Counter-examples to a conjecture of Karpenko for spin groups

On the stable Andreadakis problem

Over any smooth algebraic variety over a p-adic local field k, we construct the de Rham comparison isomorphisms for the étale cohomology with partial compact support of de Rham \({\mathbb {Z}}_p\)-local systems, and show that they are compatible with Poincaré duality and with the canonical morphisms among such cohomology. We deduce these results from their analogues for rigid analytic varieties that are Zariski open in some proper smooth rigid analytic varieties over k. In particular, we prove finiteness of étale cohomology with partial compact support of any \({\mathbb {Z}}_p\)-local systems, and establish the Poincaré duality for such cohomology after inverting p.

Let G be a split semisimple algebraic group over a field k and let X be the flag variety (i.e., the variety of Borel subgroups) of G twisted by a generic G‐torsor. We start a systematic study of the conjecture, raised in in form of a question, that the canonical epimorphism of the Chow ring of X onto the associated graded ring of the topological filtration on the Grothendieck ring of X is an isomorphism. Since the topological filtration in this case is known to coincide with the computable gamma filtration, this conjecture indicates a way to compute the Chow ring. We reduce its proof to the case of . For simply‐connected or adjoint G, we reduce the proof to the case of simple G. Finally, we provide a list of types of simple groups for which the conjecture holds. Besides of some classical types considered previously (namely, A, C, and the special orthogonal groups of types B and D), the list contains the exceptional types G2, F4, and simply‐connected E6.

We consider all Bott-Samelson varieties BS(s) for a fixed connected semisimple complex algebraic group with maximal torus T as the class of objects of some category. The class of morphisms of this category is an extension of the class of canonical (inserting the neutral element) morphisms BS(s)↪BS(s′), where s is a subsequence of s′. Every morphism of the new category induces a map between the T-fixed points but not necessarily between the whole varieties. We construct a contravariant functor from this new category to the category of graded \(\phantom {\dot {i}\!}H^{\bullet }_{T}(\text {pt})\)-modules coinciding on the objects with the usual functor \(\phantom {\dot {i}\!}H_{T}^{\bullet }\) of taking T-equivariant cohomologies. We also discuss the problem how to define a functor to the category of T-spaces from a smaller subcategory. The exact answer is obtained for groups whose root systems have simply laced irreducible components by explicitly constructing morphisms between Bott-Samelson varieties (different from the canonical ones).

A remark on connective K-theory

We show a canonical injective morphism from the quantum cohomology ring QH∗(G/P ) to the associated graded algebra of QH∗(G/B), which is with respect to a nice filtration on QH∗(G/B) introduced by Leung and the author. This tells us the vanishing of a lot of genus zero, three-pointed Gromov-Witten invariants of flag varieties G/P .

Let X be the variety of Borel subgroups of a split semisimple algebraic group G over a field, twisted by a generic G-torsor. Conjecturally, the canonical epimorphism of the Chow ring $$\mathop {\mathrm {CH}}\nolimits X$$ onto the associated graded ring GK(X) of the topological filtration on the Grothendieck ring K(X) is an isomorphism. We prove the new cases $$G={\text {Spin}}(11)$$ and $$G={\text {Spin}}(12)$$ of this conjecture. On an equivalent note, we compute the Chow ring $$\mathop {\mathrm {CH}}\nolimits Y$$ of the highest orthogonal grassmannian Y for the generic 11- and 12-dimensional quadratic forms of trivial discriminant and Clifford invariant. In particular, we describe the torsion subgroup of the Chow group $$\mathop {\mathrm {CH}}\nolimits Y$$ and determine its order which is equal to $$16\;777\; 216$$ . On the other hand, we show that the Chow group $$\mathop {\mathrm {CH}}\nolimits _0Y$$ of 0-cycles on Y is torsion-free.

