Alternating signs of quiver coefficients

We prove a formula for the Grothendieck class of a quiver variety, which generalizes the cohomological component formulas of Knutson, Miller, and Shimozono. Our formula implies that the -theoretic quiver coefficients have alternating signs and gives an explicit combinatorial formula for these coefficients. We also prove some new variants of the factor sequences conjecture and a conjecture of Knutson, Miller, and Shimozono, which states that their double ratio formula agrees with the original quiver formulas of the author and Fulton. For completeness we include a short proof of the ratio formula.

     

Positivity of quiver coefficients through Thom polynomials

We use the Thom Polynomial theory developed by Fehér and Rimányi to prove the component formula for quiver varieties conjectured by Knutson, Miller, and Shimozono. This formula expresses the cohomology class of a quiver variety as a sum of products of Schubert polynomials indexed by minimal lace diagrams, and implies that the quiver coefficients of Buch and Fulton are non-negative. We also apply our methods to give a new proof of the component formula from the Gröbner degeneration of quiver varieties, and to give generating moves for the KMS-factorizations that form the index set in K-theoretic versions of the component formula.     

The quantum orbifold cohomology of weighted projective spaces

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S ¹-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.     

Four positive formulae for type A quiver polynomials

We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99 ]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, meaning that their summands have coefficient 1 and correspond bijectively to components of a torus-invariant scheme. The remaining (presently non-geometric) formula is a variant of the conjecture of Buch and Fulton in terms of factor sequences of Young tableaux [BF99 ]; our proof of it proceeds by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in “doubled” versions, two for double quiver polynomials, and the other two for their stable limits, the double quiver functions, where setting half the variables equal to the other half specializes to the ordinary case. Our method begins by identifying quiver polynomials as multidegrees [BB82 , Jos84 , BB85 , Ros89 ] via equivariant Chow groups [EG98 ]. Then we make use of Zelevinsky’s map from quiver loci to open subvarieties of Schubert varieties in partial flag manifolds [Zel85 ]. Interpreted in equivariant cohomology, this lets us write double quiver polynomials as ratios of double Schubert polynomials [LS82 ] associated to Zelevinsky permutations; this is our first formula. In the process, we provide a simple argument that Zelevinsky maps are scheme-theoretic isomorphisms (originally proved in [LM98 ]). Writing double Schubert polynomials in terms of pipe dreams [FK96 ] then provides another geometric formula for double quiver polynomials, via [KM05 ]. The combinatorics of pipe dreams for Zelevinsky permutations implies an expression for limits of double quiver polynomials in terms of products of Stanley symmetric functions [Sta84 ]. A degeneration of quiver loci (orbit closures of GL on quiver representations) to unions of products of matrix Schubert varieties [Ful92 , KM05 ] identifies the summands in our Stanley function formula combinatorially, as lacing diagrams that we construct based on the strands of Abeasis and Del Fra in the representation theory of quivers [AD80 ]. Finally, we apply the combinatorial theory of key polynomials to pass from our lacing diagram formula to a double Schur function formula in terms of peelable tableaux [RS95a , RS98 ], and from there to our formula of Buch–Fulton type.     

Unramified cohomology of degree 3 and Noether’s problem

Let G be a finite group and W be a faithful representation of G over C. The group G acts on the field of rational functions C(W). The question whether the field of invariant functions C(W) ^G is purely transcendental over C goes back to Emmy Noether. Using the unramified cohomology group of degree 2 of this field as an invariant, Saltman gave the first examples for which C(W) ^G is not rational over C. Around 1986, Bogomolov gave a formula which expresses this cohomology group in terms of the cohomology of the group G. In this paper, we prove a formula for the prime to 2 part of the unramified cohomology group of degree 3 of C(W) ^G . Specializing to the case where G is a central extension of an F _p -vector space by another, we get a method to construct nontrivial elements in this unramified cohomology group. In this way we get an example of a group G for which the field C(W) ^G is not rational although its unramified cohomology group of degree 2 is trivial. Dedicated to Jean-Louis Colliot-Thélène.     

