Smooth Projective Toric Variety Research Articles

Computations of Gromov–Witten invariants of toric varieties

We present the Julia package ToricAtiyahBott.jl, providing an easy way to perform the Atiyah–Bott formula on the moduli space of genus 0 stable maps M‾0,m(X,β) where X is any smooth projective toric variety, and β is any effective 1-cycle. The list of the supported cohomological cycles contains the most common ones, and it is extensible. We provide a detailed explanation of the algorithm together with many examples and applications. The toric variety X, as well as the cohomology class β, must be defined using the package Oscar.jl.

Cohomology Rings of Toric Bundles and the Ring of Conditions

AbstractThe celebrated BKK Theorem expresses the number of roots of a system of generic Laurent polynomials in terms of the mixed volume of the corresponding system of Newton polytopes. In Pukhlikov and Khovanskiĭ (Algebra i Analiz 4(4):188–216, 1992), Pukhlikov and the second author noticed that the cohomology ring of smooth projective toric varieties over $${\mathbb {C}}$$ C can be computed via the BKK Theorem. This complemented the known descriptions of the cohomology ring of toric varieties, like the one in terms of Stanley–Reisner algebras. In Sankaran and Uma (Comment Math Helv 78(3):540–554, 2003), Sankaran and Uma generalized the “Stanley–Reisner description” to the case of toric bundles, i.e., equivariant compactifications of (not necessarily algebraic) torus principal bundles. We provide a description of the cohomology ring of toric bundles which is based on a generalization of the BKK Theorem, and thus extends the approach by Pukhlikov and the second author. Indeed, for every cohomology class of the base of the toric bundle, we obtain a BKK-type theorem. Furthermore, our proof relies on a description of graded-commutative algebras which satisfy Poincaré duality. From this computation of the cohomology ring of toric bundles, we obtain a description of the ring of conditions of horospherical homogeneous spaces as well as a version of Brion–Kazarnovskii theorem for them. We conclude the manuscript with a number of examples. In particular, we apply our results to toric bundles over a full flag variety G/B. The description that we get generalizes the corresponding description of the cohomology ring of toric varieties as well as the one of full flag varieties G/B previously obtained by Kaveh (J Lie Theory 21(2):263–283, 2011).

Curve classes on Calabi–Yau complete intersections in toric varieties

AbstractWe prove the integral Hodge conjecture for curve classes on smooth varieties of dimension at least three constructed as a complete intersection of ample hypersurfaces in a smooth projective toric variety, such that the anticanonical divisor is the restriction of a nef divisor. In particular, this includes the case of smooth anticanonical hypersurfaces in toric Fano varieties. In fact, using results of Casagrande and the toric minimal model program, we prove that in each case, is generated by classes of rational curves.

What makes a complex a virtual resolution?

Virtual resolutions are homological representations of finitely generated Pic ( X ) \text {Pic}(X) -graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex of graded free modules over the Cox ring is a virtual resolution. We then turn our attention to the saturation of Fitting ideals by the irrelevant ideal of the Cox ring and prove some results that mirror the classical theory of Fitting ideals for Noetherian rings.

Homological and combinatorial aspects of virtually Cohen–Macaulay sheaves

When studying a graded module $M$ over the Cox ring of a smooth projective toric variety $X$, there are two standard types of resolutions commonly used to glean information: free resolutions of $M$ and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study. The theory of virtual resolutions includes a notion of a virtually Cohen--Macaulay property, though tools for assessing which modules are virtually Cohen--Macaulay have only recently started to be developed. In this paper, we continue this research program in two related ways. The first is that, when $X$ is a product of projective spaces, we produce a large new class of virtually Cohen--Macaulay Stanley--Reisner rings, which we show to be virtually Cohen--Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety $X$, we develop homological tools for assessing the virtual Cohen--Macaulay property. Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. We also use these tools to establish relationships among the arithmetically, geometrically, and virtually Cohen--Macaulay properties.

A construction of SKT manifolds using toric geometry

We produce infinite families of SKT manifolds by using methods of toric geometry like the J-construction. These SKT manifolds are total spaces of certain principal G-bundles over smooth projective toric varieties, where G is an even dimensional compact connected Lie group.

