Let $G$ be a connected affine algebraic group and $X$ a regular $G$-variety (in the sense of Bifet-De Concini-Procesi) with open orbit $G/H$ and boundary divisor $D$. We show the vanishing of the $G$-equivariant Chern classes of the bundle of differential forms on $X$ with logarithmic poles along $D$, in degrees larger than $\dim(X) - \rk(G) + \rk(H)$. Our motivation comes from Gieseker's degeneration method to prove the Newstead-Ramanan conjecture on the vanishing of the top Chern classes of the moduli space of stable vector bundles on a curve.
In this paper, we shall discuss possible theories of defining equivariant singular Bott-Chern classes and corresponding uniqueness property. By adding a natural axiomatic characterization to the usual ones of equivariant Bott-Chern secondary characteristic classes, we will see that the construction of Bismut's equivariant Bott-Chern singular currents provides a unique way to define a theory of equivariant singular Bott-Chern classes. This generalizes J. I. Burgos Gil and R. Liţcanu's discussion to the equivariant case. As a byproduct of this study, we shall prove a concentration formula which …
For a complex variety with a torus action we propose a new method of computing Chern-Schwartz-MacPherson classes. The method does not apply resolution of singularities. It is based on Localization Theorem in equivariant cohomology.
Consider the natural torus action on a partial flag manifold $Fl$. Let $Ω_I\subset Fl$ be an open Schubert variety, and let $c^{sm}(Ω_I)\in H_T^*(Fl)$ be its torus equivariant Chern-Schwartz-MacPherson class. We show a set of interpolation properties that uniquely determine $c^{sm}(Ω_I)$, as well as a formula, of `localization type', for $c^{sm}(Ω_I)$. In fact, we proved similar results for a class $κ_I\in H_T^*(Fl)$ --- in the context of quantum group actions on the equivariant cohomology groups of partial flag varieties. In this note we show that $c^{SM}(Ω_I)=κ_I$.
We study a K-theoretic characteristic class of singular varieties, namely the equivariant motivic Chern class. We prove that the motivic Chern class is characterized by an axiom system inspired by that of "K-theoretic stable envelopes," recently defined by Okounkov and studied in relation with quantum group actions on the K-theory algebra of moduli spaces. We also give explicit formulas for the equivariant motivic Chern classes of Schubert cells and matrix Schubert cells. Lastly, we calculate the equivariant motivic Chern class of the orbits of the A2 quiver representation, which yields formul…
We describe the equivariant Chow ring of the wonderful compactification $X$ of a symmetric space of minimal rank, via restriction to the associated toric variety $Y$. Also, we show that the restrictions to $Y$ of the tangent bundle $T_X$ and its logarithmic analogue $S_X$ decompose into a direct sum of line bundles. This yields closed formulae for the equivariant Chern classes of $T_X$ and $S_X$, and, in turn, for the Chern classes of reductive groups considered by Kiritchenko.
We define the equivariant Chern-Schwartz-MacPherson class of a possibly singular algebraic variety with a group action over the complex number field (or a field of characteristic 0). In fact, we construct a natural transformation from the equivariant constructible function functor to the equivariant homology functor (in the sense of Totaro-Edidin-Graham), which may be regarded as MacPherson's transformation for (certain) quotient stacks. We discuss on other type Chern/Segre classes and give some applications generalizing orbifold Euler characteristics and Thom polynomials of singularities. The…
We prove the Bogomolov-Gieseker type inequality conjectured by Bayer, Macri and Toda for threefolds with semistable tangent bundles and vanishing Chern classes in any characteristic, which was originally proved by Bayer, Macri and Stellari in characteristic zero. This gives the existence of Bridgeland stability conditions on such threefolds. As applications, we obtain Reider type theorem and confirm Fujita's conjecture for such threefolds in any characteristic.