Twisted Bimodules and Hochschild Cohomology for Self-injective Algebras of Class A n , II

Up to derived equivalence, the representation-finite self-injective algebras of class A _n are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Möbius algebras. In Part I (Forum Math. 11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Möbius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Möbius algebras.     

The Geisser-Levine method revisited and algebraic cycles over a finite field

We reformulate part of the arguments of T. Geisser and M. Levine relating motivic cohomology with finite coefficients to truncated étale cohomology with finite coefficients [9,10]. This reformulation amounts to a uniqueness theorem for motivic cohomology, and shows that the Geisser-Levine method can be applied generally to compare motivic cohomology with other types of cohomology theories. We apply this to prove an equivalence between conjectures of Tate and Beilinson on cycles in characteristic p and a vanishing conjecture for continuous étale cohomology. Received: 23 November 2000 / Published online: 5 September 2002     

Geometric cycles with local coefficients and the cohomology of arithmetic subgroups of the exceptional group <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

We study the cohomology of a compact locally symmetric space attached to an arithmetic subgroup of a rational form of a group of type G ₂ with values in a finite dimensional irreducible representation E of G ₂. By constructing suitable geometric cycles and parallel sections of the bundle [(E)\tilde]{\tilde{E}} we prove non-vanishing results for this cohomology. We prove all possible non-vanishing results compatible with the known vanishing theorems regarding unitary representations with non-zero cohomology in the case of the short fundamental weight of G ₂. A decisive tool in our approach is a formula for the intersection numbers with local coefficients of two geometric cycles.     

Decomposition theorem for the cd-index of Gorenstein<Superscript>*</Superscript> posets

We prove a decomposition theorem for the cd-index of a Gorenstein^* poset analogous to the decomposition theorem for the intersection cohomology of a toric variety. From this we settle a conjecture of Stanley that the cd-index of Gorenstein* lattices is minimized on Boolean algebras.     

On Combinatorics of Quiver Component Formulas

Buch and Fulton [9] conjectured the nonnegativity of the quiver coefficients appearing in their formula for a quiver cycle. Knutson, Miller and Shimozono [24] proved this conjecture as an immediate consequence of their component formula. We present an alternative proof of the component formula by substituting combinatorics for Gröbner degeneration [23, 24]. We relate the component formula to the work of Buch, Kresch, Tamvakis and the author [10] where a splitting formula for Schubert polynomials in terms of quiver coefficients was obtained. We prove analogues of this latter result for the type BCD-Schubert polynomials of Billey and Haiman [4]. The form of these analogues indicate that it should be interesting to pursue a geometric context that explains them.     

The Asymptotic Behavior of the Joint Linear Complexity Profile of Multisequences

We prove a conjecture on the asymptotic behavior of the joint linear complexity profile of random multisequences over a finite field. This conjecture was previously shown only in the special cases of single sequences and pairs of sequences. We also establish an asymptotic formula for the expected value of the nth joint linear complexity of random multisequences over a finite field. Some more precise results are shown for triples of sequences.     

The integral Novikov conjectures for S-arithmetic groups I

We prove the integral Novikov conjecture for torsion free S-arithmetic subgroups Γ of linear reductive algebraic groups G of rank 0 over a global field k. They form a natural class of groups and are in general not discrete subgroups of Lie groups with finitely many connected components. Since many natural S-arithmetic subgroups contain torsion elements, we also prove a generalized integral Novikov conjecture for S-arithmetic subgroups of such algebraic groups, which contain torsion elements. These S-arithmetic subgroups also provide a natural class of groups with cofinite universal spaces for proper actions. Partially Supported by NSF grants DMS 0405884 and 0604878.     

