Let $G$ be an algebraic group and let $X$ be a smooth $G$-variety with two orbits: an open orbit and a a closed orbit of codimension $1$. We give an algebraic description of the category of $G$-equivariant vector bundles on $X$ under a mild technical hypothesis. We deduce simpler classifications in the special cases of line bundles and vector bundles which are generically local systems. We apply our results to the study of admissible representations of semisimple Lie groups. Our main result gives a new set of constraints on the associated cycles of unipotent representations.
We give a classification of embedded smooth projective varieties swept out by rational homogeneous varieties whose Picard number and codimension are one.
We extend results of our previous papers, on ordinary multiple points of curves, and on the computation of their conductor, to ordinary multiple subvarieties of codimension one.
We classify almost homogeneous normal varieties of Albanese codimension $1$, defined over an arbitrary field. We prove that such a variety has a unique normal equivariant completion. Over a perfect field, the group scheme of automorphisms of this completion is smooth, except in one case in characteristic $2$, and we determine its (reduced) neutral component.
We obtain an explicit presentation of the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation of the ordinary cobordism ring. Another application is an equivariant Schubert calculus in cobordism. We also describe the rational cobordism rings of wonderful symmetric varieties of minimal rank.
For a complete toric variety, we obtain an explicit formula for the localized equivariant Todd class in terms of the combinatorial data -- the fan. This is based on the equivariant Riemann-Roch theorem and the computation of the equivariant cohomology and equivariant homology of toric varieties.
We study the equivariant cobordism rings for the action of a torus $T$ on smooth varieties over an algebraically closed field of characteristic zero. We prove a theorem describing the rational $T$-equivariant cobordism rings of smooth projective $G$-spherical varieties with the action of a maximal torus $T$ of $G$. As an application, we obtain explicit presentations for the rational equivariant cobordism rings of smooth projective horospherical varieties of Picard number one.
Based on the celebrated result on zeros of holomorphic 1-forms on complex varieties of general type by Popa and Schnell, we study holomorphic 1-forms on $n$-dimensional varieties of Kodaira dimension $n-1$. We show that a complex minimal smooth projective variety $X$ of Kodaira dimension $κ(X)=\dim X-1$ admits a holomorphic 1-form without zero if and only if there is a smooth morphism from $X$ to an elliptic curve. Furthermore, for a general smooth projective variety (not necessarily minimal) $X$ of Kodaira codimension one, we give a structure theorem for $X$ given that $X$ admits a holomorphi…
This expository note surveys some results on equivariant K-theory of varieties with a torus action, focusing on recent work with Sam Payne and Richard Gonzales. It is based on my contribution to the Clifford Lectures at Tulane University in March 2015.
We characterize embeddability of algebraic varieties into smooth toric varieties and prevarieties. Our embedding results hold also in an equivariant context and thus generalize a well known embedding theorem of Sumihiro on quasiprojective G-varieties. The main idea is to reduce the embedding problem to the affine case. This is done by constructing equivariant affine conoids, a tool which extends the concept of an equivariant affine cone over a projective G-variety to a more general framework.
Ran proved that smooth codimension 2 varieties in ${\bf P}^{m+2}$ are $j$-normal if $(j+1)(3j-1)\le m-1$, in this paper we extend this result to small codimension projective varieties. Let $X$ be a r codimension subvariety of $\pro$, we prove that if the set $Σ_{(j+1)}$ of $(j+1)$-secants to $X$ through a generic external point is not empty, $2(r+1)j\leq m-r$ and $(j+1)((r+1)j-1)\leq m-1$ then $X$ is $j$-normal. If $X$ is given by the zero locus of a section of a rank $r$ vector bundle $E$ on $\pro$, we prove that $\textrm{deg} Σ_{j+1}=\frac{1}{(j+1)!}\prod_{i=0}^{j}c_{r}(E(-i))$. Moreover we …
We prove the Tate conjecture for divisor classes and the Mumford-Tate conjecture for the cohomology in degree 2 for varieties with $h^{2,0}=1$ over a finitely generated field of characteristic 0, under a mild assumption on their moduli. As an application of this general result, we prove the Tate and Mumford-Tate conjectures for some classes of algebraic surfaces with $p_g=1$.