According to a conjecture attributed to Hartshorne and Lichtenbaum and proven by Ellingsrud and Peskine, the smooth rational surfaces in $\mathbb{P}^4$ belong to only finitely many families. We formulate and study a collection of analogous problems in which $\mathbb{P}^4$ is replaced by a smooth fourfold $X$ with vanishing first integral Chern class. We embed such $X$ into a smooth ambient variety and count families of smooth surfaces which arise in $X$ from the ambient variety. We obtain various finiteness results in such settings. The central technique is the introduction of a new numerical …
Let $Y$ be a smooth projective curve of genus $g\ge 2$ and let $M_{r,d}(Y)$ be the moduli space of stable vector bundles of rank $r$ and degree $d$ on $Y$. A classical conjecture of Newstead and Ramanan states that $ c_i(M_{2,1}(Y))=0$ for $i>2(g-1)$ i.e. the top $2g-1$ Chern classes vanish. The purpose of this paper is to generalize this vanishing result to the rank 3 case by generalizing Gieseker's degeneration method. More precisely, we prove that $c_i(M_{3,1}(Y))=0$ for $i>6g-5$. In other words, the top $3g-3$ Chern classes vanish. Notice that we also have $c_i(M_{3,2}(Y))=0$ for $i>6g-5$.
In this survey paper we review recent advances in the calculus of Chern-Schwartz-MacPherson, motivic Chern, and elliptic classes of classical Schubert varieties. These three theories are one-parameter ($\hbar$) deformations of the notion of fundamental class in their respective extraordinary cohomology theories. Examining these three classes in conjunction is justified by their relation to Okounkov's stable envelope notion. We review formulas for the $\hbar$-deformed classes originating from Tarasov-Varchenko weight functions, as well as their orthogonality relations. As a consequence, explici…
Consider a complex algebraic group $G$ acting on a smooth variety $M$ with finitely many orbits, and let $Ω$ be an orbit. The following three invariants of $Ω\subset M$ can be characterized axiomatically: (1) the equivariant fundamental class $[\overlineΩ, M]\in H^*_G(M)$, (2) the equivariant Chern-Schwartz-MacPherson class $c(Ω, M)\in H^*_G(M)$, and (3) the equivariant motivic Chern class $mC(Ω, M) \in K_G(M)[y]$. The axioms for Chern-Schwartz-MacPherson and motivic Chern classes are motivated by the axioms for cohomological and K-theoretic stable envelopes of Okounkov and his coauthors. For …
We study motivic Chern classes of cones. First we show examples of projective cones of smooth curves such that their various $K$-classes (sheaf theoretic, push-forward and motivic) are all different. Then we show connections between the torus equivariant motivic Chern class of a projective variety and of its affine cone, generalizing results on projective Thom polynomials.
We develop Pieri type as well as Murnaghan--Nakayama type formulas for equivariant Chern--Schwartz--MacPherson classes of Schubert cells in the classical flag variety. These formulas include as special cases many previously known multiplication formulas for Chern--Schwartz--MacPherson classes or Schubert classes. We apply the equivariant Murnaghan--Nakayama formula to the enumeration of rim hook tableaux.
We obtain A- and B- series of flag manifolds for T-equivariant tangent bundles and equivariant Chern polynomial following the arguments of Lian, Liu, Yau.
Given two equivariant vector bundles over an algebraic GKM manifold with the same equivariant Chern classes, we show that the genus zero equivariant Gromov--Witten theory of their projective bundles are naturally isomorphic.
Kawamata proposed a conjecture predicting that every nef and big line bundle on a smooth projective variety with trivial first Chern class has nontrivial global sections. We verify this conjecture for several cases, including (i) all hyperkähler varieties of dimension $\leq 6$; (ii) all known hyperkähler varieties except for O'Grady's 10-dimensional example; (iii) general complete intersection Calabi-Yau varieties in certain Fano manifolds (e.g. toric ones). Moreover, we investigate the effectivity of Todd classes of hyperkähler varieties and Calabi-Yau varieties. We prove that the fourth Todd…
We discuss a relationship between Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds, Fomin-Kirillov algebra, and the generalized nil-Hecke algebra. We show that nonnegativity conjecture in Fomin-Kirillov algebra implies the nonnegativity of the Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of the Bott-Samelson varieties. W…
We prove the conjectures of Graham-Kumar and Griffeth-Ram concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant K-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homol…
The Chern-Schwartz-MacPherson class (CSM) and the Segre-Schwartz-MacPherson class (SSM) are deformations of the fundamental class of an algebraic variety. They encode finer enumerative invariants of the variety than its fundamental class. In this paper we offer three contributions to the theory of equivariant CSM/SSM classes. First, we prove an interpolation characterization for CSM classes of certain representations. This method---inspired by recent works of Maulik-Okounkov and Gorbounov-Rimanyi-Tarasov-Varchenko---does not require a resolution of singularities and often produces explicit (no…
We give a reduction of the conjecture that for terminal projective threefolds whose anticanonical divisors are nef, the second Chern classes are pseudo-effective. On the other hand, some effective non-vanishing results are obtained as applications of the pseudo-effectivity of the second Chern classes.