1991 Mathematics Subject Classification 32C38.

A Riemannian manifold (M, g) is called Einstein, if it has constant Ricci curvature. It is well known that if (M=G/K, g) is a compact homogeneous Riemannian manifold, then the G-invariant \tl{Einstein} metrics of unit volume, are the critical points of the scalar curvature function restricted to the space of all G-invariant metrics with volume 1. For a G-invariant Riemannian metric the Einstein equation reduces to a system of algebraic equations. The positive real solutions of this system are the $G$-invariant Einstein metrics on M. An important family of compact homogeneous spaces consists of the generalized flag manifolds. These are adjoint orbits of a compact semisimple Lie group. Flag manifolds of a compact connected semisimple Lie group exhaust all compact and simply connected homogeneous Kahler manifolds and are of the form G/C(S), where C(S) is the centralizer (in G) of a torus S in G. Such homogeneous spaces admit a finite number of G-invariant complex structures, and for any such complex structure there is a unique compatible G-invariant Kahler-Einstein metric. In this thesis we classify all flag manifolds M=G/K of a compact simple Lie group G, whose isotropy representation decomposes into 2 or 4, isotropy summands. For these spaces we solve the (homogeneous) Einstein equation, and we obtain the explicit form of new G-invariant Einstein metrics. For most cases we give the classification of homogeneous Einstein metrics. We also examine the isometric problem. For the construction of the Einstein equation on certain flag manifolds with four isotropy summands, we apply for first time the twistor fibration of a flag manifold over an isotropy irreducible symmetric space of compact type. This method is new and it can be used also for other flag manifolds. For flag manifolds with two isotropy summands, we use the restricted Hessian and we characterize the new Einstein metrics as local minimum points of the scalar curvature function restricted to the space of G-invariant Riemannian metrics of volume 1. We mention that the classification of flag manifolds with two isotropy summands gives us new examples of homogeneous spaces, for which the motion of a charged particle under the electromagnetic field, and the geodesics curves, are completely determined.

We revisit the classical two-dimensional McKay correspondence in two respects: The first one, which is the main point of this work, is that we take into account of the multiplicative structure given by the orbifold product; second, instead of using cohomology, we deal with the Chow motives. More precisely, we prove that for any smooth proper two-dimensional orbifold with projective coarse moduli space, there is an isomorphism of algebra objects, in the category of complex Chow motives, between the motive of the minimal resolution and the orbifold motive. In particular, the complex Chow ring (resp. Grothendieck ring, cohomology ring, topological K-theory) of the minimal resolution is isomorphic to the complex orbifold Chow ring (resp. Grothendieck ring, cohomology ring, topological K-theory) of the orbifold surface. This confirms the two-dimensional Motivic Crepant Resolution Conjecture.

We describe the point class and Todd class in the Chow ring of a moduli space of quiver representations, building on a result of Ellingsrud–Strømme. This, together with the presentation of the Chow ring by the second author, makes it possible to compute integrals on quiver moduli. To do so, we construct a canonical morphism of universal representations in great generality, and along the way point out its relation to the Kodaira–Spencer morphism. We illustrate the results by computing some invariants of some “small” Kronecker moduli spaces. We also prove that the first non-trivial (6-dimensional) Kronecker moduli space is isomorphic to the zero locus of a general section of $\mathcal{Q}^{\vee }(1)$ on $\textrm{Gr}(2,8)$.

Let F be the complete flag variety over Spec(Z) with the tautological filtration 0 \subset E_1 \subset E_2 \subset ... \subset E_n=E of the trivial bundle E over F. The trivial hermitian metric on E(\C) induces metrics on the quotient line bundles L_i(\C). Let \hat{c}_1(L_i) be the first Chern class of L_i in the arithmetic Chow ring \hat{CH}(F) and x_i = -\hat{c}_1(L_i). Let h(X_1,...,X_n) be a polynomial with integral coefficients in the ideal generated by the elementary symmetric polynomials e_i. We give an effective algorithm for computing the arithmetic intersection h(x_1,...,x_n) in \hat{CH}(F), as the class of a SU(n)-invariant differential form on F(\C). In particular we show that all the arithmetic Chern numbers one obtains are rational numbers. The results are true for partial flag varieties and generalize those of Maillot for grassmannians. An `arithmetic Schubert calculus' is established for an `invariant arithmetic Chow ring' which specializes to the Arakelov Chow ring in the grassmannian case.