We construct a basis of the equivariant $K$-theory of Bott towers, and we describe precisely the multiplicative structure of these algebras. We deduce similar results for Bott-Samelson varieties. Thanks to the link between flag varieties and Bott-Samelson varieties, we give a method to compute the structure constants of the equivariant $K$-theory of flag varieties in the basis constructed by Kostant and Kumar.
In this paper we extend to the singular setting the theory of Fano foliations developed in our previous paper. A Q-Fano foliation on a complex projective variety X is a foliation F whose anti-canonical class is an ample Q-Cartier divisor. In the spirit of Kobayashi-Ochiai Theorem, we prove that under some conditions the index i of a Q-Fano foliation is bounded by the rank r of F, and classify the cases in which i=r. Next we consider Q-Fano foliations F for which i=r-1. These are called del Pezzo foliations. We classify codimension 1 del Pezzo foliations on mildly singular varieties.
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.
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.
Let X\subsetneq\mathbb{P}_{\mathbb{C}}^{N} be an n-dimensional nondegenerate smooth projective variety containing an m-dimensional subvariety Y. Assume that either m>\frac{n}{2} and X is a complete intersection or that m\geq\frac{N}{2}, we show deg(X)|deg(Y) and codim_{span(Y)}Y\geq codim_{\mathbb{P}^{N}}X, where span(Y) is the linear span of Y. These bounds are sharp. As an application, we classify smooth projective n-dimensional quadratic varieties swept out by m\geq[\frac{n}{2}]+1 dimensional quadrics passing through one point.
We prove that torsion codimension 2 algebraic cycles modulo rational equivalence on supersingular abelian varieties are algebraically equivalent to zero. As a consequence, we prove that homological equivalence coincides with algebraic equivalence for algebraic cycles of codimension 2 on supersingular abelian varieties over the algebraic closure of finite fields.
We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we obtain a Lê-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the 1- form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last se…
We show that Gorenstein singularities that are cones over singular Fano varieties provided by so-called flag quivers are smoothable in codimension three. Moreover, we give a precise characterization about the smoothability in codimension three of the Fano variety itself.
We prove that the only rational homogeneous varieties with Picard number 1 of the exceptional algebraic groups admitting irreducible equivariant Ulrich vector bundles are the Cayley plane $E_6/P_1$ and the $E_7$-adjoint variety $E_7/P_1$. From this result, we see that a general hyperplane section $F_4/P_4$ of the Cayley plane also has an equivariant but non-irreducible Ulrich bundle.
Let G be a semisimple affine algebraic group and P a parabolic subgroup of G. We classify all flag varieties G/P which admit an action of the commutative unipotent group G_a^n with an open orbit.
We show the existence of an equivariant weight filtration on the equivariant homology of real algebraic varieties equipped with a finite group action, by applying group homology onto the weight complex of McCrory and Parusiński. The group action enriches the information contained in the induced equivariant weight spectral sequence from which we can not extract finite additivity directly if the group is of even order. Still, we manage to construct additive invariants from its elementary bricks, and prove that they are induced from the very equivariant geometry of real algebraic varieties with a…
Let X be a T-variety, where T is an algebraic torus. We describe a fully faithful functor from the category of T-equivariant vector bundles on X to a certain category of filtered vector bundles on a suitable quotient of X by T. We show that if X is factorial, this functor gives an equivalence of categories. This generalizes Klyachko's description of equivariant vector bundles on toric varieties. We apply our machinery to show that vector bundles of low rank on projective space which are equivariant with respect to special subtori of the maximal torus must split, generalizing a theorem of Kaney…
In this note we derive a formalism for describing equivariant sheaves over toric varieties. This formalism is a generalization of a correspondence due to Klyachko, which states that equivariant vector bundles on toric varieties are equivalent to certain sets of filtrations of vector spaces. We systematically construct the theory from the point of view of graded ring theory and this way we clarify earlier constructions of Kaneyama and Klyachko. We also connect the formalism to the theory of fine-graded modules over Cox' homogeneous coordinate ring of a toric variety. As an application we constr…
We construct Q-factorial terminal Fano varieties, starting in dimension 4, whose nef cone jumps when the variety is deformed. It follows that de Fernex and Hacon's results on deformations of 3-dimensional Fanos are optimal. The examples are based on the existence of high-dimensional flips which deform to isomorphisms, generalizing the Mukai flop. We also extend de Fernex and Hacon's positive results on deformations of Fano varieties. Toric Fano varieties which are smooth in codimension 2 and Q-factorial in codimension 3 are rigid. The divisor class group is deformation-invariant for klt Fanos …
We determine generators for the codimension 1 Chow group of the moduli spaces of genus zero stable maps to flag varieties G/P. In the case of SL flags, we find all relations between our generators, showing that they essentially come from $\bar M_{0,n}$. In addition, we analyze the codimension 2 classes on the moduli spaces of stable maps to Grassmannians and prove a new codimension 2 relation. This will lead to a partial reconstruction theorem for the Grassmannian of 2 planes.
