Showing papers in "Inventiones Mathematicae in 2007"

Global well-posedness for the critical 2D dissipative quasi-geostrophic equation

Abstract: We give an elementary proof of the global well-posedness for the critical 2D dissipative quasi-geostrophic equation. The argument is based on a non-local maximum principle involving appropriate moduli of continuity.

The homotopy theory of dg-categories and derived Morita theory

Abstract: The main purpose of this work is the study of the homotopy theory of dg-categories up to quasi-equivalences. Our main result provides a natural description of the mapping spaces between two dg-categories $C$ and $D$ in terms of the nerve of a certain category of $(C,D)$-bimodules. We also prove that the homotopy category $Ho(dg-Cat)$ is cartesian closed (i.e. possesses internal Hom's relative to the tensor product). We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dg-categories of modules over two dg-categories $C$ and $D$ as the dg-category of $(C,D)$-bi-modules. Finally, we give three applications of our results. The first one expresses Hochschild cohomology as endomorphisms of the identity functor, as well as higher homotopy groups of the \emph{classifying space of dg-categories} (i.e. the nerve of the category of dg-categories and quasi-equivalences between them). The second application is the existence of a good theory of localization for dg-categories, defined in terms of a natural universal property. Our last application states that the dg-category of (continuous) morphisms between the dg-categories of quasi-coherent (resp. perfect) complexes on two schemes (resp. smooth and proper schemes) is quasi-equivalent to the dg-category of quasi-coherent complexes (resp. perfect) on their product.

Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems

Abstract: We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic non-linear Schrodinger equation in a suitable scaling limit. The result is extended to k-particle density matrices for all positive integer k.

Conservation laws for conformally invariant variational problems

Abstract: We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations,..., etc.) in divergence form. These divergence-free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE (see [Hel]). It enables us also to establish new results. In particular we solve a conjecture by E. Heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by S. Hildebrandt [Hil1] claiming that critical points of continuously differentiable elliptic conformally invariant Lagrangian in two dimensions are continuous.

Knot Floer homology detects fibred knots

Abstract: Ozsvath and Szabo conjectured that knot Floer homology detects fibred knots in S^3. We will prove this conjecture for null-homologous knots in arbitrary closed 3-manifolds. Namely, if K is a knot in a closed 3-manifold Y, Y-K is irreducible, and \hat{HFK}(Y,K) is monic, then K is fibred. The proof relies on previous works due to Gabai, Ozsvath–Szabo, Ghiggini and the author. A corollary is that if a knot in S^3 admits a lens space surgery, then the knot is fibred.

The Kähler-Ricci flow on surfaces of positive Kodaira dimension

Abstract: 4 Estimates 9 4.1 The zeroth order and volume estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 A partial second order estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3 Gradient estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.4 The second order estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

The topological structure of scaling limits of large planar maps

Abstract: We discuss scaling limits of large bipartite planar maps. If p≥2 is a fixed integer, we consider, for every integer n≥2, a random planar map Mn which is uniformly distributed over the set of all rooted 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of Mn, equipped with the graph distance rescaled by the factor n-1/4, converges in distribution as n→∞ towards a limiting random compact metric space, in the sense of the Gromov–Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p and of the subsequence, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4.

A central limit theorem for convex sets

Abstract: We show that there exists a sequence \(\varepsilon_n\searrow0\) for which the following holds: Let K⊂ℝn be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in ℝn, t0∈ℝ and σ>0 such that $$\sup_{A\subset\mathbb{R}}\left|\textit{Prob}\,\{\langle X,\theta\rangle\in A\}-\frac{1}{\sqrt{2\pi\sigma}}\int_Ae^{-\frac{(t - t_0)^2}{2\sigma^2}} dt\right|\leq\varepsilon_n,\qquad{(\ast)}$$ where the supremum runs over all measurable sets A⊂ℝ, and where 〈·,·〉 denotes the usual scalar product in ℝn. Furthermore, under the additional assumptions that the expectation of X is zero and that the covariance matrix of X is the identity matrix, we may assert that most unit vectors θ satisfy (*), with t0=0 and σ=1. Corresponding principles also hold for multi-dimensional marginal distributions of convex sets.

Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups

Abstract: We prove that if a countable discrete group Γ is w-rigid, i.e. it contains an infinite normal subgroup H with the relative property (T) (e.g. $\Gamma=SL(2,\mathbb{Z})\ltimes\mathbb{Z}^2$ , or Γ=H×H’ with H an infinite Kazhdan group and H’ arbitrary), and $\mathcal{V}$ is a closed subgroup of the group of unitaries of a finite separable von Neumann algebra (e.g. $\mathcal{V}$ countable discrete, or separable compact), then any $\mathcal{V}$ -valued measurable cocycle for a measure preserving action $\Gamma\curvearrowright X$ of Γ on a probability space (X,μ) which is weak mixing on H and s-malleable (e.g. the Bernoulli action $\Gamma\curvearrowright[0,1]^{\Gamma}$ ) is cohomologous to a group morphism of Γ into $\mathcal{V}$ . We use the case $\mathcal{V}$ discrete of this result to prove that if in addition Γ has no non-trivial finite normal subgroups then any orbit equivalence between $\Gamma\curvearrowright X$ and a free ergodic measure preserving action of a countable group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ≃Λ.

