# Compositio Mathematica

Online ISSN: 1570-5846
Print ISSN: 0010-437X
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Fix a prime number . We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro- completion of the fundamental group of the thrice punctured projective line. Similar techniques are also used to prove part of a conjecture of Goncharov, also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line. The main technical tool is the weighted completion of a profinite group with respect to a reductive representation (and other appropriate data).

For any finite group G we define the moduli space of pointed admissible G-covers and the concept of a G-equivariant cohomological field theory (G-CohFT), which, when G is the trivial group, reduce to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that by taking the "quotient" by G, a G-CohFT reduces to a CohFT. We also prove that a G-CohFT contains a G-Frobenius algebra, a G-equivariant generalization of a Frobenius algebra, and that the "quotient" by G agrees with the obvious Frobenius algebra structure on the space of G-invariants, after rescaling the metric. We also introduce the moduli space of G-stable maps into a smooth, projective variety X with G action. Gromov-Witten-like invariants of these spaces provide the primary source of examples of G-CohFTs. Finally, we explain how these constructions generalize (and unify) the Chen-Ruan orbifold Gromov-Witten invariants of the global quotient [X/G] as well as the ring H*(X,G) of Fantechi and Goettsche.

We prove modularity lifting theorems for l-adic Galois representations of any dimension satisfying a unitary type condition and a Fontaine-Laffaille type condition at l. This extends the results of Clozel, Harris and Taylor, and the subsequent work by Taylor. The proof uses the Taylor-Wiles method, as improved by Diamond, Fujiwara, Kisin and Taylor, applied to Hecke algebras of unitary groups, and results of Labesse on stable base change and descent from unitary groups to GL_n. Comment: 32 pages, final version incorporating referee's suggestions

For non-singular intersections of pairs of quadrics in 11 or more variables, we prove an asymptotic for the number of rational points in an expanding box.

Our main result is the description of generators of the total coordinate ring of the blow-up of $\mathbb{P}^n$ in any number of points that lie on a rational normal curve. As a corollary we show that the algebra of invariants of the action of a two-dimensional vector group introduced by Nagata is finitely generated by certain explicit determinants. We also prove the finite generation of the algebras of invariants of actions of vector groups related to T-shaped Dynkin diagrams introduced by Mukai.

We show that the torsion in the group of indecomposable (2,1)-cycles on a smooth projective variety over an algebraically closed field is isomorphic to a twist of its Brauer group, away from the characteristic. In particular, this group is infinite as soon as b_2-\rho>0. We derive a new insight into Roitman's theorem on torsion 0-cycles over a surface.

The rank 4 locus of a general skew-symmetric 7x7 matrix gives the pfaffian variety in P^20 which is not defined as a complete intersection. Intersecting this with a general P^6 gives a Calabi-Yau manifold. An orbifold construction seems to give the 1-parameter mirror-family of this. However, corresponding to two points in the 1-parameter family of complex structures, both with maximally unipotent monodromy, are two different mirror-maps: one corresponding to the general pfaffian section, the other to a general intersection of G(2,7) in P^20 with a P^13. Apparently, the pfaffian and G(2,7) sections constitute different parts of the A-model (Kahler structure related) moduli space, and, thus, represent different parts of the same conformal field theory moduli space.

Let K be a function field and C a non-isotrivial curve of genus g >= 2 over K. In this paper, we will show that if C has a global stable model with only geometrically irreducible fibers, then Bogomolov conjecture over function fields holds.

Over a perfect field $k$ of characteristic $p > 0$, we construct a Witt vector cohomology with compact supports'' for separated $k$-schemes of finite type, extending (after tensorisation with $\mathbb{Q}$) the classical theory for proper $k$-schemes. We define a canonical morphism from rigid cohomology with compact supports to Witt vector cohomology with compact supports, and we prove that it provides an identification between the latter and the slope $< 1$ part of the former. Over a finite field, this allows one to compute congruences for the number of rational points in special examples. In particular, the congruence modulo the cardinality of the finite field of the number of rational points of a theta divisor on an abelian variety does not depend on the choice of the theta divisor. This answers positively a question by J.-P. Serre.

