P. Sankaran, V. Uma
We describe the singular cohomology ring, the K-ring of complex vector bundles, the Chow ring, and the Grothendieck ring of coherent sheaves of the total space of the fibre bundle with base space an irreducible nonsingular complete Noetherian scheme and fibre a nonsingular projective T-toric variety associated to a prinicipal T-bundle over the field of complex numbers.
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We compute the cohomology rings of smooth real toric varieties and of real toric spaces, which are quotients of real moment-angle complexes by freely acting subgroups of the ambient 2-torus. The differential graded algebra we present is in fact an equivariant dga model, valid for arbitrary coefficients. We deduce from our description that smooth toric varieties are M-varieties.
Topologically, compact toric varieties can be constructed as identification spaces: they are quotients of the product of a compact torus and the order complex of the fan. We give a detailed proof of this fact, extend it to the non-compact case and draw several, mostly cohomological conclusions. In particular, we show that the equivariant integral cohomology of a toric variety can be described in terms of piecewise polynomials on the fan if the ordinary integral cohomology is concentrated in even degrees. This generalizes a result of Bahri-Franz-Ray to the non-compact case. We also investigate …
Cohomology of smooth toric varieties: naturality
本文研究光滑环面簇及矩角复形部分商上同调与 Stanley-Reisner 环挠积同构的自然性，证明当 2 可逆时同构关于环面态射自然，并显式描述一般情形下的形变项。
Let $X_Σ$ be a smooth, not necessarily compact toric variety. We show that a certain complex, defined in terms of the fan $Σ$, computes the integral cohomology of $X_Σ$, including the module structure over the homology of the torus. In some cases we can also give the product. As a corollary we obtain that the cycle map from Chow groups to integral Borel-Moore homology is split injective for smooth toric varieties. Another result is that the differential algebra of singular cochains on the Borel construction of $X_Σ$ is formal.
We describe the quantum cohomology ring of a toric Fano variety $X$ in terms of the usual topological cohomology ring for an auxiliary infinite-dimensional scheme. This scheme is a part of an algebro-geometric model for the universal cover of the space of free loops with values in $X$.
In this article we study the equivariant elliptic cohomology of complex toric varieties. We prove a partial reconstruction theorem showing that equivariant elliptic cohomology encodes considerable non-trivial information on the equivariant 1-skeleton of a toric variety X (although it stops short of being a complete invariant of its GKM graphs). Elliptic cohomology is supposed to encode higher categorical geometric data, and proposals have been made linking elliptic cocycles to categorified bundles. In particular, contrary to ordinary cohomology and K-theory, elliptic cohomology is expected not…
Cohomology Rings of Toric Wonderful Model
本文利用偏序集爆破、环上Gröbner基和容许函数，刻画了任意建筑集（含非良连通情形）的环面奇妙模型的上同调环。
The simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and more generally, polyhedral products. In this paper we extend the analysis to include toric orbifolds. Our main results yield infinite families of toric orbifolds, derived from a given one, whose integral cohomology is free of torsion and is concentrated in even degrees, a property which might be termed \emph{integrally equivariantly formal}. In all cases, it …
In this paper we construct a spectral sequence computing a modified version of morphic cohomology of a toric variety (even when it is singular) in terms of combinatorial data coming from the fan of the toric variety.
We study the Fulton-Macpherson Chow cohomology of affine toric varieties. In particular, we prove that the Chow cohomology vanishes in positive degree. We prove an analogous result for the operational $K$-theory defined by Anderson and Payne.
We introduce the notion of a \emph{conic sequence} of a convex polytope. It is a way of building up a polytope starting from a vertex and attaching faces one by one with certain regulations. We apply this to a toric variety to obtain an iterated cofibration structure on it. This allows us to prove several vanishing results in the rational cohomology of a toric variety and to calculate Poincaré polynomials for a large class of singular toric varieties.
We prove that the cohomology of the moduli space of morphisms of a fixed finite degree from a smooth projective curve $C$ of genus $g$ to a complete simplicial toric variety $\mathbb{P}(Σ)$, denoted by the rational polyhedral fan $Σ$, stabilizes. As an arithmetic consequence we obtain a resolution of the Batyrev-Manin conjecture for toric varieties over global function fields in all but finitely many characteristics.
We explain a method for calculating the cohomology of line bundles on a toric variety in terms of the cohomology of certain constructible sheaves on the polytope. We show its effective use by means of some examples.
