Showing papers in "Compositio Mathematica in 2009"
TL;DR: In this paper, the present problem has been suggested by Miss Esther Klein in connection with the following proposition: "Our present problem is the same problem as the one suggested by the author of this paper."
Abstract: Our present problem has been suggested by Miss Esther Klein in connection with the following proposition.
1,556 citations
TL;DR: In this paper, the authors investigated cluster-tilting objects in triangulated 2-Calabi-Yau and related categories, including pre-projective algebras of non-Dynkin quivers.
Abstract: We investigate cluster-tilting objects (and subcategories) in triangulated 2-Calabi–Yau and related categories. In particular, we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi–Yau categories contains, as special cases, the cluster categories and the stable categories of preprojective algebras of Dynkin graphs. For these 2-Calabi–Yau categories, we construct cluster-tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We discuss connections with cluster algebras and subcluster algebras related to unipotent groups, in both the Dynkin and non-Dynkin cases.
266 citations
TL;DR: In this paper, the authors show that the non-crossing partitions associated with a finite Coxeter group form a lattice, which is a natural bijection with the cluster tilting objects in the associated cluster category.
Abstract: We situate the noncrossing partitions associated with a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated with a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver Q without oriented cycles are in natural bijection with the cluster tilting objects in the associated cluster category. We also show that these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition.
185 citations
TL;DR: In this article, it was shown that there is a hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others.
Abstract: We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P. Albers and P. Biran-O. Cornea) as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.
153 citations
TL;DR: For a fixed parabolic subalgebra p of gl(n, C) as discussed by the authors, it is shown that the center of the principal block O-0(p) of the parabolic category 0 is naturally isomorphic to the cohomology ring H*(B-p), which corresponds under localisation and the Riemann-Hilbert correspondence to a full projective-injective module in the corresponding category O 0(p).
Abstract: For a fixed parabolic subalgebra p of gl(n, C) we prove that the centre of the principal block O-0(p) of the parabolic category 0 is naturally isomorphic to the cohomology ring H*(B-p) of the corresponding Springer fibre. We give a. diagrammatic description of O-0(p) for maximal parabolic p and give an explicit isomorphism to Braden's description of the category Perv(B)(G(k,,n)) of Schubert-constructible perverse sheaves on Grassmannians. As a consequence Khovanov's algebra, H-n is realised as the endomorphism ring of some object from Perv(B)(G(n, n)) which corresponds under localisation and the Riemann-Hilbert correspondence to a full projective-injective module in the corresponding category O-0(p). From there one can deduce that Khovanov's tangle invariants are obtained from the more general functorial invariants in [C. Stroppel, Catgorification of the Temperley Lieb category, tangles, and cobordisms via projective functors, Duke Math. J. 126(3) (2005), 547-596] by restriction.
108 citations
TL;DR: In this article, the authors give a rigorous definition of tropical fans and their morphisms, and show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point.
Abstract: We give a rigorous definition of tropical fans (the ‘local building blocks for tropical varieties’) and their morphisms. For a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point; a statement that can be viewed as one of the important first steps of tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some ℝr) together with the evaluation and forgetful morphisms. Using our results this gives new, easy and unified proofs of various tropical independence statements, e.g. of the fact that the numbers of rational tropical curves (in any ℝr) through given points are independent of the points.
104 citations
TL;DR: In this article, the Brauer-Manin obstruction for the existence of integral points on schemes defined over the integers was shown to be the only obstruction to integral points in linear algebraic groups.
Abstract: An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.
85 citations
TL;DR: In this article, the authors generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being ℚ-Gorenstein or the pair being log √ − Gorenstein.
Abstract: In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being ℚ-Gorenstein or the pair being log ℚ-Gorenstein. The main features of the theory extend to this setting in a natural way.
78 citations
TL;DR: In this article, a re-normalization of the Reshetikhin-Turaev quantum invariants of links, by modified quantum dimensions, is presented, which leads to non-trivial link invariants.
Abstract: In this paper we give a re-normalization of the Reshetikhin-Turaev quantum invariants of links, by modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum dimensions. More interestingly we will give two examples where the usual quantum dimensions vanish but the modified quantum dimensions are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras previously defined by the first two authors. These link invariants are multivariable and generalize the multivariable Alexander polynomial. The second example, is a hierarchy of link invariants arising from nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev's quantum dilogarithm invariants of knots.
78 citations
TL;DR: In this article, it was shown that given an overconvergent F-isocrystal on a variety over a field of positive characteristic, one can pull back along a suitable generically finite cover to obtain an isocrystal which extends, with logarithmic singularities and nilpotent residues, to some complete variety.
