Showing papers on "Toric variety published in 2022"
TL;DR: In this paper , a new interpretation of the Sottile-Sturmfels construction is proposed, based on the notion of poset polytopes introduced by Fang-Fourier-Litza-Pegel.
Abstract: Abstract We study toric degenerations of semi-infinite Grassmannians (a.k.a. quantum Grassmannians). While the toric degenerations of the classical Grassmannians are well studied, the only known example in the semi-infinite case is due to Sottile and Sturmfels. We start by providing a new interpretation of the Sottile–Sturmfels construction by finding a poset such that their degeneration is the toric variety of the order polytope of the poset. We then use our poset to construct and study a new toric degeneration in the semi-infinite case. Our construction is based on the notion of poset polytopes introduced by Fang–Fourier–Litza–Pegel. As an application, we introduce semi-infinite PBW-semistandard tableaux, giving a basis in the homogeneous coordinate ring of a semi-infinite Grassmannian.
6 citations
TL;DR: In this paper , the authors compare entropic regularization with log-barrier methods, with reciprocal linear spaces, starting at the analytic center, and develop the use of optimal conic couplings.
3 citations
TL;DR: In this article , the authors studied combinatorics of string polytopes when [Formula: see text], and presented a sufficient condition on the string polytope such that the toric variety of the stringpolytope has a small toric resolution.
Abstract: Let G be a semisimple algebraic group over [Formula: see text]. For a reduced word [Formula: see text] of the longest element in the Weyl group of G and a dominant integral weight [Formula: see text], one can construct the string polytope [Formula: see text], whose lattice points encode the character of the irreducible representation [Formula: see text]. The string polytope [Formula: see text] is singular in general and combinatorics of string polytopes heavily depends on the choice of [Formula: see text]. In this paper, we study combinatorics of string polytopes when [Formula: see text], and present a sufficient condition on [Formula: see text] such that the toric variety [Formula: see text] of the string polytope [Formula: see text] has a small toric resolution. Indeed, when [Formula: see text] has small indices and [Formula: see text] is regular, we explicitly construct a small toric resolution of the toric variety [Formula: see text] using a Bott manifold. Our main theorem implies that a toric variety of any string polytope admits a small toric resolution when [Formula: see text]. As a byproduct, we show that if [Formula: see text] has small indices then [Formula: see text] is integral for any dominant integral weight [Formula: see text], which in particular implies that the anticanonical limit toric variety [Formula: see text] of a partial flag variety [Formula: see text] is Gorenstein Fano. Furthermore, we apply our result to symplectic topology of the full flag manifold [Formula: see text] and obtain a formula of the disk potential of the Lagrangian torus fibration on [Formula: see text] obtained from a flat toric degeneration of [Formula: see text] to the toric variety [Formula: see text].
3 citations
TL;DR: In this paper , a uniqueness criterion for additive action on a complete toric variety is given, where the additive action is defined as a regular effective action Gan×X→X with an open orbit of the commutative unipotent group Gan.
3 citations
TL;DR: In this article, it was shown that negative curves are rational in many cases and that the Cox ring of the blow-up of a toric variety at the point (1, 1, 1, …, 1 ) coincides with the extended symbolic Rees ring of an ideal of a polynomial ring.
2 citations
TL;DR: In this paper , it was shown that the class of staged tree models with a toric structure extends far outside of the balanced case, if a change of coordinates is allowed, and that it is an open problem whether all staged trees models have toric structures.
2 citations
01 Jan 2022
TL;DR: In this article , Harada and Kaveh described how one incorporates a symplectic structure into the degeneration process, providing a very useful tool for solving certain problems in symplectic geometry.
Abstract: Toric degeneration in algebraic geometry is a process of degenerating a given projective variety into a toric one. Then one can obtain information about the original variety via analyzing the toric one, which is a much easier object to study. Harada and Kaveh described how one incorporates a symplectic structure into this process, providing a very useful tool for solving certain problems in symplectic geometry. Below we present two applications of this method: questions about the Gromov width, and cohomological rigidity problems.
2 citations
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TL;DR: In this paper , the authors illustrate toric quasifolds and their atlases by describing some notable examples, and conclude with a number of considerations, including the need to consider convex polytopes that are not rational.
Abstract: Toric quasifolds are highly singular spaces that were first introduced in order to address, from the symplectic viewpoint, the longstanding open problem of extending the classical constructions of toric geometry to those simple convex polytopes that are not rational. We illustrate toric quasifolds, and their atlases, by describing some notable examples. We conclude with a number of considerations.
2 citations
TL;DR: In this article , the authors identify a moduli space of real degenerations with the secondary polytope of a projective toric variety associated to the secondary fan of the integer points in the polytopes corresponding to the toric varieties.
