Showing papers in "Compositio Mathematica in 2001"

Reconstruction of a variety from the derived category and groups of autoequivalences

Abstract: We consider smooth algebraic varieties with ample either canonical or anticanonical sheaf. We prove that such a variety is uniquely determined by its derived category of coherent sheaves. We also calculate the group of exact autoequivalences for these categories. The technics of ample sequences in Abelian categories is used.

Geometry of the Moment Map for Representations of Quivers

Abstract: We study the moment map associated to the cotangent bundle of the space of representations of a quiver, determining when it is flat, and giving a stratification of its Marsden–Weinstein reductions. In order to do this we determine the possible dimension vectors of simple representations of deformed preprojective algebras. In an appendix we use deformed preprojective algebras to give a simple proof of much of Kac's Theorem on representations of quivers in characteristic zero.

Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

Abstract: We prove the genus zero part of the generalized Witten conjecture, relating moduli spaces of higher spin curves to Gelfand–Dickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdVr equations. We formulate axioms for a cohomology class on this moduli space which allow one to construct a cohomological field theory of rank r−1 in all genera. In genus zero it produces a Frobenius manifold which is isomorphic to the Frobenius manifold structure on the base of the versal deformation of the singularity Ar−1. We prove analogs of the puncture, dilaton, and topological recursion relations by drawing an analogy with the construction of Gromov–Witten invariants and quantum cohomology.

Local Systems on $\mathbb P$1 − S for S a Finite Set

Abstract: I give the necessary and sufficient conditions for the existence of Unitary local systems with prescribed local monodromies on P1 − S where S is a finite set. This is used to give an algorithm to decide if a rigid local system on P1 − S has finite global monodromy, thereby answering a question of N. Katz. The methods of this article (use of Harder–Narasimhan filtrations) are used to strengthen Klyachko's theorem on sums of Hermitian matrices. In the Appendix, I give a reformulation of Mehta–Seshadri theorem in the SU(n) setting.

Derived picard groups of finite dimensional hereditary algebras

Abstract: Let A be a finite-dimensional algebra over a field k. The derived Picard group DPick(A) is the group of triangle auto-equivalences of D> b( mod A) induced by two-sided tilting complexes. We study the group DPick(A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPick, as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPick(A) on a certain infinite quiver Γirr. This representation is faithful when the quiver Δ of A is a tree, and then DPick(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPick(A). When A is hereditary, DPick(A) coincides with the full group of k-linear triangle auto-equivalences of Db( mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to Db( mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.

Gröbner Flags and Gorenstein Algebras

Abstract: The goal of this paper is to study the Koszul property and the property of having a Grobner basis of quadrics for classical varieties and algebras as canonical curves, finite sets of points and Artinian Gorenstein algebras with socle in low degree. Our approach is based on the notion of Grobner flags and Koszul filtrations. The main results are the existence of a Grobner basis of quadrics for the ideal of the canonical curve whenever it is defined by quadrics, the existence of a Grobner basis of quadrics for the defining ideal of s ≤ 2n points in general linear position in P n , and the Koszul property of the ‘generic’ Artinian Gorenstein algebra of socle degree 3.

The p-Rank of Ramified Covers of Curves

Abstract: In this paper we study the p-rank of Abelian prime-to-p covers of the generic r-pointed curve of genus g. There is an obvious bound on the p-rank of the cover. We show that it suffices to compute the p-rank of cyclic prime-to-p covers of the generic r-pointed curve of genus zero. In that situation, we show that, for large p, the p-rank of the cover is equal to the bound.

Syzygies of Veronese Embeddings

Abstract: We prove that the Veronese embedding ϕOℙn(d):ℙn↪ ℙN with n≥2, d≥3 does not satisfy property Np (according to Green and Lazarsfeld) if p≥3d−2. We make the conjecture that also the converse holds. This is true for n=2 and for n=d=3.

On Birational Maps and Jacobian Matrices

Abstract: One is concerned with Cremona-like transformations, i.e., rational maps from ℙn to ℙm that are birational onto the image Y ⊂ ℙm and, moreover, the inverse map from Y to ℙn lifts to ℙm. We establish a handy criterion of birationality in terms of certain syzygies and ranks of appropriate matrices and, moreover, give an effective method to explicitly obtaining the inverse map. A handful of classes of Cremona and Cremona-like transformations follow as applications.

Some Results on Secant Varieties Leading to a Geometric Flip Construction

Abstract: We study the relationship between the equations defining a projective variety and properties of its secant varieties. In particular, we use information about the syzygies among the defining equations to derive smoothness and normality statements about SecX and also to obtain information about linear systems on the blow up of projective space along a variety X. We use these results to geometrically construct, for varieties of arbitrary dimension, a flip first described in the case of curves by M. Thaddeus via Geometric Invariant Theory.