We prove that for smooth projective threefolds whose anticanonical divisors are nef, the second Chern classes are pseudo-effective under a weak assumption. As an application, the pseudo-effectivity of the second Chern classes implies that Kawamata's Effective Non-vanishing Conjecture holds for such threefolds.
We give a complete answer to the question of (semi)stability of tangent bundle of any nonsingular projective complex toric variety with Picard number 2 by using combinatorial crietrion of (semi)stability of an equivariant sheaf. We also give a complete answer to the question of (semi)stability of tangent bundle of all toric Fano 4-folds with Picard number (\leq) 3 which are classified by Batyrev \cite{batyrev}. We have constructed a collection of equivariant indecomposable rank 2 vector bundles on Bott tower and pseudo-symmetric toric Fano varieties. Further in case of Bott tower, we have show…
The cohomology of the Hilbert schemes of points on smooth projective surfaces can be approached both with vertex algebra tools and equivariant tools. Using the first tool, we study the existence and the structure of universal formulas for the Chern classes of the tangent bundle over the Hilbert scheme of points on a projective surface. The second tool leads then to nice generating formulas in the particular case of the Hilbert scheme of points on the affine plane.
We develop a theory of ``ad hoc'' Chern characters for twisted matrix factorizations associated to a scheme $X$, a line bundle ${\mathcal L}$, and a regular global section $W \in Γ(X, {\mathcal L})$. As an application, we establish the vanishing, in certain cases, of $h_c^R(M,N)$, the higher Herbrand difference, and, $η_c^R(M,N)$, the higher codimensional analogue of Hochster's theta pairing, where $R$ is a complete intersection of codimension $c$ with isolated singularities and $M$ and $N$ are finitely generated $R$-modules. Specifically, we prove such vanishing if $R = Q/(f_1, \dots, f_c)$ h…
Let X be a smooth complete complex toric variety such that the boundary is a simple normal crossing divisor, and let E be a holomorphic vector bundle on X. We prove that E admits an equivariant structure if and only if E admits a logarithmic connection singular over D. More precisely, we show that an equivariant vector bundle on X has a tautological integrable logarithmic connection singular over D. This is used in computing the Chern classes of the equivariant vector bundles on X. We also prove a version of the result for holomorphic vector bundles on log parallelizable G-pairs (X,D), where G…
We study complex Chern-Simons theory on a Seifert manifold $M_3$ by embedding it into string theory. We show that complex Chern-Simons theory on $M_3$ is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the dimensional reduction of this theory to 2d gives the low energy dynamics of vortices in four-dimensional gauge theory, the fact apparently overlooked in the vortex literature. We also generalize the relations between 1) the Verlinde algebra, 2) quantum cohomology of the Grassmanni…
We calculate the equivariant motivic Chern class for configuration space of a quasiprojective (maybe singular) variety and the space of vectors with different directions. We prove the formulas for generating series of these classes. We generalize the localization theorems results about Bialynicki-Birula decomposition to acquire some stability for the motivic Chern classes of configuration spaces.
In this paper we aim at the description of foliations having tangent sheaf $T\mathcal F$ with $c_1(T\mathcal F)=c_2(T\mathcal F)=0$ on non-uniruled projective manifolds. We prove that the universal covering of the ambient manifold splits as a product, and that the Zariski closure of a general leaf of $\mathcal F$ is an Abelian variety. It turns out that the analytic type of the Zariski closures of leaves may vary from leaf to leaf. We discuss how this variation is related to arithmetic properties of the tangent sheaf of the foliation.