We introduce and study a filtration on the representation ring $R(G)$ of an affine algebraic group $G$ over a field. This filtration, which we call Chow filtration, is an analogue of the coniveau filtration on the Grothendieck ring of a smooth variety. We compare it with the other known filtrations on $R(G)$ and show that all three define on $R(G)$ the same topology. For any $n\geq 1$, we compute the Chow filtration on $R(G)$ for the special orthogonal group $G:=O^+(2n+1)$. In particular, we show that the graded group associated with the filtration is torsion-free. On the other hand, the Chow ring of the classifying space of $G$ over any field of characteristic $\ne 2$ is known to contain non-zero torsion elements. As a consequence, any sufficiently good approximation of the classifying space yields an example of a smooth quasi-projective variety $X$ such that its Chow ring is generated by Chern classes and at the same time contains non-zero elements vanishing under the canonical homomorphism onto the graded ring associated with the coniveau filtration on the Grothendieck ring of $X$.

The classical cycle class map for a smooth complex variety sends cycles in the Chow ring to cycles in the singular cohomology ring. We study two cycle class maps for smooth real varieties: the map from the I-cohomology ring to singular cohomology induced by the signature, and a new cycle class map defined on the Chow-Witt ring. For both maps, we establish basic compatibility results like compatibility with pullbacks, pushforwards and cup products. As a first application of these general results, we show that both cycle class maps are isomorphisms for cellular varieties.

The Chow Moving Lemma is a theorem which asserts that a given algebraic s-cycle on a smooth algebraic variety X can be moved within its rational equivalence class to intersect properly a given r-cycle on X provided that r + s ≥ dim(X) (cf. [Chow], [S2]). In the past few years, there has been considerable interest in studying spaces of algebraic cycles rather than simply cycles modulo an equivalence relation. With this in mind, it is natural to ask whether one can move a given “bounded family” of s-cycles on the smooth variety X to intersect properly a given “bounded family” of r-cycles. The main point of this paper is to formulate and prove just such a result. In Theorem 3.1, we demonstrate that for any integer e and any smooth projective variety X, one can simultaniously and algebraically “move” all effective s-cycles of degree ≤ e on X so that each such cycle meets every effective r-cycle of degree ≤ e on X in proper dimension. The primary motivation for this Moving Lemma for Cycles of Bounded Degree was the possibility of a duality theorem between cohomology and homology theories defined in terms of homotopy groups of cycle spaces. Using Theorem 3.1, we have proved such a duality theorem for complex quasi-projective varieties in [F-L2]. We prove our Moving Lemma for varieties over an arbitary infinite field, permitting a proof in [F-V] of a duality theorem for “motivic cohomology and homology”. The reader will find that our Moving Lemma has numerous good properties. First of all, the move is given as an algebraic move (parametrized by a punctured projective line) on Chow varietes. Although this move is “good” only for s-cycles of bounded degree, it is defined on all effective s-cycles. Moreover, the move starts at “time 0” by expressing an effective s-cycle Z as a difference of effective s-cycles both of which have intersection properties no worse than Z. Finally, our Moving Lemma is applicable to smooth quasiprojective varieties, for it is stated for a possibly singular projective variety X resulting in a conclusion of proper intersection off the singular locus of X. The classical motivation for the moving lemma was to define an intersection product on algebraic cycles modulo rational equivalence, thereby establishing the Chow ring A∗(X). Some of the classical literature overlooked the question of whether or not intersection of cycles defined via a moving lemma is independent of the move (e.g., [Chow], [R], [S2]; on the other hand, cf. [Chev], [S3])). One direct consequence of our Moving Lemma is a proof for smooth quasi-projective varieties that the intersection product is indeed well defined independent of the choice of move (Theorem 3.4). Of course, the intersection product now has an intrinsic formulation for all smooth algebraic varieties due to Fulton