Fake projective planes

Abstract: A fake projective plane is a complex surface different from but has the same Betti numbers as the complex projective plane. It is a complex hyperbolic space form and has the smallest Euler Poincare characteristic among smooth surfaces of general type. The first example was constructed by Mumford. Later on two more examples were found by Ishida and Kato. A fourth possible one was recently proposed by Keum. In a recent joint work with Gopal Prasad, we showed that there are seventeen non-empty classes of fake projective planes and there can at most be four more specific classes. Higher dimensional analogues and examples were also obtained. The main purpose of the talk is to explain the joint work with Prasad and other related results such as arithmeticity of the lattices involved obtained earlier by Klingler and Yeung independently.

Tannakian duality for Anderson–Drinfeld motives and algebraic independence of Carlitz logarithms

Abstract: We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.

On the classification of simple inductive limit C*-algebras, II: The isomorphism theorem

Abstract: In this article, it is proved that the invariant consisting of the scaled ordered K-group and the space of tracial states, together with the natural pairing between them, is a complete invariant for the class of unital simple C *-algebras which can be expressed as the inductive limit of a sequence $$A_1\to A_2\to\cdots\to A_n\to\cdots$$ with $A_n=\bigoplus_{i=1}^{t_n}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$ , where X n,i is a compact metrizable space and P n,i is a projection in M [n,i](C(X n,i )) for each n and i, and the spaces X n,i are of uniformly bounded finite dimension. Note that the C *-algebras in the present class are not assumed to be of real rank zero, as they were in [EG2].

On the spectral gap for finitely-generated subgroups of SU(2)

Abstract: We prove the spectral gap property for free subgroups of SU(2) generated by elements satisfying a noncommutative diophantine property, in particular for free subgroups generated by elements with algebraic entries.

The Kodaira dimension of the moduli of K3 surfaces

Abstract: The global Torelli theorem for projective K3 surfaces was first proved by Piatetskii-Shapiro and Shafarevich 35 years ago, opening the way to treating moduli problems for K3 surfaces. The moduli space of polarised K3 surfaces of degree 2d is a quasi-projective variety of dimension 19. For general d very little has been known hitherto about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for d>61 and for d=46, 50, 54, 57, 58, 60.

A reduction theorem for the McKay conjecture

Abstract: The McKay conjecture asserts that for every finite group G and every prime p, the number of irreducible characters of G having p’-degree is equal to the number of such characters of the normalizer of a Sylow p-subgroup of G. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups. We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold for a finite group G if every simple group involved in G satisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL2(q) for all prime powers q≥4, and for the Suzuki groups Sz(q) and Ree groups R(q), where q=2 e or q=3 e respectively, and e>1 is odd. Since our conditions are also satisfied by the sporadic simple group J 1, it follows that the McKay conjecture holds (for all primes p) for every finite group having an abelian Sylow 2-subgroup.

Representation theory of mathcal{W} -algebras

Abstract: We study the representation theory of the $\mathcal{W}$ -algebra $\mathcal{W}_k(\bar{\mathfrak{g}})$ associated with a simple Lie algebra $\bar{\mathfrak{g}}$ at level k. We show that the “-” reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k∈ℂ. Moreover, we show that the character of each irreducible highest weight representation of $\mathcal{W}_k(\bar{\mathfrak{g}})$ is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra $\mathfrak{g}$ of $\bar{\mathfrak{g}}$ . As a consequence we complete (for the “-” reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of $\mathcal{W}$ -algebras.

Tropical varieties for non-archimedean analytic spaces

Abstract: Generalizing the construction from tropical algebraic geometry, we associate to every (irreducible d-dimensional) closed analytic subvariety of $\mathbb{G}_{m}^{n}$ a tropical variety in ℝ n with respect to a complete non-archimedean place. By methods of analytic and formal geometry, we prove that the tropical variety is a totally concave locally finite union of d-dimensional polytopes. For an algebraic morphism f:X’→A to a totally degenerate abelian variety A, we give a bound for the dimension of f(X’) in terms of the singularities of a strictly semistable model of X’. A closed d-dimensional subvariety X of A induces a periodic tropical variety. A generalization of Mumford’s construction yields models of X and A which can be handled with the theory of toric varieties. For a canonically metrized line bundle L on A, the measures c 1(L| X )∧d are piecewise Haar measures on X. Using methods of convex geometry, we give an explicit description of these measures in terms of tropical geometry. In a subsequent paper, this is a key step in the proof of Bogomolov’s conjecture for totally degenerate abelian varieties over function fields.