We establish the analogue of the Friedlander-Mazur conjecture for Teh's reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasi-projective variety $X$ vanishes in homological degrees larger than the dimension of $X$ in all weights. As an application we obtain a vanishing of homotopy groups of the mod-2 topological groups of averaged cycles and a characterization in a range of indices of the motivic cohomology of a real variety as homotopy groups of the complex of averaged equidimensional cycles. We also establish an equivariant Poincare dua…
Let $X$ be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that $X$ admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if $X$ has toroidal singularities, if $X$ has finite quotient singularities, and if the second cohomology group of its tangent sheaf vanishes.
Let $\mathrm{R}$ be a real closed field. We prove that for any fixed $d$, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $\mathrm{R}^k$ defined by polynomials of degrees bounded by $d$ vanishes in dimensions $d$ and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of $d^{O(d)} s^d k^{\lfloor d/2 \rfloor-1} $ on the equivariant Betti numbers of closed symmetric semi-algebraic subsets of $\mathrm{R}^k$ defined by quantifier-free formulas involving $s$ symmetric polynomials of degrees bounded by $d$, wher…
We observe that certain equivariant intersection numbers of Chern characters of tautological sheaves on Hilbert schemes for suitable circle actions can be computed using the Bloch-Okounkov formula, hence they are related to Gromov-Witten invariants of elliptic curves and its operator formalism in terms of operators on the Fock space.
Newstead and Ramanan conjectured the vanishing of the top (2g-1) Chern classes of the moduli space of stable, odd degree vector bundles of rank 2 on a Riemann surface of genus g. This was proved by Gieseker [G], while an analogue in rank 3 was recently settled by Kiem and Li [KL]. We generalise this to the vanishing of the top (g-1)r rational Chern classes of the moduli space M of stable principal bundles with semi-simple structure group of rank r, whenever M is a compact orbifold.
The aim of this paper is to classify simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology. It is shown that there is a one-to-one correspondence between equivariant diffeomorphism types of these manifolds and 3-valent labelled graphs, called torus graphs introduced by Maeda-Masuda-Panov. Using this correspondence and combinatorial arguments, we prove that a simply connected 6-dimensional torus manifold with vanishing odd degree cohomology is equivariantly diffeomorphic to the 6-dimensional sphere or an equivariant connected sum of copies of 6-dimensional quasitor…
We obtain refined generating series formulae for equivariant characteristic classes of external and symmetric products of singular complex quasi-projective varieties. More concretely, we study equivariant versions of Todd, Chern and Hirzebruch classes for singular spaces, with values in delocalized Borel-Moore homology of external and symmetric products. As a byproduct, we recover our previous characteristic class formulae for symmetric products, and obtain new equivariant generalizations of these results, in particular also in the context of twisting by representations of the symmetric group.
In this paper, I construct noncompact analogs of the Chern classes of equivariant vector bundles over complex reductive groups. For the tangent bundle, these Chern classes yield an adjunction formula for the Euler characteristic of complete intersections in reductive groups. In the case where the complete intersection is a curve, this formula gives an explicit answer for the Euler characteristic and the genus of the curve.
We find presentations by generators and relations for the equivariant quantum cohomology rings of the maximal isotropic Grassmannians of types B,C and D, and we find polynomial representatives for the Schubert classes in these rings. These representatives are given in terms of the same Pfaffian formulas which appear in the theory of factorial $P$- and $Q$-Schur functions. After specializing to equivariant cohomology, we interpret the resulting presentations and Pfaffian formulas in terms of Chern classes of tautological bundles.
We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincaré duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern classes vanishes below the dimension of the moduli space, in analogy with the Newstead-Ramanan conjecture for stable bundles.
The purpose of this paper is to prove the localization theorem for torus actions in equivariant intersection theory. Using the theorem we give another proof of the Bott residue formula for Chern numbers of bundles on smooth complete varieties. In addition, our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions. This class is rather special, but it includes some interesting examples such as complete intersections and Schubert varieties.