We obtain a decomposition formula of a representation of Sp(p,q) and SO^\ast(2n) unitarily induced from a derived functor module, which enables us to reduce the problem of irreducible decompositions to the study of derived functor modules. In particular, we show such an induced representation is decomposed into a direct sum of irreducible unitarily induced modules from derived functor modules under some regularity condition on the parameters. In particular, representations of SO^\ast(2n) and Sp(p,q) induced from one-dimensional unitary representations of their parabolic subgroups are irreducible.

Let CS n be the flag manifold SO(2n)/U(n). We give a partial classification for the endomorphisms of the cohomology ring H *(CS n ; Z) which is very close to a homotopy classification of all selfmaps of CS n . Applications concerning the geometry of the space are discussed.

We continue the work of Braun and Floystad, and Cook bounding the degree of smooth surfaces in P4 not of general type using generic initial ideal theory.

We prove that the multiplicity of an arbitrary dominant weight for an integrable highest weight representation of the affine Kac-Moody algebra $A_{r}^{(1)}$ is a polynomial in the rank $r$. In the process we show that the degree of this polynomial is less than or equal to the depth of the weight with respect to the highest weight. These results allow weight multiplicity information for small ranks to be transferred to arbitrary ranks.

We deal with a divisorial contraction in dimension 3 which contracts its exceptional divisor to a cA_1 point. We prove that any such contraction is obtained by a suitable weighted blow-up.

We identify the Poisson boundary of the dual of the universal compact quantum group A_u(F) with a measurable field of ITPFI factors.

We construct a map between Bloch's higher Chow groups and Deligne homology for smooth, complex quasiprojective varieties on the level of complexes. For complex projective varieties this results in a formula which generalizes at the same time the classical Griffiths Abel-Jacobi map and the Borel/Beilinson/Goncharov regulator type maps.

We compute relations of rational equivalence among special codimensional two cycles on families of abelian surfaces using elements of higher chow groups.

The purpose of this paper is to show how generalizations of generic vanishing theorems to a -divisor setting can be used to study the geometric properties of pluritheta divisors on a principally polarized Abelian variety (PPAV for short).

We prove various characterizations of the period torsor of abelian varieties. This is the submitted version.

For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In particular, for a fixed prime p of K, we determine the probability that p splits into r primes in a random G-extension of K that is unramified at p. We find that these probabilities are nicely behaved and mostly independent. This is in analogy to Chebotarev's density theorem, which gives the probability that in a fixed extension a random prime of K splits into r primes in the extension. We also give the asymptotics for the number of G-extensions with bounded conductor. In fact, we give a class of extension invariants, including conductor, for which we obtain the same counting and probabilistic results. In contrast, we prove that that neither the analogy with the Chebotarev probabilities nor the independence of probabilities holds when extensions are ordered by discriminant. Comment: 28 pages, submitted

Let L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring Z_(p)[G] that annihilates the p-part of the class group of L. Comment: further revised; 22 pages. To appear in Compositio Mathematica.

We show that, with some technical conditions, an Abelian monoidal category admits a monoidal embedding into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail.

For a closed d-dimensional subvariety X of an abelian variety A and a canonically metrized line bundle L on A, Chambert-Loir has introduced measures $c_1(L|_X)^{\wedge d}$ on the Berkovich analytic space associated to A with respect to the discrete valuation of the ground field. In this paper, we give an explicit description of these canonical measures in terms of convex geometry. We use a generalization of the tropicalization related to the Raynaud extension of A and Mumford's construction. The results have applications to the equidistribution of small points.

In this work we use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the \textit{schematization functor} $X \mapsto (X\otimes \mathbb{C})^{sch}$, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a \textit{Hodge decomposition} on $(X\otimes\mathbb{C})^{sch}$. This Hodge decomposition is encoded in an action of the discrete group $\mathbb{C}^{\times \delta}$ on the object $(X\otimes \mathbb{C})^{sch}$ and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro-algebraic fundamental group as defined by C.Simpson, and in the simply connected case, the Hodge decomposition on the complexified homotopy groups as defined by J.Morgan and R. Hain. This Hodge decomposition is shown to satisfy a purity property with respect to a weight filtration, generalizing the fact that the higher homotopy groups of a simply connected projective manifold have natural mixed Hodge structures. As a first application we construct a new family of examples of homotopy types which are not realizable as complex projective manifolds. Our second application is a formality theorem for the schematization of a complex projective manifold. Finally, we present conditions on a complex projective manifold $X$ under which the image of the Hurewitz morphism of $\pi_{i}(X) \to H_{i}(X)$ is a sub-Hodge structure.