Howe Duality for Lie Superalgebras

Abstract: We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicity-free decomposition of a symmetric and skew-symmetric algebra (in the super sense) under the action of the dual pair and present explicit formulas for the highest-weight vectors in each isotypic subspace of the symmetric algebra. We give an explicit multiplicity-free decomposition into irreducible gl(m|n)-modules of the symmetric and skew-symmetric algebras of the symmetric square of the natural representation of gl(m|n). In the former case, we also find explicit formulas for the highest-weight vectors. Our work unifies and generalizes the classical results in symmetric and skew-symmetric models and admits several applications.

Euler–Poincaré Pairings and Elliptic Representations of Weyl Groups and p-Adic Groups

Abstract: The space of elliptic virtual representations of a p-adic group is endowed with a natural inner product EP( , ), defined analytically by Kazhdan and homologically by Schneider–Stuhler. Arthur has computed EP in terms of analytic R-groups. For Iwahori spherical representations, we show that EP can also be expressed in terms of a corresponding inner product on space of elliptic virtual representations of Weyl groups. This leads to an explicit description of both elliptic representation theories, in terms of the Kazhdan–Lusztig and Springer correspondences

The Sebastiani–Thom Isomorphism in the Derived Category

Abstract: We prove that the Sebastiani–Thom isomorphism for Milnor fibres and their monodromies exists as a natural isomorphism between vanishing cycles in the derived category.

On Central Elements in the Universal Enveloping Algebras of the Orthogonal Lie Algebras

Abstract: We present an analogy of the famous formula that the square of the Pfaffian is equal to the determinant for an alternating matrix for the case where the entries are the generators of the orthogonal Lie algebras. This identity clarifies the relation between the two sets of central elements in the enveloping algebra of the orthogonal Lie algebras. We employ systematically the exterior calculus for the proof.

Tamely Ramified Towers and Discriminant Bounds for Number Fields

Abstract: The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R 2m be the minimal root discriminant for totally complex number fields of degree 2m, and put α0 = lim inf m R 2m . One knows that α0 ≥ 4πe γ ≈ 22.3, and, assuming the Generalized Riemann Hypothesis, α0 ≥ 8πe γ ≈ 44.7. It is of great interest to know if the latter bound is sharp. In 1978, Martinet constructed an infinite unramified tower of totally complex number fields with small constant root discriminant, demonstrating that α0 < 92.4. For over twenty years, this estimate has not been improved. We introduce two new ideas for bounding asymptotically minimal root discriminants, namely, (1) we allow tame ramification in the tower, and (2) we allow the fields at the bottom of the tower to have large Galois closure. These new ideas allow us to obtain the better estimate α0 < 83.9.

Equivariant Tamagawa Numbers and Galois Module Theory I

Abstract: Let L/K be a finite Galois extension of number fields. We use complexes arising from the etale cohomology of Z on open subschemes of Spec OL to define a canonical element of the relative algebraic K-group K0Z[Gal(L/K)], R. We establish some basic properties of this element, and then use it to reinterpret and refine conjectures of Stark, of Chinburg and of Gruenberg, Ritter and Weiss. Our results precisely explain the connection between these conjectures and the seminal work of Bloch and Kato concerning Tamagawa numbers. This provides significant new insight into these important conjectures and also allows one to use powerful techniques from arithmetic algebraic geometry to obtain new evidence in their favour.

A Family of Irreducible Representations of the Witt Lie Algebra with Infinite-Dimensional Weight Spaces

Abstract: We define a 4-parameter family of generically irreducible and inequivalent representations of the Witt Lie algebra on which the infinitesimal rotation operator acts semisimply with infinite-dimensional eigenspaces. They are deformations of the (generically indecomposable) representations on spaces of polynomial differential operators between two spaces of tensor densities on S1, which are constructed by composing each such differential operator with the action of a rotation by a fixed angle.

Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique ≤2; Galois cohomology of quasi-split groups over fields of cohomological dimension ≤ 2

Abstract: Let k be a perfect field with cohomological dimension ≤ 2. Serre's conjecture II claims that the Galois cohomology set H1(k,G) is trivial for any simply connected semi-simple algebraic G/k and this conjecture is known for groups of type 1A n after Merkurjev–Suslin and for classical groups and groups of type F4 and G2 after Bayer–Parimala. For any maximal torus T of G/k, we study the map H1(k, T) → H1(k, G) using an induction process on the type of the groups, and it yields conjecture II for all quasi-split simply connected absolutely almost k-simple groups with type distinct from E8. We also have partial results for E8 and for some twisted forms of simply connected quasi-split groups. In particular, this method gives a new proof of Hasse principle for quasi-split groups over number fields including the E8-case, which is based on the Galois cohomology of maximal tori of such groups.