Some projective wonderful models for the complement of a toric arrangement in a n-dimensional algebraic torus T were constructed in [3]. In this paper we describe their integer cohomology rings by generators and relations.
We give several mild conditions on a toric bundle on a nonsingular toric variety under which the projectivization of the toric bundle is Frobenius split.
On the integral cohomology of real toric manifolds
本文研究实 toric 流形的整数上同调，通过组合数据推导出上同调群公式，并提出环结构的组合公式，进一步给出判断 spin^c 流形的准则。
We show that the moduli space of rank three toric vector bundles on smooth projective toric varieties satisfies Murphy's law, that is, every singularity type of finite type arises somewhere on this space.
We give a criterion for a projectivized toric vector bundle to be a Mori dream space and describe its Cox ring using generators and relations. Both of these results are in terms of the matroids of all symmetric powers of the bundle. We also give a criterion for a toric vector bundle to be big and describe several interesting examples of toric vector bundles which highlights how positivity properties for toric vector bundles are more complicated than for toric line bundles.
This article uses homological methods for evaluating compactly supported cohomology groups of noncompact toric surfaces
We explicitly describe cohomology of the sheaf of differential forms with poles along a semiample divisor on a complete simplicial toric variety. As an application, we obtain a new vanishing theorem which is an analogue of the Bott-Steenbrink-Danilov vanishing theorem.
We describe the weight filtration in the cohomology of toric varieties. We present a role of the Frobenius automorphism in an elementary way. We prove that equivariant intersection homology of an arbitrary toric variety is pure. We obtain results concerning Koszul duality: nonequivariant intersection cohomology is equal to the cohomology of the Koszul complex $IH_T^*(X)\otimes H^*(T)$. We also describe the weight filtration in $IH^*(X)$.
We show that the equivariant Chow cohomology ring of a toric variety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan. This gives a large class of singular spaces for which localization holds in equivariant Chow cohomology with integer coefficients. We also compute the equivariant Chow cohomology of toric prevarieties and general complex hypertoric varieties in terms of piecewise polynomial functions.
Equivariant $K$-theory of cellular toric bundles and related spaces
本文推广了光滑射影环面丛的K-理论结果，刻画了以T-胞腔环面簇为纤维的环面丛的等变与普通拓扑K-环，并将其应用于环面horospherical嵌入的等变拓扑K-环计算。
We compare the cohomology ring of the flag variety $FL_n$ and the Chow cohomology ring of the Gelfand-Zetlin toric variety $X_{GZ}$. We show that $H^*(FL_n, \mathbb{Q})$ is the Gorenstein quotient of the subalgebra $L$ of $A^*(X_{GZ}, \mathbb{Q})$ generated by degree $1$ elements. We compute these algebras for $n=3$ to see that, in general, the subalgebra $L$ does not have Poincare duality.
In this paper, using Klyachko's classification theorem we study positivity and semi-stability of toric vector bundles on a class of nonsingular projective toric varieties, known as Bott towers. In particular, we give a criterion of $s$-jet ampleness of line bundles and characterize nef and big line bundles on Bott towers using Bott numbers. We obtain a criterion for the ampleness of discriminant zero semi-stable toric vector bundles on nonsingular projective varieties. We also describe toric subbundles of toric vector bundles on nonsingular toric varieties.
We prove a weak version of a bigness criterion for equivariant vector bundles on toric varieties.
We introduce extended toric Deligne-Mumford stacks. We use an extended toric Deligne-Mumford stack to get the toric stack bundle and compute its orbifold Chow ring. Finally we generalize one result of Borisov, Chen and Smith so that the orbifold Chow ring of the toric stack bundle and the Chow ring of its crepant resolution are fibres of a flat family.
Simplicial toric stack bundles are smooth Deligne-Mumford stacks over smooth varieties with fibre a toric Deligne-Mumford stack. We compute the Grothendieck $K$-theory of simplicial toric stack bundles and study the Chern character homomorphism.
In this note we prove that any toric Fano manifold with nef tangent bundle is a product of projective spaces. In particular, it implies that Campana-Peternell conjecture hold for toric manifolds.
The automorphism group of a projective bundle P(E) over a simplicial toric variety is described when the bundle E is a direct sum of line bundles. Applications to study of moduli of complete intersections on toric varieties, including Calabi Yau m-folds and Del Pezzo surfaces, are indicated.
