic schemes). In particular, we show that for so-called toroidal actions the quotient is again a toroidal scheme. Our main result constructs for an arbitrary action a canonical torification by an equivariant blowings up. This extends earlier results of Abramovich–de Jong, Abramovich–Karu–Matsuki–Włodarczyk, and Gabber in various aspects. [ABSTRACT FROM AUTHOR]

This paper constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth 4-dimensional toric Fano varieties. The tilting bundles lead to a large class of explicit Calabi–Yau-5 algebras, obtained as the corresponding rolled-up helix algebra. A database of the full strong exceptional collections can be found in the package QuiversToricVarieties for the computer algebra system Macaulay2 . [ABSTRACT FROM AUTHOR]

The Behrend function of a C -scheme X is a constructible function ν X : X (C) → Z introduced by Behrend, intrinsic to the scheme structure of X. It is a (subtle) invariant of singularities of X , playing a prominent role in enumerative geometry. To date, only a handful of general properties of the Behrend function are known. In this paper, we compute it for a large class of fat points (schemes supported at a single point). We first observe that, if X ↪ A N is a fat point, ν X is the sum of the multiplicities of the irreducible components of the exceptional divisor E X A N in the blowup Bl X A N. Moreover, we prove that ν X can be computed explicitly through the normalisation of Bl X A N. The proofs of our explicit formulas for the Behrend function of a fat point in A 2 rely heavily on toric geometry techniques. Along the way, we find a formula for the number of irreducible components of E X A 2 , where X ↪ A 2 is a fat point such that Bl X A 2 is normal. [ABSTRACT FROM AUTHOR]

We show that any compact convex simple lattice polytope is the moment polytope of a Kähler-Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang-Zhu Theorem (Wang and Zhu in Adv Math 188:47-103, ) giving the existence of a Kähler-Ricci soliton on any toric monotone manifold on any compact convex simple labeled polytope satisfying the combinatoric condition corresponding to monotonicity. We obtain that any compact convex simple polytope [InlineEquation not available: see fulltext.] admits a set of inward normals, unique up to dilatation, such that there exists a symplectic potential satisfying the Guillemin boundary condition (with respect to these normals) and the Kähler-Einstein equation on [InlineEquation not available: see fulltext.]. We interpret our result in terms of existence of singular Kähler-Einstein metrics on toric manifolds. [ABSTRACT FROM AUTHOR]

Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of PDEs. We explain these conjectures and verify them in some cases. [ABSTRACT FROM AUTHOR]