Embedding of a maximal curve in a Hermitian variety

Abstract: Let X be a projective, geometrically irreducible, non-singular, algebraic curve defined over a finite field Fq2 of order q2 If the number of Fq2-rational points of X satisfies the Hasse–Weil upper bound, then X is said to be Fq2-maximal For a point P0 ∈ X(Fq2), let π be the morphism arising from the linear series D: = |(q + 1)P0|, and let N: = dim(D) It is known that N ≥ 2 and that π is independent of P0 whenever X is Fq2-maximal

Hilbert Functions, Residual Intersections, and Residually S2 Ideals

Abstract: Let R be a homogeneous ring over an in¢nite ¢eld, I R a homogeneous ideal, and a I an ideal generated by s forms of degrees d1; ... ;ds so that codimOa : IUXs.We give broad conditionsfor whenthe Hilbertfunction ofR=aorofR=Oa : IUis determinedby I and the degrees d1; ... ;ds.These conditions are expressed in terms of residual intersections of I, culminating in the notion of residually S2 ideals. We prove that the residually S2 property is implied by the vanishing of certain Ext modules and deduce that generic projections tend to produce ideals with this property.

Rational hypergeometric functions

Abstract: Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dimension at most three. Toric residues are applied to show that every toric resultant appears in the denominator of some rational hypergeometric function.

Isomorphism Classes of A-Hypergeometric Systems

Abstract: Given a finite set A of integral vectors and a parameter vector, Gel'fand, Kapranov, and Zelevinskii defined a system of differential equations, called an A-hypergeometric (or a GKZ hypergeometric) system. Classifying the parameters according to the D-isomorphism classes of their corresponding A-hypergeometric systems is one of the most fundamental problems in the theory. In this paper we give a combinatorial answer for the problem under the assumption that the finite set A lies in a hyperplane off the origin, and illustrate it in two particularly simple cases: the normal case and the monomial curve case.

Annihilators and Associated Varieties of Aq(λ) Modules for U(p, q)

Abstract: Vogan has conjectured that the cohomologically induced modules A q(λ) in the weakly fair range exhaust all unitary representations of U(p, q) with certain kinds of real integral infinitesimal character. To prove a statement like this, it is essential to identify these modules among the set of all irreducible Harish-Chandra modules. Barbasch and Vogan have parametrized this latter set in terms of their annihilators and asymptotic supports (or, equivalently, associated varieties). In this paper, we identify the weakly fair A q(λ) in this parametrization by combining known results about their asymptotic supports together with an explicit computation of their annihilators. In particular, this determines all vanishing and coincidences among the A q(λ) in the weakly fair range, and gives the Langlands parameters of these modules.

Estimation de sommes multiples de fonctions arithmétiques

Abstract: We estimate some sums of the shape S(X β1,..., X βm ):=∑1 ≤ d1 ≤ Xβ1...∑1 ≤ dm ≤Xβm f(d 1,..., d m ) when m ∈ N and f is a nonnegative arithmetical function. We relate them to the behaviour of the associated Dirichlet series F(s 1,..., s m ) = ∞∑ d1 = 1 ... ∞∑ dm = 1 f(d 1,..., d m )/d 1 s1 ... d m sm. The main aim of this work is to develop analytic tools to count the rational points of bounded height on toric varieties.

Gromov–Witten Invariants of Blow-ups Along Surfaces

Abstract: In this paper, using the gluing formula of Gromov–Witten invariants for symplectic cutting developed by Li and Ruan, we established some relations between Gromov–Witten invariants of a semipositive symplectic manifold M and its blow-ups along a smooth surface.

A Filtration on the Chow Groups of a Complex Projective Variety

Abstract: Let X/C be a projective algebraic manifold, and further let CHk(X)Q be the Chow group of codimension k algebraic cycles on X, modulo rational equivalence. By considering Q-spreads of cycles on X and the corresponding cycle map into absolute Hodge cohomology, we construct a filtration {Fl}l ≥ 0 on CHk(X)Q of ‘Bloch-Beilinson’ type. In the event that a certain conjecture of Jannsen holds (related to the Bloch-Beilinson conjecture on the injectivity, modulo torsion, of the Abel–Jacobi map for smooth proper varieties over Q), this filtration truncates. In particular, his conjecture implies that Fk+1 = 0.

A generalisation of Nagata's theorem on ruled surfaces

Abstract: We prove a generalisation of a theorem of Nagata on ruled surface to the case of the fiber bundle E/P → X, associated to a principal G-bundle E. Using this we prove boundedness for the isomorphism classes of semi-stable G-bundles in all characteristics.

Cancellation Theorems for Projective Modules over a Two-Dimensional Ring and its Polynomial Extensions

Abstract: We show that over polynomial extensions of normal affine domains of dimension two over perfect fields (char. ≠ 2) of cohomological dimension ≤ 1, all finitely generated projective modules are cancellative, thus answering a question of Weibel affirmatively in the case of polynomial extensions.

Long Time Behaviour of Leafwise Heat Flow for Riemannian Foliations

Abstract: For any Riemannian foliation F on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about the space of bundle-like metrics on M, about the dimension of the space of leafwise harmonic forms, and mainly about the second term of the differentiable spectral sequence of F.