Abstract: The space of torus translations and degenerations of a projective toric variety forms a toric variety associated to the secondary fan of the integer points in the polytope corresponding to the toric variety. This is used to identify a moduli space of real degenerations with the secondary polytope. A configuration $${{\mathcal {A}}}$$ of real vectors gives an irrational projective toric variety in a simplex. We identify a space of translations and degenerations of the irrational projective toric variety with the secondary polytope of $${{\mathcal {A}}}$$ . For this, we develop a theory of irrational toric varieties associated to arbitrary fans. When the fan is rational, the irrational toric variety is the nonnegative part of the corresponding classical toric variety. When the fan is the normal fan of a polytope, the irrational toric variety is homeomorphic to that polytope.
2 citations
2 citations
30 Jun 2022
TL;DR: In this paper , the mirror symmetry for smooth toric stacks was shown to hold for coherent sheaves on a toric stack, and a clean new proof of mirror symmetry was given for the smooth torric stacks.
Abstract: The Cox construction presents a toric variety as a quotient of affine space by a torus. The category of coherent sheaves on the corresponding stack thus has an evident description as invariants in a quotient of the category of modules over a polynomial ring. Here we give the mirror to this description, and in particular, a clean new proof of mirror symmetry for smooth toric stacks.
TL;DR: In this paper , an extension of the log-local principle holds for X $X$ a (not necessarily smooth) Q $\mathbb {Q}$ -factorial projective toric variety, D $D$ the toric boundary, and descendant point insertions.
Abstract: Let X $X$ be a smooth projective complex variety and let D = D 1 + ⋯ + D l $D=D_1+\cdots +D_l$ be a reduced normal crossing divisor on X $X$ with each component D j $D_j$ smooth, irreducible and numerically effective. The log–local principle put forward in van Garrel et al. (Adv. Math. 350 (2019) 860–876) conjectures that the genus 0 log Gromov–Witten theory of maximal tangency of ( X , D ) $(X,D)$ is equivalent to the genus 0 local Gromov–Witten theory of X $X$ twisted by ⨁ j = 1 l O ( − D j ) $\bigoplus _{j=1}^{l}\mathcal {O}(-D_j)$ . We prove that an extension of the log–local principle holds for X $X$ a (not necessarily smooth) Q $\mathbb {Q}$ -factorial projective toric variety, D $D$ the toric boundary, and descendant point insertions.
TL;DR: In this article , the cylindrical contact homology invariants of a Q-Gorenstein toric contact manifold are related to the Ehrhart (quasi-)polynomial of its toric diagram.
TL;DR: In this paper , it was shown that it is possible to find a toric hypersurface Calabi-Yau in which there are more than h 1,1 (Y 4 ) nonperturbative superpotential terms with trivial intermediate Jacobian.
Abstract: A bstract In this note, we prove combinatorial formulas for the Hodge number h 2 , 1 of prime toric divisors in an arbitrary toric hypersurface Calabi-Yau fourfold Y 4 . We show that it is possible to find a toric hypersurface Calabi-Yau in which there are more than h 1,1 ( Y 4 ) non-perturbative superpotential terms with trivial intermediate Jacobian. Hodge numbers of divisors in toric complete intersection Calabi-Yaus are the subjects of the sequel.
TL;DR: In this paper , an algorithm relying on elimination theory for finding generators of the vanishing ideal of a subgroup parameterized by a matrix was proposed, which can be used to study algebraic geometric codes arising from the subgroup.
Abstract: Let $K$ be a finite field and $X$ be a complete simplicial toric variety over $K$ with split torus $T_X\cong (K^*)^n$. We give an algorithm relying on elimination theory for finding generators of the vanishing ideal of a subgroup $Y_Q$ parameterized by a matrix $Q$ which can be used to study algebraic geometric codes arising from $Y_Q$. We give a method to compute the lattice $L$ whose ideal $I_L$ is exactly $I(Y_Q)$ under a mild condition. As applications, we give precise descriptions for the lattices corresponding to some special subgroups. We also prove a Nullstellensatz type theorem valid over finite fields, and share Macaulay2* codes for our algorithms.
TL;DR: In this article , the authors study families of polynomial dynamical systems inspired by biochemical reaction networks, which have also been called toric, and propose algorithms to detect the disguised toric locus.
TL;DR: In this paper , the authors studied log Calabi-Yau varieties obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary and showed 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.
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.
TL;DR: In this paper , the authors illustrate an extension of polar duality between Fano toric varieties to a more general duality called ''framed'' duality, so giving rise to a powerful and unified method of producing mirror partners of hypersurfaces and complete intersections in toric variety, of any Kodaira dimension.
