Let $G$ be a connected linear algebraic group over a field $k$ of characteristic zero. For a principal $G$-bundle $π: E \to X$ over a scheme $X$ of finite type over $k$ and a parabolic subgroup $P$ of $G$, we describe the rational algebraic cobordism and higher Chow groups of the flag bundle $E/P \to X$ in terms of the cobordism of $X$ and that of the classifying space of a maximal torus of $G$ contained in $P$. As a consequence, we also obtain the formula for the cobordism and higher Chow groups of the principal bundles over the scheme $X$. If $X$ is smooth, this describes the cobordism ring of these bundles in terms of the cobordism ring of $X$.
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.
Lee and Pandharipande studied a "double point" algebraic cobordism theory of varieties equipped with vector bundles, and speculated that some features of that story might extend to the case of varieties with principal G-bundles. This note shows that this expectation holds rationally, and more generally after inverting the torsion index of the group, for reductive G. We show that (after inverting the torsion index) the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism, that the theory for a point is dual to Omega^*(BG), and describe how the theory f…
We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We also construct actions of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.
The double point relation defines a natural theory of algebraic cobordism for bundles on varieties. We construct a simple basis (over the rationals) of the corresponding cobordism groups over Spec(C) for all dimensions of varieties and ranks of bundles. The basis consists of split bundles over products of projective spaces. Moreover, we prove the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism.
We prove a formula for the push-forward class of Bott-Samelson resolutions in the algebraic cobordism ring of the flag bundle. We specialise our formula to connective K-theory providing a geometric interpretation to the double beta-polynomials of Fomin and Kirillov by computing the fundamental classes of Schubert varieties. As a corollary we obtain a Thom-Porteous formula generalising those of the Chow ring and of the Grothendieck ring of vector bundles.
In this paper we provide generalisations of Pfaffian formulas for the degeneracy loci classes in the algebraic cobordism of symplectic/odd orthogonal Grassmann bundles.
The construction of double point cobordism groups of vector bundles on varieties in the work [Lee-P] (arXiv:1002.1500 [math.AG]) of Yuan-Pin Lee and Rahul Pandharipande gives immediately double point cobordism groups of filtered vector bundles on varieties. We note also that among the four basic operations -- direct sum, tensor product, dual, and Hom -- on vector bundles on varieties, only taking dual is compatible with double point cobordisms of vector bundles on varieties in general, by a demonstration on an example of vector bundles on Calabi-Yau 3-folds. A question on refined and/or higher…
In this paper, we construct stable Bott--Samelson classes in the projective limit of the algebraic cobordism rings of full flag varieties, upon an initial choice of a reduced word in a given dimension. Each stable Bott--Samelson class is represented by a bounded formal power series modulo symmetric functions in positive degree. We make some explicit computations for those power series in the case of infinitesimal cohomology. We also obtain a formula of the restriction of Bott--Samelson classes to smaller flag varieties.
The purpose of this paper is to study an extended version of bivariant derived algebraic cobordism where the cycles carry a vector bundle on the source as additional data. We show that, over a field of characteristic 0, this extends the analogous homological theory of Lee and Pandharipande constructed earlier. We then proceed to study in detail the restricted theory where only rank 1 vector bundles are allowed, and prove a weak version of projective bundle formula for bivariant cobordism. Since the proof of this theorem works very generally, we introduce precobordism theories over arbitrary No…
Based on the Brieskorn-Slodowy-Grothendieck diagram, we write the holomorphic structures (or filtrations) of the ADE Lie algebra bundles over the corresponding type ADE flag varieties, over the cotangent bundles of these flag varieties, and over the corresponding type $ADE$ singular surfaces. The main tool is the cohomology of line bundles over flag varieties and their cotangent bundles.
The main ingredient of the algebraic cobordism of M. Levine and F. Morel is a cobordism cycle of the form $(M \xrightarrow {h} X; L_1, \cdots, L_r)$ with a proper map $h$ from a smooth variety $M$ and line bundles $L_i$'s over $M$. In this paper we consider a \emph{cobordism bicycle} of a finite set of line bundles $(X \xleftarrow p V \xrightarrow s Y; L_1, \cdots, L_r)$ with a proper map $p$ and a smooth map $s$ and line bundles $L_i$'s over $V$. We will show that the Grothendieck group $\mathscr Z^*(X, Y)$ of the abelian monoid of the isomorphism classes of cobordism bicycles of finite sets …
The purpose of this article is to show that the bivariant algebraic $A$-cobordism groups considered previously by the author are independent of the chosen base ring $A$. This result is proven by analyzing the bivariant ideal generated by the so called snc relations, and, while the alternative characterization we obtain for this ideal is interesting by itself because of its simplicity, perhaps more importantly it allows us to easily extend the definition of bivariant algebraic cobordism to divisorial Noetherian derived schemes of finite Krull dimension. As an interesting corollary, we define th…
As a natural extension of the theory of uniform vector bundles on Fano manifolds, we consider uniform principal bundles, and study them by means of the associated flag bundles, as their natural projective geometric realizations. In this paper we develop the necessary background, and prove some theorems that are flag bundle counterparts of some of the central results in the theory of uniform vector bundles.