We express Nron functions and Schneider's local p-adic height pairing on an abelian variety A with split multiplicative reduction with theta functions and their automorphy factors on the rigid analytic torus uniformizing A.Moreover, we show formulas for the -splittingsof the Poincar biextension corresponding to Nron's and Schneider's local height pairings.

We give an algorithm for removing stackiness from smooth, tame Artin stacks with abelian stabilisers by repeatedly applying stacky blow-ups. The construction works over a general base and is functorial with respect to base change and compositions with gerbes and smooth, stabiliser preserving maps. As applications, we indicate how the result can be used for destackifying general Deligne-Mumford stacks in characteristic zero, and to obtain a weak factorisation theorem for such stacks. Over an arbitrary field, the method can be used to obtain a functorial algorithm for desingularising varieties with simplicial toric quotient singularities, without assuming the presence of a toroidal structure.

In this paper, we show that the Chern classes c k of the de Rham bundle H\textdR\mathcal{H}_{{\text{dR}}} defined on any [(Ag)]{\bar {\mathcal{A}_g}} of the moduli space of Abelian varieties of dimension g are zero in the rational Chow ring of [(Ag)]{\bar {\mathcal{A}_g}} , for g=4, 5 and k>0.

In the paper, we study special configurations of lines and points in the complex projective plane, so called k-nets. We describe the role of these configurations in studies of cohomology on arrangement complements. Our most general result is the restriction on k - it can be only 3,4, or 5. The most interesting class of nets is formed by 3-nets that relate to finite geometries, latin squares, loops, etc. All known examples of 3-nets in P^2 realize finite Abelian groups. We study the problem what groups can be so realized. Our main result is that, except for groups with all invariant factors under 10, realizable groups are isomorphic to subgroups of a 2-torus. This follows from the `algebraization' result asserting that in the dual plane, the points dual to lines of a net lie on a plane cubic.

Using non-abelian cohomology we introduce new obstructions to the Hasse principle. In particular, we generalize the classical descent formalism to principal homogeneous spaces under noncommutative algebraic groups and give explicit examples of application.

We provide an unconditional proof of the Andr\'e-Oort conjecture for the coarse moduli space $\mathcal{A}_{2,1}$ of principally polarized Abelian surfaces, following the strategy outlined by Pila-Zannier.

Let $A$ be an abelian variety defined over a field $k$. In this paper we define a filtration $F^{r}$ of the group $CH_{0}(A)$ and prove an isomorphism $\frac{K(k;A,...,A)}{\Sym}\otimes\mathbb{Z}[\frac{1}{r!}]\simeq F^{r}/F^{r+1}\otimes\mathbb{Z}[\frac{1}{r!}]$, where $K(k;A,...,A)$ is the Somekawa K-group attached to $r$-copies of the abelian variety $A$.\\ In the special case when $k$ is a finite extension of $\mathbb{Q}_{p}$ and $A$ has split multiplicative reduction, we compute the kernel of the map $CH_{0}(A)\otimes\Z[\frac{1}{2}]\rightarrow \rm{Hom}(Br(A),\Q/\Z)\otimes\Z[\frac{1}{2}]$, induced by the pairing $CH_{0}(A)\times Br(A)\rightarrow\mathbb{Q}/\Z$.

Let $\pi:\CA\ra S$ be an abelian scheme over a scheme $S$ which is quasi-projective over an affine noetherian scheme and let $\CL$ be a symmetric, rigidified, relatively ample line bundle on $\CA$. We show that there is an isomorphism \det(\pi_*\CL)^{\o times 24}\simeq\big(\pi_*\omega_{\CA}^{\vee}\big)^{\o times 12d} of line bundles on $S$, where $d$ is the rank of the (locally free) sheaf $\pi_*\CL$. We also show that the numbers 24 and $12d$ are sharp in the following sense: if $N>1$ is a common divisor of 12 and 24, then there are data as above such that \det(\pi_*\CL)^{\o times (24/N)}\not\simeq\big(\pi_*\omega_{\CA}^{\vee}\big)^{\o times (12d/N)}.