Abstract: The present paper is dedicated to illustrating an extension of polar duality between Fano toric varieties to a more general duality, called \emph{framed} duality, so giving rise to a powerful and unified method of producing mirror partners of hypersurfaces and complete intersections in toric varieties, of any Kodaira dimension. In particular, the class of projective hypersurfaces and their mirror partners are studied in detail. Moreover, many connections with known Landau-Ginzburg mirror models, Homological Mirror Symmetry and Intrinsic Mirror Symmetry, are discussed.
TL;DR: In this paper , the notion of integral piecewise linear maps from a fan to a toric principal G-bundle was introduced, and the class of isomorphism classes of (framed) toric vector bundles on these maps were recovered.
Abstract: We define the notion of a piecewise linear map from a fan $$\Sigma $$ to $$\tilde{\mathfrak {B}}(G)$$ , the cone over the Tits building of a linear algebraic group G. Let $$X_\Sigma $$ be a toric variety with fan $$\Sigma $$ . We show that when G is reductive the set of integral piecewise linear maps from $$\Sigma $$ to $$\tilde{\mathfrak {B}}(G)$$ classifies the isomorphism classes of (framed) toric principal G-bundles on $$X_\Sigma $$ . This in particular recovers Klyachko’s classification of toric vector bundles, and gives new classification results for the orthogonal and symplectic toric principal bundles.
10 Jun 2022
TL;DR: In this article , a mixed derived category of constructible complexes (in the sense of [BGS96]) for any toric variety associated to a fan is presented, whose heart is a mixed version of the category of perverse sheaves.
Abstract: In [BGS96], Beilinson, Ginzburg, and Soergel introduced the notion of mixed categories. This idea often underlies many interesting "Koszul dualities." In this paper, we produce a mixed derived category of constructible complexes (in the sense of [BGS96]) for any toric variety associated to a fan. Furthermore, we show that it comes equipped with a t-structure whose heart is a mixed version of the category of perverse sheaves. In chapters 2 and 3, we provide the necessary background. Chapter 2 concerns the categorical preliminaries, while chapter 3 gives the background geometry. This concerns both some basics of toric varieties as well as basics of constructible sheaves in this setting. In chapter 4, we introduce the primary category of interest, Dmix(X0) for a toric variety X0 defined over some finite field. We prove that this is a mixed version of Dbc(X), the bounded derived category of constructible complexes over X = X0 xSpec(Fq) Spec(F q), the variety obtained by extension of scalars. In chapter 5, we introduce the standard suite of functors associated to a locally closed inclusion of toric varieties, h : Y0 → X0, between the mixed categories Dmix(X0) and Dmix(Y0). We provide some functors associated to other special types of toric maps as well. Finally, we prove that some of these functors commute with the realization functor r : Dmix(X0) → DbT,m(X0). We call this being genuine.
21 Dec 2022
TL;DR: In this paper , the authors generalize Laurent monomials to toric quasifolds, a special class of highly singular spaces that extend simplicial toric varieties to the nonrational setting.
Abstract: We generalize Laurent monomials to toric quasifolds, a special class of highly singular spaces that extend simplicial toric varieties to the nonrational setting.
TL;DR: In this paper , a family of fully faithful functors from the category of torus equivariant reflexive sheaves on polarized toric orbifolds was defined, and it was shown that slope stability is preserved by these functors if and only if the pair $((X,\,L),\,G)$ satisfies a combinatorial criterion.
Abstract: For $(X,\,L)$ a polarized toric variety and $G\subset \mathrm {Aut}(X,\,L)$ a torus, denote by $Y$ the GIT quotient $X/\!\!/G$ . We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on $Y$ to the category of torus equivariant reflexive sheaves on $X$ . We show, under a genericity assumption on $G$ , that slope stability is preserved by these functors if and only if the pair $((X,\,L),\,G)$ satisfies a combinatorial criterion. As an application, when $(X,\,L)$ is a polarized toric orbifold of dimension $n$ , we relate stable equivariant reflexive sheaves on certain $(n-1)$ -dimensional weighted projective spaces to stable equivariant reflexive sheaves on $(X,\,L)$ .
22 Sep 2022
TL;DR: In this article , the authors investigated 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 $X$.
Abstract: 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 $I^{\text{sat}}/I$, where $I$ denotes an ideal generated by $n+1$ generic forms, $n$ is the dimension of $X$ and $I^{\text{sat}}$ 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: (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 give a new formula for computing toric residues of the product of two forms.
Journal Article•
07 Jan 2022TL;DR: In this article , the authors define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on a torus of automorphisms of (X,L) to the category on X, and show that slope stability is preserved by these functors if and only if the pair ((X, L), G) satisfies a combinatorial criterion.
