Chow Cohomology Research Articles

The eleventh cohomology group of

AbstractWe prove that the rational cohomology group$H^{11}(\overline {\mathcal {M}}_{g,n})$vanishes unless$g = 1$and$n \geq 11$. We show furthermore that$H^k(\overline {\mathcal {M}}_{g,n})$is pure Hodge–Tate for all even$k \leq 12$and deduce that$\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$is surprisingly well approximated by a polynomial inq. In addition, we use$H^{11}(\overline {\mathcal {M}}_{1,11})$and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and nontautological algebraic cycle classes in Chow cohomology.

Cohomology ring of the flag variety vs Chow cohomology ring of the Gelfand–Zetlin toric variety

We compare the cohomology ring of the flag variety {\mathcal{F}\ell}_n and the Chow cohomology ring of the Gelfand–Zetlin toric variety X_{\operatorname{GZ}} .We show that H^*(\mathcal{F}{\ell}_n, \mathbb{Q}) is the Poincaré duality quotient of the subalgebra of A^*(X_{\operatorname{GZ}}, \mathbb{Q}) generated by degree 1 elements. We compute these algebras for n=3 and see that, in general, this subalgebra does not have Poincaré duality.

The Chow cohomology of affine toric varieties

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.

Toric chordality

We study the geometric change of Chow cohomology classes in projective toric varieties under the Weil–McMullen dual of the intersection product with a Lefschetz element. Based on this, we introduce toric chordality, a generalization of graph chordality to higher skeleta of simplicial complexes with a coordinatization over characteristic 0, leading us to a far-reaching generalization of Kalai's work on applications of rigidity of frameworks to polytope theory. In contrast to “homological” chordality, the notion that is usually studied as a higher-dimensional analogue of graph chordality, we will show that toric chordality has several advantageous properties and applications.∘Most strikingly, we will see that toric chordality allows us to introduce a higher version of Dirac's propagation principle.∘Aside from the propagation theorem, we also study the interplay with the geometric properties of the simplicial chain complex of the underlying simplicial complex, culminating in a quantified version of the Stanley–Murai–Nevo generalized lower bound theorem.∘Finally, we apply our technique to give a simple proof of the generalized lower bound theorem in polytope theory and∘prove the balanced generalized lower bound conjecture of Klee and Novik.

Equivariant operational Chow rings of $T$-linear schemes

We study T -linear schemes, a class of objects that includes spherical and Schubert varieties. We provide a localization theorem for the equivariant Chow cohomology of these schemes that does not depend on resolution of singularities. Furthermore, we give an explicit presentation of the equivariant Chow cohomology of possibly singular complete spherical varieties admitting a smooth equivariant envelope (e.g., group embeddings).

CHOW GROUPS, CHOW COHOMOLOGY, AND LINEAR VARIETIES

AbstractWe compute the Chow groups and the Fulton–MacPherson operational Chow cohomology ring for a class of singular rational varieties including toric varieties. The computation is closely related to the weight filtration on the ordinary cohomology of these varieties. We use the computation to answer one of the open problems about operational Chow cohomology: it does not have a natural map to ordinary cohomology.