We describe an efficient algorithm for the computation of separable isogenies between abelian varieties represented in the coordinate system given by algebraic theta functions. Let A be an abelian variety of dimension g defined over a field of odd characteristic. Our algorithm comprises two principal steps. First, given a theta null point for A and a subgroup K isotropic for the Weil pairing, we explain how to compute the theta null point corresponding to the quotient abelian variety A/K. Then, from the knowledge of a theta null point of A/K, we present an algorithm to obtain a rational expression for an isogeny from A to A/K. The algorithm that results from combining these two steps can be viewed as a higher-dimensional analog of the well-known algorithm of Vélu for computing isogenies between elliptic curves. In the case where K is isomorphic to (ℤ/ℓℤ) g for ℓ∈ℕ*, the overall time complexity of this algorithm is equivalent to O(log ℓ) additions in A and a constant number of ℓth root extractions in the base field of A. In order to improve the efficiency of our algorithms, we introduce a compressed representation that allows us to encode a point of level 4ℓ of a g-dimensional abelian variety using only g(g+1)/2⋅4 g coordinates. We also give formulas for computing the Weil and commutator pairings given input points in theta coordinates.

To a symmetric, relatively ample line bundle on an Abelian scheme one can associate a linear combination of the determinant bundle and the relative canonical bundle, which is a torsion element in the Picard group of the base. We improve the bound on the order of this element found by Faltings and Chai. In particular, we obtain an optimal bound when the degree of the line bundle d is odd and the set of residue characteristics of the base does not intersect the set of primes p dividing d, such that p p \leqslant 2g - 1p \leqslant 2g - 1 , where g is the relative dimension of the Abelian scheme. Also, we show that in some cases these torsion elements generate the entire torsion subgroup in the Picard group of the corresponding moduli stack.

Let M be a projective manifold, p:M_{G} --> M a regular covering over M with a free abelian transformation group G. We describe holomorphic functions on M_{G} of an exponential growth with respect to the distance defined by a metric pulled back from M. As a corollary we obtain for such functions Cartwright and Liouville type theorems. Our approach brings together L_{2} cohomology technique for holomorphic vector bundles on complete K\"{a}hler manifolds and geometric properties of projective manifolds.

We formulate a conjecture which generalizes Darmon's "refined class number formula". We discuss relations between our conjecture and the equivariant leading term conjecture of Burns. As an application, we give another proof of the "except 2-part" of Darmon's conjecture, which was first proved by Mazur and Rubin.

We investigate the relationship between the usual and general Hodgeconjectures for abelian varieties. For certain abelian varieties A, weshow that the usual Hodge conjecture for all powers of A implies thegeneral Hodge conjecture for A.

The cones of divisors and curves defined by various positivity conditions on a smooth projective variety have been the subject of a great deal of work in algebraic geometry, and by now they are quite well understood. However the analogous cones for cycles of higher codimension and dimension have started to come into focus only recently. The purpose of this paper is to explore some of the phenomena that can occur by working out the picture fairly completely in a couple of simple but non-trivial cases. Specifically, we study cycles of arbitrary codimension on the self-product of an elliptic curve with complex multiplication, as well as two dimensional cycles on the product of a very general abelian surface with itself. Already one finds various non-classical behavior, for instance nef cycles that fail to be pseudoeffective: this answers a question raised in 1964 by Grothendieck in correspondence with Mumford. We also propose a substantial number of open problems for further investigation.

An abelian cover is a finite morphism $X\to Y$ of varieties which is the quotient map for a generically faithful action of a finite abelian group $G$. Abelian covers with $Y$ smooth and $X$ normal were studied in \cite{Pardini_AbelianCovers}. Here we study the non-normal case, assuming that $X$ and $Y$ are $S_2$ varieties that have at worst normal crossings outside a subset of codimension $\ge 2$. Special attention is paid to the case of $\Z_2^r$-covers of surfaces, which is used in arxiv:0901.4431 to construct explicitly compactifications of some components of the moduli space of surfaces of general type.