In this paper, we investigate when there exists a wild hypersurface bundle over a smooth proper toric variety in positive characteristic. In particular, we determine the possibilities for toric varieties with Picard number at most three or toric Fano varieties of dimension at most four. Moreover, we can construct wild hypersurface bundles over them.
We study Givental's Lagrangian cone for the quantum orbifold cohomology of toric stack bundles and prove that the I-function gives points in the Lagrangian cone, namely we construct an explicit slice of the Lagrangian cone defined by the genus $0$ Gromov-Witten theory of a toric stack bundle.
In this paper, we prove the Yau-Tian-Donaldson conjecture of the filtration version for toric manifolds and homogeneous toric bundles.
Moduli of toric principal bundles
本文基于Kaveh-Manon对典范主丛的分类，构建了具有给定总等变特征类的典范主丛的模空间，作为部分旗变种的积空间的局部闭子流形。
We reproduce the quantum cohomology of toric varieties (and of some hypersurfaces in projective spaces) as the cohomology of certain vertex algebras with differential. The deformation technique allows us to compute the cohomology of the chiral de Rham complex over the projective space.
Equivariant vector bundles on toric schemes over semirings
该研究定义了半环上概形的等变向量丛，证明幂等半域上环面概形的环面向量丛可等变分裂为线丛直和，还研究了等变皮卡群并给出对应Klyachko分类定理版本。
For any $n\geq 3$, we explicitly construct smooth projective toric $n$-folds of Picard number $\geq 5$, where any nontrivial nef line bundles are big.
We give a simple combinatorial proof of the toric version of Mori's theorem that the only $n$-dimensional smooth projective varieties with ample tangent bundle are the projective spaces $\mathbb{P}^n$.
We classify holomorphic as well as algebraic torus equivariant principal $G$-bundles over a nonsingular toric variety $X$, where $G$ is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric variety admits a trivialization in equivariant sense. We also obtain some splitting results.
We study line bundles on toric DM stacks $\mathbb{P}_{\mathbfΣ}$ of dimension two. We give a combinatorial criterion of when infinitely many line bundles on $\mathbb{P}_{\mathbfΣ}$ have trivial cohomology. We further discuss the structure of the set of such line bundles.
We describe explicitly the cohomology of the total complex of certain diagrams of invertible sheaves on normal toric varieties. These diagrams, called wheels, arise in the study of toric singularities associated to dimer models. Our main tool describes the generators in a family of syzygy modules associated to the wheel in terms of walks in a family of graphs.
Let X be a smooth simplicial toric variety. Let Z be the set of T-fixed points of X. We construct a filtration for A(Z), the ring of complex-valued functions on Z, such that Gr A(Z) is isomorphic to the cohomology algebra of X. This is the explanation of the general results of Carrell and Liebermann on the cohomology of T-varities, in the case of toric varieties. We give an explicit isomorphism between Gr A(Z) and Brion's description of the polytope algebra.
Using the homogeneous coordinate ring construction of a toric variety IP defined by a complete simplicial fan and the methods of local cohomology theory we develop a framework for the calculation of cohomology groups H^{*}(IP, O(p)) of twisting sheaves O(p) on IP.
We give a combinatorial criterion for the tangent bundle on a smooth toric variety to be stable with respect to a given polarisation in terms of the corresponding lattice polytope. Furthermore, we show that for a smooth toric surface and a smooth toric variety of Picard rank 2, there exists an ample line bundle with respect to which the tangent bundle is stable if and only if it is an iterated blow-up of projective space.
This paper explicitly describes Hodge structures of complete intersections of ample hypersurfaces in compact simplicial toric varieties.
Using the notion of a valuation into the semifield of piecewise linear functions, we give a classification of torus equivariant flat families of finite type over a toric variety base, by certain piecewise linear maps between fans. As a consequence we derive a classification of toric vector bundles phrased in terms of tropicalized linear spaces. We use these tools to give a characterization of the Mori dream space property for a projectivized toric vector bundle.