Chow cohomology groups of algebraic surfaces

For an algeraic surface with isolated singularities we consider its higher Chow group and Chow cohomology (the latter defined by the author). We study the canonical map from the Chow cohomology to the higher Chow group by relating it to the canonical map from the higher Chow group to Chow cohomology of the exceptional divisor of a desingularization of the surface. Introduction. For a quasi-projective variety S over a field, S. Bloch defined its higher Chow groups CH(S, n) as the homology of a certain complex Z(S, ·) called the cycle complex, [Bl 1], [Bl 2], [Bl 3]. One may view this as a Borel-Moore homology theory; for example it is covariantly functorial for proper maps, and contravariantly functorial for open immersions (more generally for flat maps). For a quasi-projective variety S over a field of characteristic zero, using resolution of singularities and Bloch’s cycle complexes, we defined the Chow cohomology groups CHC(S, n) (see [Ha 2] for details). To briefly recall it, we take a cubical hyperresolution X• → S, which gives a strict truncated simplicial scheme; it consists of smooth varieties Xa for 0 ≤ a ≤ N with some N , and the face maps di : Xn → Xn−1, i = 0, . . . , n, satisfying the usual identities. (It differs from a simplicial scheme in that there are only face maps and no degeneracies, and there are only finite many terms.) We then form the double complex Z(X0, ·) d∗ −−−→Z(X1, ·) d∗ −−−→· · · d ∗ −−−→Z(XN , ·) where the a-th column is the cycle complex of Xa, and the horizontal differentials d ∗ : Z(Xa, ·) → Z(Xa+1, ·) are the alternating sums of the pull-backs di by the face maps. (Strictly speaking one must take appropriate quasi-isomorphic subcomplexes for d∗ be defined, see §1.) The total complex of this double complex is denoted Z(X•, ·)∗, and called the cohomological cycle complex of S. Then CHC(S, n) is by definition the (−n)-th cohomology of Z(X•, ·)∗. It is proven in [Ha 2] that CHC(S, n) is well-defined up to canonical isomorphism, independent of the choice of a hyperresolution, and that the association S 7→ CHC(S, n) is contravariantly functorial for all maps. The condition the characteristic being zero is unnecessary if dimS ≤ 2, since its desingularization (and thus its cubical hyperresolution) exist. In that case we exhibit the cubical hyperresolution and the cohomological cycle complex of S explicitly in §§1 and 2. (The relationship of our Chow cohomology to the motivic cohomology of [FV] will be discussed in a separate paper.) 2010Mathematics Subject Classification. Primary 14C25; Secondary 14C15, 14C35.

On stringy invariants of GUT vacua

We investigate aspects of certain stringy invariants of singular elliptic fibrations which arise in engineering Grand Unified Theories in F-theory. In particular, we exploit the small resolutions of the total space of these fibrations provided recently in the physics literature to compute ‘stringy characteristic classes’, and find that numerical invariants obtained by integrating such characteristic classes are predetermined by the topology of the base of the elliptic fibration. Moreover, we derive a simple (dimension independent) formula for pushing forward powers of the exceptional divisor of a blowup, which one may use to reduce any integral (in the sense of Chow cohomology) on a small resolution of a singular elliptic fibration to an integral on the base. We conclude with a speculatory note on the cohomology of small resolutions of GUT vacua, where we conjecture that certain simple formulas for their Hodge numbers may be given solely in terms of the first Chern class and Hodge numbers of the base. 1Supported by NSF Grant number 1017880. ar X iv :1 21 1. 60 77 v1 [ m at h. A G ] 2 6 N ov 2 01 2

Cocycles on tropical varieties via piecewise polynomials

We use piecewise polynomials to define tropical cocycles generalising the well-known notion of tropical Cartier divisors to higher codimensions. Groups of cocycles are tropical analogues of Chow cohomology groups. We also introduce an intersection product of cocycles with tropical cycles (the counterpart of the classical cap product) and prove that this gives rise to a Poincaré duality in some cases.

Equivariant Chow cohomology of nonsimplicial toric varieties

For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne shows that the equivariant Chow cohomology of X_P is the Sym(N)--algebra C^0(P) of integral piecewise polynomial functions on P. We use the Cartan-Eilenberg spectral sequence to analyze the associated reflexive sheaf on Proj(N), showing that the Chern classes depend on subtle geometry of P and giving criteria for the splitting of the sheaf as a sum of line bundles. For certain fans associated to the reflection arrangement A_n, we describe a connection between C^0(P) and logarithmic vector fields tangent to A_n.

Tropical intersection theory from toric varieties

We apply ideas from intersection theory on toric varieties to tropical intersection theory. We introduce mixed Minkowski weights on toric varieties which interpolate between equivariant and ordinary Chow cohomology classes on compact toric varieties. These objects fit into the framework of tropical intersection theory developed by Allermann and Rau. Standard facts about intersection theory on toric varieties are applied to show that the definitions of tropical intersection product on tropical cycles in $${\mathbb{R}^n}$$ given by Allermann–Rau and Mikhalkin are equivalent. We introduce an induced tropical intersection theory on subvarieties on a toric variety. This gives a conceptual proof that the intersection of tropical ψ-classes on $${\overline{\mathcal{M}}_{0,n}}$$ used by Kerber and Markwig computes classical intersection numbers.