Let $X$ be a smooth projective curve defined over an algebraically closed field $k$, and let $E$ be a vector bundle on $X$. We compute the nef cone of any flag bundle associated to $E$.
We prove a formula for the push-forward class of Bott-Samelson resolutions in the algebraic cobordism ring of the flag bundle. We then provide a geometric interpretation to the double beta-polynomials of Fomin and Kirillov by specializing our formula to the case of connected K-theory.
We introduce analogs of the Kempf--Laksov desingularizations of Schubert bundles in (non-necessary Lagrangian) symplectic Grassmann bundles. In this setting, these are (possibly singular) irreducible flag bundles that are birational to Schubert bundles, and can be described as chains of zero-loci of regular sections in projectivized bundles. The orthogonal analogs are also presented. We immediatly derive universal Gysin formulas for isotropic Schubert bundles from these very constructions.
We give a short, geometric proof of Graham's theorem on positivity in the equivariant cohomology of a flag variety, based on a transversality argument.
In this note, we give Gysin formulas for partial flag bundles for the classical groups. We then give Gysin formulas for Schubert varieties in Grassmann bundles, including isotropic ones. All these formulas are proved in a rather uniform way by using the step-by-step construction of flag bundles and the Gysin formula for a projective bundle. In this way we obtain a comprehensive list of new universal formulas.
As an application of a recent characterization of complete flag manifolds as Fano manifolds having only ${\mathbb P}^1$-bundles as elementary contractions, we consider here the case of a Fano manifold $X$ of Picard number one supporting an unsplit family of rational curves whose subfamilies parametrizing curves through a fixed point are rational homogeneous, and we prove that $X$ is homogeneous. In order to do this, we first study minimal sections on flag bundles over the projective line, and discuss how Grothendieck's theorem on principal bundles allows us to describe a flag bundle upon some …
We construct and study a theory of bivariant cobordism of derived schemes. Our theory provides a vast generalization of the algebraic bordism theory of characteristic 0 algebraic schemes, constructed earlier by Levine and Morel, and a (partial) non-$\Ab^1$-invariant refinement of the motivic cohomology theory $MGL$ in Morel--Voevodsky's stable motivic homotopy theory. Our main result is that bivariant cobordism satisfies the projective bundle formula. As applications of this, we construct cobordism Chern classes of vector bundles, and establish a strong connection between the cobordism cohomol…
We study vector bundles on flag varieties over an algebraically closed field $k$. In the first part, we suppose $G=G_k(d,n)$ $(2\le d\leq n-d)$ to be the Grassmannian manifold parameterizing linear subspaces of dimension $d$ in $k^n$, where $k$ is an algebraically closed field of characteristic $p>0$. Let $E$ be a uniform vector bundle over $G$ of rank $r\le d$. We show that $E$ is either a direct sum of line bundles or a twist of a pull back of the universal bundle $H_d$ or its dual $H_d^{\vee}$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles…
We prove that every toric quiver flag variety $Y$ is isomorphic to a fine moduli space of cyclic modules over the algebra $\text{End}(T)$ for some tilting bundle $T$ on $Y$. This generalises the well known fact that $\mathbb{P}^n$ can be recovered from the endomorphism algebra of $\bigoplus_{0\leq i\leq n} \mathcal{O}_{\mathbb{P}^n}(i)$.
We prove that the complex cobordism class of any hyper-Kähler manifold of dimension $2n$ is a unique combination with rational coefficients of classes of products of punctual Hilbert schemes of $K3$ surfaces. We also prove a similar result using the generalized Kummer varieties instead of punctual Hilbert schemes. As a key step, we establish a closed formula for the top Chern character of their tangent bundles.