Abstract: We complete our proof that given an overconvergent F-isocrystal on a variety over a field of positive characteristic, one can pull back along a suitable generically finite cover to obtain an isocrystal which extends, with logarithmic singularities and nilpotent residues, to some complete variety. We also establish an analogue for F-isocrystals overconvergent inside a partial compactification. By previous results, this reduces to solving a local problem in a neighborhood of a valuation of height 1 and residual transcendence degree zero. We do this by studying the variation of some numerical invariants attached to p-adic differential modules, analogous to the irregularity of a complex meromorphic connection. This allows for an induction on the transcendence defect of the valuation, i.e., the discrepancy between the dimension of the variety and the rational rank of the valuation.
65 citations
TL;DR: In this article, the authors considered the problem of preserving stability under the Fourier-Mukai transform on an abelian surface and a K3 surface, and they showed that the stability with respect to these polarizations is preserved under ℱℰ, if the degree of stable sheaves on X is sufficiently large.
Abstract: We consider the problem of preservation of stability under the Fourier–Mukai transform ℱℰ:D(X)→D(Y ) on an abelian surface and a K3 surface. If Y is the moduli space of μ-stable sheaves on X with respect to a polarization H, we have a canonical polarization on Y and we have a correspondence between (X,H) and . We show that the stability with respect to these polarizations is preserved under ℱℰ, if the degree of stable sheaves on X is sufficiently large.
TL;DR: For the p-adic Galois representation associated to a Hilbert modular form, Carayol has shown that, under a certain assumption, its restriction to the local Galois group at a finite place not dividing p is compatible with the local Langlands correspondence as mentioned in this paper.
Abstract: For the p-adic Galois representation associated to a Hilbert modular form, Carayol has shown that, under a certain assumption, its restriction to the local Galois group at a finite place not dividing p is compatible with the local Langlands correspondence. Under the same assumption, we show that the same is true for the places dividing p, in the sense of p-adic Hodge theory, as is shown for an elliptic modular form. We also prove that the monodromy-weight conjecture holds for such representations.
TL;DR: In this paper, the authors apply local methods involving the theory of arithmetic differential equations to prove quantitative versions of a similar statement, which applies also to certain infinite-rank subgroups, as well as to the situation where the set of CM-points is replaced by certain isogeny classes of points on the modular curve.
Abstract: In the predecessor to this article, we used global equidistribution theorems to prove that given a correspondence between a modular curve and an elliptic curve A, the intersection of any finite-rank subgroup of A with the set of CM-points of A is finite. In this article we apply local methods, involving the theory of arithmetic differential equations, to prove quantitative versions of a similar statement. The new methods apply also to certain infinite-rank subgroups, as well as to the situation where the set of CM-points is replaced by certain isogeny classes of points on the modular curve. Finally, we prove Shimura-curve analogues of these results.
TL;DR: In this paper, the authors introduce the notion of an S-constructible stack of categories on a space X and prove that the 2-category of Sconstructible stacks on X is equivalent to 2-functors 2Funct(EP≤2(X,S),Cat) from the exit-path 2 category to the 2 category of small categories.
Abstract: For a Whitney stratification S of a space X (or, more generally, a topological stratification in the sense of Goresky and MacPherson) we introduce the notion of an S-constructible stack of categories on X. The motivating example is the stack of S-constructible perverse sheaves. We introduce a 2-category EP≤2(X,S), called the exit-path 2-category, which is a natural stratified version of the fundamental 2-groupoid. Our main result is that the 2-category of S-constructible stacks on X is equivalent to the 2-category of 2-functors 2Funct(EP≤2(X,S),Cat) from the exit-path 2-category to the 2-category of small categories.
TL;DR: Wildeshaus et al. as mentioned in this paper proposed a method to construct intrinsically in DMgm(k) a motivic version of interior cohomology of smooth, but possibly non-projective schemes.
Abstract: In a recent paper, Bondarko [Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), Preprint (2007), 0704.4003] defined the notion of weight structure, and proved that the category DMgm(k) of geometrical motives over a perfect field k, as defined and studied by Voevodsky, Suslin and Friedlander [Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000)], is canonically equipped with such a structure. Building on this result, and under a condition on the weights avoided by the boundary motive [J. Wildeshaus, The boundary motive: definition and basic properties, Compositio Math. 142 (2006), 631–656], we describe a method to construct intrinsically in DMgm(k) a motivic version of interior cohomology of smooth, but possibly non-projective schemes. In a sequel to this work [J. Wildeshaus, On the interior motive of certain Shimura varieties: the case of Hilbert–Blumenthal varieties, Preprint (2009), 0906.4239], this method will be applied to Shimura varieties.
TL;DR: Using a p-adic analogue of the convolution method of Rankin-Selberg and Shimura, the authors constructed the two-variable padic L-function of a Hida family of Hilbert modular eigenforms of parallel weight.