We prove for a large family of rings R that their lambda-pure global dimension is greater than one for each infinite regular cardinal lambda. This answers in negative a problem posed by Rosicky. The derived categories of such rings then do not satisfy the Adams lambda-representability for morphisms for any lambda. Equivalently, they are examples of well generated triangulated categories whose lambda-abelianization in the sense of Neeman is not a full functor for any lambda. In particular we show that given a compactly generated triangulated category, one may not be able to find a Rosicky functor among the lambda-abelianization functors.

We consider the moduli space M of stable principal G-bundles over a compact Riemann surface C of genus g 2, G being any reductive algebraic group and give an explicit description of the generic fibre of the Hitchin map H: T*M K. If T G is a fixed maximal torus with Weyl group W, for each given generic element K one may construct a W-Galois covering ~C of C and consider the generalized Prym variety P=HomW(X(T),J(~C)), where X(T) denotes the group of characters on T and J(C) the Jacobian. The connected component P0 P which contains the trivial element is an abelian variety. In the present paper we use the classical theory of representations of finite groups to compute dim P = dim M. Next, by means of mostly elementary techniques, we explicitly construct a finite map F from each connected component H–1()c of the Hitchin fibre to P0 and study its degree. In case G=PGl(2) one has that the generic fibre of F:H–1()c P0 is a principal homogeneous space with respect to a product of (2d-2) copies of Z/2Z where d is the degree of the canonical bundle over C. However if the Dynkin diagram of G does not contain components of type Bl, l 1 or when the commutator subgroup (G,G) is simply connected the map F is injective.

We consider the moduli space of stable principal G-bundles over a compact Riemann surface C of genus >1, with G a reductive algebraic group. We explicitly construct a map F from the generic fibre of the Hitchin map to a generalized Prym variety associated to a suitable Galois covering of C. The map F has finite fibres. In case G=PGl(2) one can check that the generic fibre of F is a principal homogeneous space with respect to a product of 2d-2 copies of Z/2Z where d is the degree of the canonical bundle over C. However in case the Dynkin diagram of G does not contain components of type $B_{n}$ n>0, or when the commutator subgroup (G,G) is simply connected the map F is injective.

In this paper, we investigate the action of the [^(X)]\widehat X of a compact Shimura Variety S() on the -cohomology of S()> under a cup product. We use this to split the cohomology of S() into a direct sum of (not necessarily irreducible) -Hodge structures. As an application, we prove that for the class of arithmetic subgroups of the unitary groups U(p,q) arising from Hermitian forms over CM fields, the Mumford–Tate groups associated to certain holomorphic cohomology classes on S() are Abelian. As another application, we show that all classes of Hodge type (1,1) in H2 of unitary four-folds associated to the group U(2,2) are algebraic.

Motivated by a result of Bost, we use the relationship between Faltings' heights of abelian varieties with complex multiplication and logarithmic derivatives of Artin L-functions at s=0 to investigate these heights. In particular, we prove that the height of an elliptic curve with complex multiplication by Q-d is bounded from below by an effective affine function of log d.

Let $k$ be an algebraically closed field of characteristic zero, $F$ be an algebraically closed extension of $k$ of transcendence degree one, and $G$ be the group of automorphisms over $k$ of the field $F$. The purpose of this note is to calculate the group of continuous automorphisms of $G$.

We combine Deligne's global invariant cycle theorem, and the algebraicity theorem of Cattani, Deligne and Kaplan, for the connected components of the locus of Hodge classes, to conclude that under simple assumptions these components are defined over number fields (assuming the initial family is), as expected from the Hodge conjecture. We also show that the Hodge conjecture for (weakly) absolute Hodge classes reduces to the Hodge conjecture for (weakly) absolute Hodge classes on varieties defined over number fields.

In this paper we solve the problemof desingularization of an absolutely isolatedsingularity of a differential equation, including thedicritical case. As an application, we prove thefiniteness of the number of dicritical points in theblowing up tree of an absolutely isolated singularity.

In this paper, we establish that complete Kac-Moody groups over finite fields are abstractly simple. The proof makes an essential use of Mathieu-Rousseau's construction of complete Kac-Moody groups over fields. This construction has the advantage that both real and imaginary root spaces of the Lie algebra lift to root subgroups over arbitrary fields. A key point in our proof is the fact, of independent interest, that both real and imaginary root subgroups are contracted by conjugation of positive powers of suitable Weyl group elements.

Top-cited authors
• Stony Brook University
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