The central object of investigation of this paper is the Hirzebruch class, a deformation of the Todd class, given by Hirzebruch (for smooth varieties) in his celebrated book "Topological Methods in Algebraic Geometry". The generalization for singular varieties is due to Brasselet-Schürmann-Yokura. Following the work of Weber, we investigate its equivariant version for (possibly singular) toric varieties. The local decomposition of the Hirzebruch class to the fixed points of the torus action and a formula for the local class in terms of the defining fan are mentioned. After this review part, we…
In this paper we give a lower bound for the codimension of the Andreotti-Mayer loci in the moduli space of principally polarized complex abelian varieties. We also present a conjecture on this codimension.
We describe classes of toric varieties of codimension 2 which are either minimally defined by 3 binomial equations over any algebraically closed field, or are set-theoretic complete intersections in exactly one positive characteristic.
We describe the equivariant cobordism ring of smooth toric varieties. This equivariant description is used to compute the ordinary cobordism ring of such varieties.
We propose a novel sufficient condition establishing that a piecewise affine variety has the same topology as a variety of the sphere $\mathbb{S}^n$ defined by positively homogeneous $C^1$ functions. This covers the case of $C^1$ varieties in the projective space $\mathbb{P}^n$. We prove that this condition is sufficient in the case of codimension one and arbitrary dimension. We describe an implementation working for homogeneous polynomials in arbitrary dimension and codimension and give experimental evidences that our condition might still be sufficient in codimension greater than one.
In the present paper we provide a general algorithm to compute multiplicative cohomological operations on algebraic oriented cohomology of projective homogeneous G-varieties, where G is a split reductive algebraic group over a field of characteristic 0. More precisely, we extend such operations to the respective T-equivariant (T is a maximal split torus of G) oriented theories, and then compute them using equivariant Schubert calculus techniques. This generalizes an approach suggested by Garibaldi-Petrov-Semenov for Steenrod operations. We also show that operations on the theories of additive …
In this paper we investigate special arrangements of lines in multiprojective spaces. In particular, we characterize codimensional two arithmetically Cohen-Macaulay (ACM) varieties in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$, called varieties of lines. We also describe their ACM property from combinatorial algebra point of view.
For a pair of affine toric varieties X and Y defined by dual cones, we define an equivalence between two triangulated categories. The first is a mixed version of the equivariant derived category of X and the second is a mixed version of the derived category of sheaves on Y which are locally constant with unipotent monodromy on each orbit. This equivalence satisfies the Koszul duality formalism of Beilinson, Ginzburg, and Soergel. A similar duality was constructed in math.AG/0308216; this new approach is more canonical.
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…
Let A be a tame quasi-tilted algebra and d the dimension vector of an indecomposable A-module. In the paper we prove that each irreducible component of the variety of A-modules of dimension vector d is regular in codimension one.