Coniveau 2 Complete Intersections and Effective Cones

Griffiths computation of the Hodge filtration on the cohomology of a smooth hypersurface X of degree d in \mathbbPⁿ{\mathbb{P}^n} shows that it has coniveau ≥ c once n ≥ dc. The generalized Hodge conjecture (GHC) predicts that the cohomology of X is then supported on a closed algebraic subset of codimension at least c. This is essentially unknown for c ≥ 2. In the case where c = 2, we exhibit a geometric phenomenon in the variety of lines of X explaining the estimate for the coniveau, and show that (GHC) would be implied in this case by the following conjecture on effective cones of cycles of intermediate dimension: Very moving subvarieties have their class in the interior of the effective cone.     

Linear recurrence relations for cluster variables of affine quivers

We prove that the frieze sequences of cluster variables associated with the vertices of an affine quiver satisfy linear recurrence relations. In particular, we obtain a proof of a recent conjecture by Assem–Reutenauer–Smith.     

The structure of 2D semi-simple field theories

I classify the cohomological 2D field theories based on a semi-simple complex Frobenius algebra A. They are controlled by a linear combination of κ-classes and by an extension datum to the Deligne–Mumford boundary. Their effect on the Gromov–Witten potential is described by Givental’s Fock space formulae. This leads to the reconstruction of Gromov–Witten (ancestor) invariants from the quantum cup-product at a single semi-simple point and the first Chern class of the manifold, confirming Givental’s higher-genus reconstruction conjecture. This in turn implies the Virasoro conjecture for manifolds with semi-simple quantum cohomology. The classification uses the Mumford conjecture, proved by Madsen and Weiss (European Congress of Mathematics, pp. 283–303, 2005).     

Curves of genus g on an abelian variety of dimension g

In this paper we prove a general theorem concerning the number of translation classes of curves of genus g belonging to a fixed cohomology class in a polarized abelian variety of dimension g. For g = 2 we recover results of Göttsche and Bryan-Leung. For g = 3 we deduce explicit numbers for these classes.     

Homological stability for the mapping class groups of non-orientable surfaces

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the infinite loop space of a Thom spectrum built from the canonical bundle over the Grassmannians of 2-planes in ℝ ⁿ⁺². In particular, we show that the stable rational cohomology is a polynomial algebra on generators in degrees 4i – this is the non-oriented analogue of the Mumford conjecture.     

Bisymplectic Grassmannians of planes

The bisymplectic Grassmannian \({{\,\mathrm{I_2Gr}\,}}(k,V)\) parametrizes k-dimensional subspaces of a vector space V which are isotropic with respect to two general skew-symmetric forms; it is a Fano projective variety which admits an action of a torus with a finite number of fixed points. In this work, we study its equivariant cohomology with complex coefficients when \(k=2\); the central result of the paper is an equivariant Chevalley formula for the multiplication of the hyperplane class by any Schubert class. Moreover, we study in detail the case of \({{\,\mathrm{I_2Gr}\,}}(2, {\mathbb {C}}^6)\), which is a quasi-homogeneous variety, we analyse its deformations, and we give a presentation of its cohomology.

     

On Maximal Diagonalizable Lie Subalgebras of the First Hochschild Cohomology

Let A be a basic connected finite dimensional algebra over an algebraically closed field, with ordinary quiver without oriented cycles. Given a presentation of A by quiver and admissible relations, Assem and de la Peña have constructed an embedding of the space of additive characters of the fundamental group of the presentation into the first Hochschild cohomology group of A. We compare the embeddings given by the different presentations of A. In some situations, we characterise the images of these embeddings in terms of (maximal) diagonalizable subalgebras of the first Hochschild cohomology group (endowed with its Lie algebra structure).     

Ramification theory for varieties over a local field

We define generalizations of classical invariants of wild ramification for coverings on a variety of arbitrary dimension over a local field. For an ?-adic sheaf, we define its Swan class as a 0-cycle class supported on the wild ramification locus. We prove a formula of Riemann-Roch type for the Swan conductor of cohomology together with its relative version, assuming that the local field is of mixed characteristic. We also prove the integrality of the Swan class for curves over a local field as a generalization of the Hasse-Arf theorem. We derive a proof of a conjecture of Serre on the Artin character for a group action with an isolated fixed point on a regular local ring, assuming the dimension is 2.