Abstract: Using a p-adic analogue of the convolution method of Rankin–Selberg and Shimura, we construct the two-variable p-adic L-function of a Hida family of Hilbert modular eigenforms of parallel weight. It is shown that the conditions of Greenberg–Stevens [R. Greenberg and G. Stevens, p-adic L-functions and p-adic periods of modular forms, Invent. Math. 111 (1993), 407–447] are satisfied, from which we deduce special cases of the Mazur–Tate–Teitelbaum conjecture in the Hilbert modular setting.
TL;DR: In this paper, the authors used convex bodies to study line bundles in the setting of Arakelov theory, which is parallel to [Yu2] but the content is independent, and they used the usual convex body ∆(LK) ⊂ R of the generic fibre LK viewed as a line bundle on the projective variety XK.
Abstract: This paper uses convex bodies to study line bundles in the setting of Arakelov theory. The treatment is parallel to [Yu2], but the content is independent. The method of constructing a convex body in Euclidean space, now called “Okounkov body”, from a given algebraic linear series was due to Okounkov [Ok1, Ok2], and was explored systematically by Kaveh–Khovanskii [KK] and Lazarsfeld–Mustaţǎ [LM]. Many important results of algebraic geometry can be derived from convex geometry through the bridge that the volume of the convex body gives the volume of the linear series. Let K be a number field, X be an arithmetic variety of relative dimension d over OK , and L be a hermitian line bundle over X . There are two important arithmetic invariants ĥ(L) and χ(L). Their growth under tensor powers are measured respectively by vol(L) and volχ(L). In [Yu2], we have introduced the Okounkov body ∆(L) ⊂ R of L, whose volume computes vol(L). It is a natural arithmetic analogue of the construction in [LM]. In the current paper, we use the usual Okounkov body ∆(LK) ⊂ R of the generic fibre LK viewed as a line bundle on the projective variety XK . Then we introduce the Chebyshev ∗The author is fully supported by a research fellowship of the Clay Mathematics Institute.
TL;DR: This article showed that universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers of q. As an application, they obtained a relation between the rank and crank of a partition.
Abstract: We show that some q-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers of q. We also prove that certain linear sums of q-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.
TL;DR: The main goal of as mentioned in this paper is to provide asymptotic expansions for the numbers #{p≤x:p-prime,sq(p)=k} for k close to ((q−1)/2)log qx, where sq(n) denotes the q-ary sum-of-digits function.
Abstract: The main goal of this paper is to provide asymptotic expansions for the numbers #{p≤x:p prime,sq(p)=k} for k close to ((q−1)/2)log qx, where sq(n) denotes the q-ary sum-of-digits function. The proof is based on a thorough analysis of exponential sums of the form (where the sum is restricted to p prime), for which we have to extend a recent result by the second two authors.
TL;DR: In this paper, it was shown that there is an equivalence between the 2-category of smooth Deligne-Mumford stacks with torus embeddings and actions and the 1-categories of stacky fans.
Abstract: In this paper, we show that there is an equivalence between the 2-category of smooth Deligne–Mumford stacks with torus embeddings and actions and the 1-category of stacky fans. To this end, we prove two main results. The first is related to a combinatorial aspect of the 2-category of toric algebraic stacks defined by I. Iwanari [Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Preprint (2007)]; we establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second result provides a geometric characterization of toric algebraic stacks. Logarithmic geometry in the sense of Fontaine–Illusie plays a central role in obtaining our results.
TL;DR: In this paper, the authors classify two-dimensional split trianguline representations of p-adic fields using B-pairs as defined by Berger, which is a generalization of a result of Colmez who classified two-dimensions of trianguloine representations for p≠2 by using (φ, Γ)-modules over a Robba ring.
Abstract: The aim of this article is to classify two-dimensional split trianguline representations of p-adic fields. This is a generalization of a result of Colmez who classified two-dimensional split trianguline representations of for p≠2 by using (φ,Γ)-modules over a Robba ring. In this article, for any prime p and for any p-adic field K, we classify two-dimensional split trianguline representations of using B-pairs as defined by Berger.
TL;DR: In this paper, it was shown that any low-weight crystalline deformation of ρ unramified outside a finite set of primes will be modular under the assumption that ρ has a large image and admits a lowweight modular deformation, and the main ingredient is an Ihara-type lemma for the local component at ρ of the middle degree cohomology of a Hilbert modular variety.
Abstract: Let ρ be a two-dimensional modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρ has a large image and admits a low-weight crystalline modular deformation we show that any low-weight crystalline deformation of ρ unramified outside a finite set of primes will be modular. We follow the approach of Wiles as generalized by Fujiwara. The main new ingredient is an Ihara-type lemma for the local component at ρ of the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraic p-part of the value at one of the adjoint L-function associated with a Hilbert modular newform to the cardinality of the corresponding Selmer group.
