Journal: Compositio Mathematica

Abbreviation

Compos. Math.

Publisher

Cambridge University Press

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ISSN

0010-437X
1570-5846

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Publications1 - 10 of 21
  • The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves
    Item type: Journal Article
    Molcho, Samouil; Pandharipande, Rahul; Schmitt, Johannes (2023)
    Compositio Mathematica
    We bound from below the complexity of the top Chern class of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also cannot be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton's double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for on the moduli of curves after log blow-ups.
  • Gaudin subalgebras and stable rational curves
    Item type: Journal Article
    Aguirre, Leonardo; Felder, Giovanni; Veselov, Alexander P. (2011)
    Compositio Mathematica
  • On the complexity of a putative counterexample to the p-adic Littlewood conjecture
    Item type: Journal Article
    Badziahin, Dmitry; Bugeaud, Yann; Einsiedler, Manfred; et al. (2015)
    Compositio Mathematica
  • Uniform models and short curves for random 3-manifolds
    Item type: Journal Article
    Feller, Peter; Sisto, Alessandro; Viaggi, Gabriele (2025)
    Compositio Mathematica
    We provide two constructions of hyperbolic metrics on 3-manifolds with Heegaard splittings that satisfy certain topological conditions, which both apply to random Heegaard splitting with asymptotic probability 1. These constructions provide a lot of control on the resulting metric, allowing us to prove various results about the coarse growth rate of geometric invariants, such as diameter and injectivity radius, and about arithmeticity and commensurability in families of random 3-manifolds. For example, we show that the diameter of a random Heegaard splitting grows coarsely linearly in the length of the associated random walk. The constructions only use tools from the deformation theory of Kleinian groups, that is, we do not rely on the solution of the geometrization conjecture by Perelman. In particular, we give a proof of Maher’s result that random 3-manifolds are hyperbolic that bypasses geometrization.
  • Bimodules and branes in deformation quantization
    Item type: Journal Article
    Calaque, Damien; Felder, Giovanni; Ferrario, Andrea; et al. (2011)
    Compositio Mathematica
    We prove a version of Kontsevich’s formality theorem for two subspaces (branes) of a vector space X. The result implies, in particular, that the Kontsevich deformation quantizations of S(X*) and ∧(X) associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet’s recent paper on Koszul duality in deformation quantization.
  • The semisimplicity conjecture for A-motives
    Item type: Journal Article
    Stalder, Nicolas (2010)
    Compositio Mathematica
    We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V𝔭(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G𝔭(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G𝔭(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V𝔭(M). The second states that the connected component of G𝔭(M) is reductive if M is semisimple and has a separable endomorphism algebra.
  • Vassiliev invariants of quasipositive knots
    Item type: Journal Article
    Baader, Sebastian (2006)
    Compositio Mathematica
    Quasipositive knots are transverse intersections of complex plane curves with the standard sphere S3⊂C2. It is known that any Alexander polynomial of a knot can be realized by a quasipositive knot. As a consequence, the Alexander polynomial cannot detect quasipositivity. In this paper we prove a similar result about Vassiliev invariants: for any oriented knot K and any natural number n there exists a quasipositive knot Q whose Vassiliev invariants of order less than or equal to n coincide with those of K.
  • Classification of universal formality maps for quantizations of Lie bialgebras
    Item type: Journal Article
    Merkulov, Sergei; Willwacher, Thomas (2020)
    Compositio Mathematica
    We settle several fundamental questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. Moreover, we settle these questions in a greater generality: we give a complete classification of the associated universal formality maps. An important new technical ingredient introduced in this paper is a polydifferential endofunctor D in the category of augmented props with the property that for any representation of a prop P in a vector space V the associated prop DP admits an induced representation on the graded commutative algebra circle dot V-center dot given in terms of polydifferential operators. Applying this functor to the minimal resolution (Lieb) over cap (infinity) of the genus completed prop (Lieb) over cap of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in one-to-one correspondence with morphisms of dg props F : Assb(infinity) -> D (Lieb) over cap (infinity) satisfying certain boundary conditions, where Assb(infinity) is a minimal resolution of the prop of associative bialgebras. We prove that the set of such formality morphisms is nonempty. The latter result is used in turn to give a short proof of the formality theorem for universal quantizations of arbitrary Lie bialgebras which says that for any Drinfeld associator A there is an associated Lie(infinity) quasi-isomorphism between the Lie(infinity) algebras Def(AssB(infinity) -> End(circle dot center dot V)) and Def(LieB -> End(V)) controlling, respectively, deformations of the standard bialgebra structure in circle dot V and deformations of any given Lie bialgebra structure in V. We study the deformation complex of an arbitrary universal formality morphism Def(Assb(infinity) -> F D (Lieb) over cap (infinity)) and prove that it is quasi-isomorphic to the full (i.e. not necessary connected) version of the graph complex introduced Maxim Kontsevich in the context of the theory of deformation quantizations of Poisson manifolds. This result gives a complete classification of the set {FA} of gauge equivalence classes of universal Lie connected formality maps: it is a torsor over the Grothendieck-Teichmuller group GRT = GRT1 (sic) K* and can hence can be identified with the set {U} of Drinfeld associators.
  • Local spectral equidistribution for Siegel modular forms and applications
    Item type: Journal Article
    Kowalski, Emmanuel; Kowalski, Emmanuel; Kowalski, Emmanuel; et al. (2012)
    Compositio Mathematica
    We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson’s formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.
  • Kloosterman paths and the shape of exponential sums
    Item type: Journal Article
    Kowalski, Emmanuel; Sawin, William F. (2016)
    Compositio Mathematica
    We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums Klp(a), as a varies over Fp× and as p tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.
Publications1 - 10 of 21