In this paper we show that any smoothable complex projective variety, smooth in codimension two, with klt singularities and numerically trivial canonical class admits a finite cover, étale in codimension one, that decomposes as a product of an abelian variety, and singular analogues of irreducible Calabi-Yau and irreducible symplectic varieties.
In this article we develop a new way of systematically constructing infinitely many families of smooth subvarieties $X$ of any given dimension $m$, $m \geq 3$, and any given codimension in $\mathbb P^N$, embedded by complete subcanonical linear series, and, in particular, in the range of Hartshorne's conjecture. We accomplish this by showing the existence of everywhere non--reduced schemes called ropes, embedded in $\mathbb P^N$, and by smoothing them. In the range $3 \leq m < N/2$, we construct smooth subvarieties, embedded by complete subcanonical linear series, that are not complete interse…
In this article we obtain many results on the multiplicative structure constants of $T$-equivariant Grothendieck ring of the flag variety $G/B$. We do this by lifting the classes of the structure sheaves of Schubert varieties in $K_{T}(G/B)$ to $R(T)\otimes R(T)$, where $R(T)$ denotes the representation ring of the torus $T$. We further apply our results to describe the multiplicative structure constants of $K(X)_{\mathbb Q}$ where $X$ is the wonderful compactification of the adjoint group of $G$, in terms of the structure constants of Schubert varieties in the Grothendieck ring of $G/B$.
Let X be a complete toric variety with homogeneous coordinate ring S. In this article, we compute upper and lower bounds for the codimension in the critical degree of ideals of S generated by dim(X)+1 homogeneous polynomials that don't vanish simultaneously on X.
We compute Seshadri constants of a torus equivariant nef vector bundle on a projective space satisfying certain conditions. As an application, we compute Seshadri constants of tangent bundles on projective spaces. We also consider equivariant nef vector bundles on Bott towers of height 2 (i.e. Hirzebruch surfaces) and Bott towers of height 3 respectively. Assuming some conditions on the minimal slope of the restrictions of these bundles to invariant curves, we give precise values of Seshadri constant at an arbitrary point. We also give several examples illustrating our results.
In this paper, we define an action of the group of equivariant Cartier divisors on a toric variety X on the equivariant cycle groups of X, arising naturally from a choice of complement map on the underlying lattice. If X is nonsingular, this gives a lifting of the multiplication in equivariant cohomology to the level of equivariant cycles. As a consequence, one naturally obtains an equivariant cycle representative of the equivariant Todd class of any toric variety. These results extend to equivariant cohomology the results of Thomas and Pommersheim. In the case of a complement map arising from…
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 $c^{SM}(Ω_I)=κ_I$.
In this paper we use formal group rings to construct an algebraic model of the $T$-equivariant oriented cohomology of smooth toric varieties. Then we compare our model with known results of equivariant cohomology of toric varieties to justify our construction.
In our previous research, we constructed the affine varieties $Σ_{\mathbb{A}}^{13}$ and $Π_{\mathbb{A}}^{14}$ whose partial projectivizations admit $\mathbb{P}^{2}\times\mathbb{P}^{2}$-fibrations with relative Picard number one. In this paper, we produce prime quasi-smooth $\mathbb{Q}$-Fano 3-folds which are anticanonically embedded of codimension four and belong to 23 (resp.8) classes in the Graded Ring Database [GRDB], as weighted complete intersections in weighted projectivizations of $Σ_{\mathbb{A}}^{13}$ (resp.$Π_{\mathbb{A}}^{14}$ or its cone). We also show that a general member of the a…
In the present article we investigate ordinary and equivariant Rost motives. We provide an equivariant motivic decomposition of the variety X of full flags of a split semisimple algebraic group over a smooth base scheme, study torsion subgroup of the Chow group of twisted forms of X, define some equivariant Rost motives over a field and ordinary Rost motives over a general base scheme, and relate equivariant Rost motives with classifying spaces of some algebraic groups.
The aim of this text is to give an explicit formula for the restriction to fixed points of a basis of the equivariant K-theory of the flag varieties and of the Bott-Samelson 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 si…