TL;DR: In this paper, the authors define and study virtual representation spaces for vectors hav- ing both positive and negative dimensions at the vertices of a quiver without oriented cycles, and prove that they satisfy the three basic theorems: the First Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition Theorem.
Abstract: We define and study virtual representation spaces for vectors hav- ing both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the First Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition Theorem. In the special case of Dynkin quivers with n vertices this gives the fun- damental interrelationship between supports of the semi-invariants and the Tilting Triangulation of the (n 1)-sphere.
TL;DR: In this article, the F -conjecture on the moduli space of n-pointed stable curves of genus g is stratified by the topological type of the curves being parameterized: the closure of the locus of curves with k nodes has codimension k.
Abstract: The moduli space of n -pointed stable curves of genus g is stratified by the topological type of the curves being parameterized: the closure of the locus of curves with k nodes has codimension k The one-dimensional components of this stratification are smooth rational curves called F -curves These are believed to determine all ample divisors F - conjecture A divisor on is ample if and only if it positively intersects the F - curves In this paper, proving the F -conjecture on is reduced to showing that certain divisors on for N ⩽ g + n are equivalent to the sum of the canonical divisor plus an effective divisor supported on the boundary Numerical criteria and an algorithm are given to check whether a divisor is ample By using a computer program called the Nef Wizard, written by Daniel Krashen, one can verify the conjecture for low genus This is done on for g ⩽24, more than doubling the number of cases for which the conjecture is known to hold and showing that it is true for the first genera such that is known to be of general type
TL;DR: For any positive integer x,d and k with gcd (x,d) = 1 and 3 11, a large number of new ternary equations arise, which are solved by combining the Frey curve and Galois representation approach with local and cyclotomic considerations.
Abstract: We prove that for any positive integers x,d and k with gcd (x,d)=1 and 3 11, a large number of new ternary equations arise, which we solve by combining the Frey curve and Galois representation approach with local and cyclotomic considerations. Furthermore, the number of systems of equations grows so rapidly with k that, in contrast with the previous proofs, it is practically impossible to handle the various cases in the usual manner. The main novelty of this paper lies in the development of an algorithm for our proofs, which enables us to use a computer. We apply an efficient, iterated combination of our procedure for solving the new ternary equations that arise with several sieves based on the ternary equations already solved. In this way, we are able to exclude the solvability of the enormous number of systems of equations under consideration. Our general algorithm seems to work for larger values of k as well, although there is, of course, a computational time constraint.
TL;DR: In this article, the face ring of a homology manifold modulo a generic system of parameters is studied and its socle is computed and it is verified that a particular quotient of this ring is Gorenstein.
Abstract: The face ring of a homology manifold (without boundary) modulo a generic system of parameters is studied. Its socle is computed and it is verified that a particular quotient of this ring is Gorenstein. This fact is used to prove that the algebraic g-conjecture for spheres implies all enumerative consequences of its far-reaching generalization (due to Kalai) to manifolds. A special case of Kalai’s conjecture is established for homology manifolds that have a codimension-two face whose link contains many vertices.
TL;DR: In this article, a product formula and the modularity of Kudla's generating series of special cycles in Chow groups were proved for Shimura varieties of orthogonal type over real fields.
Abstract: On Shimura varieties of orthogonal type over totally real fields, we prove a product formula and the modularity of Kudla’s generating series of special cycles in Chow groups.
TL;DR: In this paper, it was shown that the Iitaka conjecture for algebraic fiber spaces holds up to dimension 6, that is, when n≤6, and when n ≥ 2.
Abstract: We prove that the Iitaka conjecture Cn,m for algebraic fibre spaces holds up to dimension six, that is, when n≤6.
TL;DR: In this article, the authors studied linear functions on fibrations whose central fiber is a linear free divisor and showed that the base space of the semi-universal unfolding of such a function carries a Frobenius manifold structure.
Abstract: We study linear functions on fibrations whose central fibre is a linear free divisor. We analyse the Gaus–Manin system associated to these functions, and prove the existence of a primitive and homogenous form. As a consequence, we show that the base space of the semi-universal unfolding of such a function carries a Frobenius manifold structure.
TL;DR: The parity conjecture for the ranks of p-power Selmer groups was proved in this paper for a class of elliptic curves defined over real number fields, and the parity conjecture was also proved for a large class of ellipses defined over totally real numbers.
Abstract: We prove the parity conjecture for the ranks of p-power Selmer groups (p⁄=2) of a large class of elliptic curves defined over totally real number fields.