Virtual resolutions for a product of projective spaces

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.

Fibration and classification of smooth projective toric varieties of low Picard number

In this paper, we show that a smooth toric variety [Formula: see text] of Picard number [Formula: see text] always admits a nef primitive collection supported on a hyperplane admitting non-trivial intersection with the cone [Formula: see text] of numerically effective divisors and cutting a facet of the pseudo-effective cone [Formula: see text], that is [Formula: see text]. In particular, this means that [Formula: see text] admits non-trivial and non-big numerically effective divisors. Geometrically, this guarantees the existence of a fiber type contraction morphism over a smooth toric variety of dimension and Picard number lower than those of [Formula: see text], so giving rise to a classification of smooth and complete toric varieties with [Formula: see text]. Moreover, we revise and improve results of Oda–Miyake by exhibiting an extension of the above result to projective, toric, varieties of dimension [Formula: see text] and Picard number [Formula: see text], allowing us to classifying all these threefolds. We then improve results of Fujino–Sato, by presenting sharp (counter)examples of smooth, projective, toric varieties of any dimension [Formula: see text] and Picard number [Formula: see text] whose non-trivial nef divisors are big, that is [Formula: see text]. Producing those examples represents an important goal of computational techniques in definitely setting an open geometric problem. In particular, for [Formula: see text], the given example turns out to be a weak Fano toric fourfold of Picard number 4.

Moment maps, strict linear precision, and maximum likelihood degree one

We study the moment maps of a smooth projective toric variety. In particular, we characterize when the moment map coming from the quotient construction is equal to a weighted Fubini-Study moment map. This leads to an investigation into polytopes with strict linear precision, and in the process we use results from and find remarkable connections between Symplectic Geometry, Geometric Modeling, Algebraic Statistics, and Algebraic Geometry.

Vanishing for Hodge ideals on toric varieties

AbstractIn this article we construct a Koszul‐type resolution of the exterior power of the sheaf of holomorphic differential forms on smooth toric varieties and use this to prove a Nadel‐type vanishing theorem for Hodge ideals associated to effective ‐divisors on smooth projective toric varieties. This extends earlier results of Mustaţă and Popa.

On derived categories of arithmetic toric varieties

We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskii and Klyachko, and toric varieties associated to Weyl fans of type $A$. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly non-toric) varieties over non-closed fields.

Segre class computation and practical applications

Let $X \subset Y$ be closed (possibly singular) subschemes of a smooth projective toric variety $T$. We show how to compute the Segre class $s(X,Y)$ as a class in the Chow group of $T$. Building on this, we give effective methods to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of $T$. Our methods may be implemented without using Groebner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used.

Cohomology of torus manifold bundles

Abstract Let X be a 2n-dimensional torus manifold with a locally standard T ≅ (S 1) n action whose orbit space is a homology polytope. Smooth complete complex toric varieties and quasitoric manifolds are examples of torus manifolds. Consider a principal T-bundle p : E → B and let π : E(X) → B be the associated torus manifold bundle. We give a presentation of the singular cohomology ring of E(X) as a H *(B)-algebra and the topological K-ring of E(X) as a K *(B)-algebra with generators and relations. These generalize the results in [17] and [19] when the base B = pt. These also extend the results in [20], obtained in the case of a smooth projective toric variety, to any smooth complete toric variety.

Twisted polytope sheaves and coherent–constructible correspondence for toric varieties

Given a smooth projective toric variety $$X_\Sigma $$ of complex dimension n, Fang–Liu–Treumann–Zaslow (Invent Math 186(1):79–114, 2011) showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves $$Coh(X_\Sigma )$$ into the dg derived category of constructible sheaves on a torus $$Sh(T^n, \Lambda _\Sigma )$$ . Recently, Kuwagaki (The nonequivariant coherent-constructible correspondence for toric stacks, 2016. arXiv:1610.03214 ) proved that the quasi-embedding is a quasi-equivalence, and generalized the result to toric stacks. Here we give a different proof in the smooth projective case, using non-characteristic deformation of sheaves to find twisted polytope sheaves that co-represent the stalk functors.