We study the problem of classifying projectivizations of rank-two vector bundles over ${\mathbb P}^2$ up to various notions of equivalence that arise naturally in ${\mathbb A}^1$-homotopy theory, namely ${\mathbb A}^1$-weak equivalence and ${\mathbb A}^1$-$h$-cobordism. First, we classify such varieties up to ${\mathbb A}^1$-weak equivalence: over algebraically closed fields having characteristic unequal to two the classification can be given in terms of characteristic classes of the underlying vector bundle. When the base field is ${\mathbb C}$, this classification result can be compared to a…
Let $V$ be a $K$-vector space of dimension $n+1$. In this paper, we focus our attention into the existence of irreducible homogeneous Ulrich bundles on flag manifolds $\FF(p, q,n)$ which parameterizes all chains of linear subspaces $L_{p} \subset L_{q} \subset \PP(V)$ of dimension $p< q$, respectively. We determine all irreducible homogeneous Ulrich bundles on $\FF(0,n-1,n)$ and we prove that there are exactly $2^{n-1}$. Similarly, we prove that $\FF(0,n-2,n)$ and $\FF(1,n-1,n)$ are also the support of irreducible homogeneous Ulrich bundles. On the other hand, we prove that $\FF(0,1,n)$ do not…
Let $P$ be a parabolic subgroup in $G=SL_n(\mathbf k)$, for $\mathbf k$ an algebraically closed field. We show that there is a $G$-stable closed subvariety of an affine Schubert variety in an affine partial flag variety which is a natural compactification of the cotangent bundle $T^*G/P$. Restricting this identification to the conormal variety $N^*X(w)$ of a Schubert divisor $X(w)$ in $G/P$, we show that there is a compactification of $N^*X(w)$ as an affine Schubert variety. It follows that $N^*X(w)$ is normal, Cohen-Macaulay, and Frobenius split.
A Fano manifold $X$ with nef tangent bundle is of flag-type if it has the same type of elementary contractions as a complete flag manifold. In this paper we present a method to associate a Dynkin diagram $\mathcal{D}(X)$ with any such $X$, based on the numerical properties of its contractions. We then show that $\mathcal{D}(X)$ is the Dynkin diagram of a semisimple Lie group. As an application we prove that Campana-Peternell conjecture holds when $X$ is a flag-type manifold whose Dynkin diagram is $A_n$ ($X$ is shown to be the complete flag in $\mathbb{P}^n$).
We give push-forward formulas for all flag bundles of types A, B, C, D. The formulas (and also the proofs) involve only Segre classes of the original vector bundles and characteristic classes of universal bundles. As an application, we provide new determinantal formulas.
Intersection rings of flag varieties and of isotropic flag varieties are generated by Chern classes of the tautological bundles modulo the relations coming from multiplicativity of total Chern classes. In this paper we describe the Groebner bases of the ideals of relations and give applications to computation of intersections, as implemented in Macaulay2.
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.
We show that there is a ${SL_n}$-stable closed subset of an affine Schubert variety in the infinite dimensional Flag variety (associated to the Kac-Moody group ${\widehat{SL_n}}$) which is a natural compactification of the cotangent bundle to the finite-dimensional Flag variety ${SL_n/B}$.
We express the diagonals of projective, Grassmann and, more generally, flag bundles of type (A) using the zero schemes of some vector bundle sections, and do the same for their single point subschemes. We discuss diagonal and point properties of these flag bundles. We study when the complex manifolds G/B for other groups have the point and diagonal properties. We discuss explicit formulas for the classes of diagonals of the varieties G/B.
In this paper, we study Segre classes in algebraic cobordism. We also prove a generalization of Kempf-Laksov formula for the degeneracy loci classes in the algebraic cobordism of the Grassmannian bundle.
Up to a factor 1/n!, the volume of a big line bundle agrees with the Euclidean volume of its Okounkov body. The latter is the convex hull of top rank valuation vectors of sections, all with respect to a single flag. In this text we give a different volume formula, valid in the ample cone, also based on top rank valuation vectors, but mixing data along several different flags.
This note gives a simple formula for the unique asymptotically conical Calabi-Yau metrics on the canonical bundle of a flag variety known to exist by the work of R. Goto and others. This is done by generalizing the well known Calabi Ansatz to general Kähler classes. We give some examples of explicit families, in particular, a formula for the two dimensional family of asymptotically conical metrics on the canonical bundle of $F_{1,2}$.
Let L be a homogeneous ample line bundle on any flag variety G/P and let T be a maximal torus of G. We prove a general necessary and sufficient condition for L to descend as a line bundle on the GIT quotient of G/P by T. We use this result to explicitly determine exactly which L descend to the GIT quotient for any simple complex algebraic group G and any parabolic subgroup P.