The cohomological rigidity problem for toric manifolds asks whether toric manifolds are diffeomorphic (or homeomorphic) if their integral cohomology rings are isomorphic. Many affirmative partial solutions to the problem have been obtained and no counterexample is known. In this paper, we study the diffeomorphism classification of toric Fano $d$-folds with $d=3,4$ or with Picard number $\ge 2d-2$. In particular, we show that those manifolds except for two toric Fano $4$-folds are diffeomorphic if their integral cohomology rings are isomorphic. The exceptional two toric Fano $4$-folds (their ID…
We introduce a collection of convex polytopes associated to a torus-equivariant vector bundle on a smooth complete toric variety. We show that the lattice points in these polytopes correspond to generators for the space of global sections and we relate edges to jets. Using the polytopes, we also exhibit toric vector bundles that are ample but not globally generated, and toric vector bundles that are ample and globally generated but not very ample.
For an affine toric variety $\mathrm{Spec}(A)$, we give a convex geometric description of the Hodge decomposition of its Hochschild cohomology. Under certain assumptions we compute the dimensions of the Hodge summands $T^1_{(i)}(A)$, generalizing the existing results about the Andre-Quillen cohomology group $T^1_{(1)}(A)$. We prove that every Poisson structure on a possibly singular affine toric variety can be quantized in the sense of deformation quantization.
On the instability of syzygy bundles on toric surfaces
本文研究环面上合冲束的斜率稳定性，证明除两类特殊环面外，总可选取极化使高次幂线丛对应的合冲束失稳。
There is a standard method to calculate the cohomology of torus-invariant sheaves $L$ on a toric variety via the simplicial cohomology of associated subsets $V(L)$ of the space $N_{\mathbb R}$ of 1-parameter subgroups of the torus. For a line bundle $L$ represented by a formal difference $Δ^+-Δ^-$ of polyhedra in the character space $M_{\mathbb R}$, [ABKW18] contains a simpler formula for the cohomology of $L$, replacing $V(L)$ by the set-theoretic difference $Δ^- \setminus Δ^+$. Here, we provide a short and direct proof of this formula.
Chern Classes of Toric Variety Bundles
本文解决了Khovanskii--Monin猜想，给出环簇丛切丛全陈类用基空间与环簇纤维陈类表示的公式，并推广至稳定复拟环流形，应用于环面螺旋球簇及Masuda公式的新证明。
We define the notion of a piecewise linear map from a fan $Σ$ to $\tilde{\mathfrak{B}}(G)$, the cone over the Tits building of a linear algebraic group $G$. Let $X_Σ$ be a toric variety with fan $Σ$. We show that when $G$ is reductive the set of integral piecewise linear maps from $Σ$ to $\tilde{\mathfrak{B}}(G)$ classifies the isomorphism classes of (framed) toric principal $G$-bundles on $X_Σ$. This in particular recovers Klyachko's classification of toric vector bundles, and gives new classification results for the orthogonal and symplectic toric principal bundles.
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.
Cox rings of projectivized toric vector bundles and toric flag Bundles
本文研究射影化环面向量丛在直和运算下何时保持Mori dream空间性质，利用全旗丛关系给出判定条件，并计算了射影空间切丛的全旗丛Cox环表示。
We investigate the equivariant intersection cohomology of a toric variety. Considering the defining fan of the variety as a finite topological space with the subfans being the open sets (that corresponds to the "toric" topology given by the invariant open subsets), equivariant intersection cohomology provides a sheaf (of graded modules over a sheaf of graded rings) on that "fan space". We prove that this sheaf is a "minimal extension sheaf", i.e., that it satisfies three relatively simple axioms which are known to characterize such a sheaf up to isomorphism. In the verification of the second o…
We investigate analytic properties of height zeta functions of toric bundles over flag varieties.
We give a complete classification of equivariant vector bundles of rank two over smooth complete toric surfaces and construct moduli spaces of such bundles. This note is a direct continuation of an earlier note where we developed a general description of equivariant sheaves on toric varieties. Here we give a first application of that description.
We study the local cohomology modules H^i_B(R) for a reduced monomial ideal B in a polynomial ring R=k[X_1,...,X_n]. We consider a grading on R which is coarser than the Z^n-grading such that each component of H^i_B(R) is finite dimensional and we give an effective way to compute these components. Using Cox's description for sheaves on toric varieties, we apply these results to compute the cohomology of coherent sheaves on toric varieties. We give algorithms for this computation which have been implemented in the Macaulay 2 system. We obtain also a topological description for the cohomology of…
In the mid 70's, Hartshorne conjectured that, for all n > 7, any rank 2 vector bundles on P^n is a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain P^n
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