On Mori cone of Bott towers

A Bott tower of height r is a sequence of projective bundlesXr⟶πrXr−1⟶πr−1⋯⟶π2X1=P1⟶π1X0={pt}, where Xi=P(OXi−1⊕Li−1) for a line bundle Li−1 over Xi−1 for all 1≤i≤r and P(−) denotes the projectivization. These are smooth projective toric varieties and we refer to the top object Xr also as a Bott tower. In this article, we study the Mori cone and numerically effective (nef) cone of Bott towers, and we classify Fano, weak Fano and log Fano Bott towers. We prove some vanishing theorems for the cohomology of tangent bundle of Bott towers.

Quantum 𝒟-modules for toric nef complete intersections

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

Computing characteristic classes of subschemes of smooth toric varieties

Let XΣ be a smooth complete toric variety defined by a fan Σ and let V=V(I) be a subscheme of XΣ defined by an ideal I homogeneous with respect to the grading on the total coordinate ring of XΣ. We show a new expression for the Segre class s(V,XΣ) in terms of the projective degrees of a rational map specified by the generators of I when each generator corresponds to a numerically effective (nef) divisor. Restricting to the case where XΣ is a smooth projective toric variety and dehomogenizing the total homogeneous coordinate ring of XΣ via a dehomogenizing ideal we also give an expression for the projective degrees of this rational map in terms of the dimension of an explicit quotient ring. Under an additional technical assumption we construct what we call a general dehomogenizing ideal and apply this construction to give effective algorithms to compute the Segre class s(V,XΣ), the Chern–Schwartz–MacPherson class cSM(V) and the topological Euler characteristic χ(V) of V. These algorithms can, in particular, be used for subschemes of any product of projective spaces Pn1×⋯×Pnj or for subschemes of many other projective toric varieties. Running time bounds for several of the algorithms are given and the algorithms are tested on a variety of examples. In all applicable cases our algorithms to compute these characteristic classes are found to offer significantly increased performance over other known algorithms.

Projective toric polynomial generators in the unitary cobordism ring

According to Milnor and Novikov's classical result, the unitary cobordism ring is isomorphic to a graded polynomial ring with countably many generators: , . In this paper we solve the well-known problem of constructing geometric representatives for the among smooth projective toric varieties, , . Our proof uses a family of equivariant modifications (birational isomorphisms) of an arbitrary complex manifold of complex dimension (, ). The key fact is that the change of the Milnor number under these modifications depends only on the dimension and the number and does not depend on the manifold itself. Bibliography: 22 titles.

Toric polynomial generators of complex cobordism

Although it is well-known that the complex cobordism ring is a polynomial ring $\Omega_{*}^{U}\cong\mathbb{Z}\left[\alpha_{1},\alpha_{2},\ldots\right]$, an explicit description for convenient generators $\alpha_{1},\alpha_{2},\ldots$ has proven to be quite elusive. The focus of the following is to construct complex cobordism polynomial generators in many dimensions using smooth projective toric varieties. These generators are very convenient objects since they are smooth connected algebraic varieties with an underlying combinatorial structure that aids in various computations. By applying certain torus-equivariant blow-ups to a special class of smooth projective toric varieties, such generators can be constructed in every complex dimension that is odd or one less than a prime power. A large amount of evidence suggests that smooth projective toric varieties can serve as polynomial generators in the remaining dimensions as well.

Equivariant principal bundles and logarithmic connections on toric varieties

Let $M$ be a smooth complex projective toric variety equipped with an action of a torus $T$, such that the complement $D$ of the open $T$--orbit in $M$ is a simple normal crossing divisor. Let $G$ be a complex reductive affine algebraic group. We prove that an algebraic principal $G$--bundle $E_G\to M$ admits a $T$--equivariant structure if and only if $E_G$ admits a logarithmic connection singular over $D$. If $E_H\to M$ is a $T$-equivariant algebraic principal $H$--bundle, where $H$ is any complex affine algebraic group, then $E_H$ in fact has a canonical integrable logarithmic connection singular over $D$.