Piecewise polynomials, Minkowski weights, and localization on toric varieties

We use localization to describe the restriction map from equivariant Chow cohomology to ordinary Chow cohomology for complete toric varieties in terms of piecewise polynomial functions and Minkowski weights. We compute examples showing that this map is not surjective in general, and that its kernel is not always generated in degree one. We prove a localization formula for mixed volumes of lattice polytopes and, more generally, a Bott residue formula for toric vector bundles.

Intersection theory on toric varieties

The operational Chow cohomology classes of a complete toric variety are identified with certain functions, called Minkowski weights, on the corresponding fan. The natural product of Chow cohomology classes makes the Minkowski weights into a commutative ring; the product is computed by a displacement in the lattice, which corresponds to a deformation in the toric variety. We show that, with rational coefficients, this ring embeds in McMullen's polytope algebra, and that the polytope algebra is the direct limit of these Chow rings, over all compactifications of a given torus. In the nonsingular case, the Minkowski weight corresponding to the Todd class is related to a certain Ehrhart polynomial.

Characteristic classes and quadric bundles

In this paper we construct Stiefel-Whitney and Euler classes in Chow cohomology for algebraic vector bundles with nondegenerate quadratic form. These classes are not in the algebra generated by the Chern classes of such bundles and are new characteristic classes in algebraic geometry. On complex varieties, they correspond to classes with the same name pulled back from the cohomology of the classifying space BSO(N,C). The classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups ([T2], [EG]). We begin by using the geometry of quadric bundles to study Chern classes of maximal isotropic subbundles. If V → X is a vector bundle with quadratic form, and if E and F are maximal isotropic subbundles of V then we prove (Theorem 1) that ci(E) and ci(F ) are equal mod 2. Moreover, if the rank of V is 2n, then cn(E) = ±cn(F ), proving a conjecture of Fulton. We define Stiefel-Whitney and Euler classes as Chow cohomology classes which pull back to Chern classes of maximal isotropic subbundles of the pullback bundle. Using the above theorem we show (Theorem 2) that these classes exist and are unique, even though V need not have a maximal isotropic subbundle. These constructions also make it possible to give “Schubert” presentations,

Intersection theory of projective linear spaces

We study the intersection theory of a class of projective linear spaces (generalizations of projective space bundles in which the fibres are linear but of varying dimensions). In particular we give exact sequences for the Chow and Chow cohomology groups reminiscent of those for regular blowups.

Intersection theory of incidence varieties and polygon spaces

In this paper smooth parameterizing spaces for polygons in projective space are introduced and their intersection theory is studied. In particular we give an expression for the Chow ring as a quotient of a polynomial ring. In addition the Chow cohomology rings of various incidence varieties are computed.

Equivariant Chow cohomology of toric varieties

We show that the equivariant Chow cohomology ring of a toric vari- ety 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 lo- calization holds in equivariant Chow cohomology with integer coe!cients. We also compute the equivariant Chow cohomology of toric prevarieties and general complex hypertoric varieties in terms of piecewise polynomial functions. If X = X(!) is a smooth, complete complex toric variety then the follow- ing rings are canonically isomorphic: the equivariant singular cohomology ring H ! T (X), the equivariant Chow cohomology ring A ! (X), the Stanley-Riesner ring SR(!), and the ring of integral piecewise polynomial functions PP ! (!). If X is simplicial but not smooth then H ! (X) may have torsion and the natural map from SR(!) takes monomial generators to piecewise linear functions with ra- tional, but not necessarily integral, coecients. In such cases, these rings are not isomorphic, but they become isomorphic after tensoring with Q. When X is not simplicial, there are still natural maps between these rings, for instance from A ! (X)Q to H ! (X)Q and from H ! T (X) to PP ! (!), but these maps are far