Abstract: For (X,L) a polarized toric variety and G a torus of automorphisms of (X,L), denote by Y the GIT quotient X/G. We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on Y to the category of torus equivariant reflexive sheaves on X. We show, under a genericity assumption on G, that slope stability is preserved by these functors if and only if the pair ((X, L), G) satisfies a combinatorial criterion. As an application, when (X,L) is a polarized toric orbifold of dimension n, we relate stable equivariant reflexive sheaves on (X, L) to stable equivariant reflexive sheaves on certain (n-1)-dimensional weighted projective spaces.
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TL;DR: Toric Kato manifolds as mentioned in this paper generalize Tsuchihashi and Oda's manifold constructions to complex dimension 2 and obtain the properly blown-up Inoue surfaces.
Abstract: We introduce and study a special class of Kato manifolds, which we call toric Kato manifolds. Their construction stems from toric geometry, as their universal covers are open subsets of toric algebraic varieties of non-finite type. This generalizes previous constructions of Tsuchihashi and Oda, and in complex dimension 2, retrieves the properly blown-up Inoue surfaces. We study the topological and analytical properties of toric Kato manifolds and link certain invariants to natural combinatorial data coming from the toric construction. Moreover, we produce families of flat degenerations of any toric Kato manifold, which serve as an essential tool in computing their Hodge numbers. In the last part, we study the Hermitian geometry of Kato manifolds. We give a characterization result for the existence of locally conformally Kähler metrics on any Kato manifold. Finally, we prove that no Kato manifold carries balanced metrics and that a large class of toric Kato manifolds of complex dimension ≥3 do not support pluriclosed metrics.
26 Apr 2022
TL;DR: In this paper , the authors generalized the canonical measures from canonical metrics to toric metrics and from discrete valuations to arbitrary non-archimedean fields, and computed the associated Monge-Amp\`ere measures for the restriction to any closed subvariety of the abelian variety.
Abstract: Toric metrics on a line bundle of an abelian variety $A$ are the invariant metrics under the natural torus action coming from Raynaud's uniformization theory. We compute here the associated Monge-Amp\`ere measures for the restriction to any closed subvariety of $A$. This generalizes the computation of canonical measures done by the first author from canonical metrics to toric metrics and from discrete valuations to arbitrary non-archimedean fields.
TL;DR: In this paper , it was shown that the iteration of normalized Nash blowups desingularizes normal toric surfaces and that the Nash blowup of a toric variety of arbitrary dimension in prime characteristic can be described combinatorially.
Abstract: Abstract We initiate the study of the resolution of singularities properties of Nash blowups over fields of prime characteristic. We prove that the iteration of normalized Nash blowups desingularizes normal toric surfaces. We also introduce a prime characteristic version of the logarithmic Jacobian ideal of a toric variety and prove that its blowup coincides with the Nash blowup of the variety. As a consequence, the Nash blowup of a, not necessarily normal, toric variety of arbitrary dimension in prime characteristic can be described combinatorially.
06 Mar 2022
TL;DR: In this paper , a simple and fast embedded resolution of varieties and principalization of ideals in the language of torus actions on ambient smooth schemes with or without SNC divisors is presented.
Abstract: We show a simple and fast embedded resolution of varieties and principalization of ideals in the language of torus actions on ambient smooth schemes with or without SNC divisors. The canonical functorial resolution of varieties in characteristic zero is given by, the introduced here, operations of cobordant blow-ups with smooth weighted centers. The centers are defined by the geometric invariant measuring the singularities on smooth schemes with SNC divisors. As the result of the procedure we obtain a smooth variety with a torus action and the exceptional divisor having simple normal crossings. Moreover, its geometric quotient is birational to the resolved variety, has abelian quotient singularities, and can be desingularized directly by combinatorial methods. The paper is based upon the ideas of the joint work with Abramovich and Temkin and a similar result by McQuillan on resolution in characteristic zero via stack-theoretic weighted blow-ups. As an application of the method we show the resolution of a certain class of isolated singularities in positive and mixed characteristic.
07 Dec 2022
TL;DR: In this paper , an equivariant version of the non-archimedean Arakelov theory of [BGS95] in the case of toric varieties was developed.
Abstract: We develop an equivariant version of the non-archimedean Arakelov theory of [BGS95] in the case of toric varieties. We define the equivariant analogues of the non-archimedean differential forms and currents appearing in loc. cit. and relate them to piecewise polynomial functions on the polyhedral complexes defining the toric models. In particular, we give combinatorial characterizations of the Green currents associated to invariant cycles and combinatorial descriptions of the arithmetic Chow groups.