In this paper, we study equivariant vector bundles on partial flag varieties arising from Schur functors. We show that a partial flag variety with three or more steps does not admit an Ulrich bundle of this form with respect to the minimal ample class. We classify Ulrich bundles of this form on two-step flag varieties F(2,n;n+1), F(2,n;n+2), F(k,k+1;n), and F(k,k+2;n). We give a conjectural description of the two-step flag varieties which admit such Ulrich bundles. Our results provide counterexamples to conjectures by Costa and Miro-Roig.
In this paper, we study equivariant vector bundles on partial flag varieties arising from Schur functors. We show that a partial flag variety with three or more steps does not admit an Ulrich bundle of this form with respect to the minimal ample class. We classify Ulrich bundles of this form on two-step flag varieties F(1,n-1;n), F(2,n-1;n), F(2,n-2;n), F(k,k+1;n) and F(k,k+2;n). We give a conjectural description of the two-step flag varieties which admit such Ulrich bundles.
Let $P$ be a parabolic subgroup in $SL_n(\mathbb C)$. We show that there is a $SL_n(\mathbb C)$-stable closed subvariety of an affine Schubert variety in an infinite dimensional partial Flag variety (associated to the Kac-Moody group ${\widehat{SL_n}(\mathbb C)}$) which is a natural compactification of the cotangent bundle to $SL_n(\mathbb C)/P$. As a consequence, we recover the Springer resolution for any orbit closure inside the variety of nilpotent matrices.
We define a variety of doubly indexed flags, this is a smooth, projective variety, and we describe it as an iterated over Grassmannian varieties. On the other hand, we consider the variety of partial flags which are stabilized by a given nilpotent endomorphism. We partition this variety into locally closed subvarieties which are vector bundles over varieties of the aforedmentioned type.
We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a vanishing theorem for pull back of line bundles to the cotangent bundle (proved for the classical groups and G_2 by Andersen and Jantzen and in characteristic zero by B. Broer (for all groups)), normality and rational singularities for the subregular nilpotent variety and good filtrations of the global sections of pull backs of line bundles to the cotangent bund…
We study the null-cone of a semi-simple algebraic group acting on a number of copies of its Lie algebra via the diagonal adjoint action. We show that the null cone has rational singularities in the case of SL3. We observe by example that the null-cone is not normal in general and that the normalization of the null-cone does not have rational singularities in general. This is achieved by computing cohomology of certain vector bundles on flag varieties.
Hartshorne in "Ample vector bundles" proved that $E$ is ample if and only if $\OOO_{P(E)}(1)$ is ample. Here we generalize this result to flag manifolds associated to a vector bundle $E$ on a complex manifold $X$: For a partition $a$ we show that the line bundle $\it Q_a^s$ on the corresponding flag manifold $\mathcal{F}l_s(E)$ is ample if and only if $ \SSS_aE $ is ample. In particular $\det Q$ on $\it{G}_r(E)$ is ample if and only if $\wedge ^rE$ is ample.\\ We give also a proof of the Ampleness Dominance theorem that does not depend on the saturation property of the Littlewood-Richardson se…
We introduce an equivariant algebraic cobordism theory Ω^G for algebraic varieties with G-action, where G is a split diagonalizable group scheme over a field k. It is done by combining the construction of the algebraic cobordism theory Ωby F. Morel and M. Levine, with the notion of (G, F)-formal group law with respect to a complete G-universe and complete G-flag F as introduced by M. Cole, J. P. C. Greenlees and I. Kriz. In particular, we use their corresponding representing ring L_G(F) in place of the Lazard ring L. We show that localization property and homotopy invariance property hold in Ω…
A real semisimple Lie group G_0 embedded in its complexification G has only finitely many orbits in any G-fag manifold Z = G/Q. The complex geometry of its open orbits D (flag domains) is studied from the point of view of compact complex submanifolds C (cycles) which arise as orbits of certain distinguished subgroups. Normal bundles E of the cycles are analyzed in some detail. It is shown that E is trivial if and only if D is holomorphically convex, in fact a product of C and a Hermitian symmetric space, and otherwise D is pseudoconcave.
We describe hypergeometric solutions of the quantum differential equation of the cotangent bundle of a gl_n partial flag variety. These hypergeometric solutions manifest the Landau-Ginzburg mirror symmetry for the cotangent bundle of a partial flag variety.