Partitions, Durfee symbols, and the Atkin–Garvan moments of ranks

Abstract: Atkin and Garvan introduced the moments of ranks of partitions in their work connecting ranks and cranks. Here we consider a combinatorial interpretation of these moments. This requires the introduction of a new representation for partitions, the Durfee symbol, and subsequent refinements. This in turn leads us to a variety of new congruences for our 'marked' Durfee symbols much in the spirit of Dyson's original conjectures on the ranks of partitions.

Right-veering diffeomorphisms of compact surfaces with boundary

Abstract: We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary. The monoid strictly contains the monoid of products of positive Dehn twists. We explain the relationship to tight contact structures and open book decompositions.

A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres

Abstract: We construct an invariant J M of integral homology spheres M with values in a completion $\widehat{\mathbb{Z}[q]}$ of the polynomial ring ℤ[q] such that the evaluation at each root of unity ζ gives the the SU(2) Witten–Reshetikhin–Turaev invariant τζ(M) of M at ζ. Thus J M unifies all the SU(2) Witten–Reshetikhin–Turaev invariants of M. It also follows that τζ(M) as a function on ζ behaves like an “analytic function” defined on the set of roots of unity.

Peripheral fillings of relatively hyperbolic groups

Abstract: In this paper a group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group G we define a peripheral filling procedure, which produces quotients of G by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3-manifold M on the fundamental group π1(M). The main result of the paper is an algebraic counterpart of Thurston’s hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of G ‘almost’ have the Congruence Extension Property and the group G is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings.

A simply connected surface of general type with p g =0 and K 2 =2

Abstract: In this paper we construct a simply connected, minimal, complex surface of general type with p g =0 and K 2=2 using a rational blow-down surgery and a ℚ-Gorenstein smoothing theory.

On the 4-rank of class groups of quadratic number fields

Abstract: We prove that the 4-rank of class groups of quadratic number fields behaves as predicted in an extension due to Gerth of the Cohen–Lenstra heuristics.

The level-rank duality for non-abelian theta functions

Abstract: We prove that the strange duality conjecture of Beauville–Donagi–Tu holds for all curves. We establish first a more extended level-rank duality, interesting in its own right, from which the standard level-rank duality follows by restriction.

Motivic Serre invariants, ramification, and the analytic Milnor fiber

Abstract: We show how formal and rigid geometry can be used in the theory of complex singularities, and in particular in the study of the Milnor fibration and the motivic zeta function. We introduce the so-called analytic Milnor fiber associated to the germ of a morphism f from a smooth complex algebraic variety X to the affine line. This analytic Milnor fiber is a smooth rigid variety over the field of Laurent series C((t)). Its etale cohomology coincides with the singular cohomology of the classical topological Milnor fiber of f; the monodromy transformation is given by the Galois action. Moreover, the points on the analytic Milnor fiber are closely related to the motivic zeta function of f, and the arc space of X. We show how the motivic zeta function can be recovered as some kind of Weil zeta function of the formal completion of X along the special fiber of f, and we establish a corresponding Grothendieck trace formula, which relates, in particular, the rational points on the analytic Milnor fiber over finite extensions of C((t)), to the Galois action on its etale cohomology. The general observation is that the arithmetic properties of the analytic Milnor fiber reflect the structure of the singularity of the germ f.

Stringy K-theory and the Chern character

Abstract: We construct two new G-equivariant rings: $\mathcal{K}(X,G)$ , called the stringy K-theory of the G-variety X, and $\mathcal{H}(X,G)$ , called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne–Mumford stack $\mathcal{X}$ , we also construct a new ring $\mathsf{K}_{\mathrm{orb}}(\mathcal{X})$ called the full orbifold K-theory of $\mathcal{X}$ . We show that for a global quotient $\mathcal{X} = [X/G]$ , the ring of G-invariants $K_{\mathrm{orb}}(\mathcal{X})$ of $\mathcal{K}(X,G)$ is a subalgebra of $\mathsf{K}_{\mathrm{orb}}([X/G])$ and is linearly isomorphic to the “orbifold K-theory” of Adem-Ruan [AR] (and hence Atiyah-Segal), but carries a different “quantum” product which respects the natural group grading. We prove that there is a ring isomorphism $\mathcal{C}\mathbf{h}:\mathcal{K}(X,G)\to\mathcal{H}(X,G)$ , which we call the stringy Chern character. We also show that there is a ring homomorphism $\mathfrak{C}\mathfrak{h}_\mathrm{orb}:\mathsf{K}_{\mathrm{orb}}(\mathcal{X}) \rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})$ , which we call the orbifold Chern character, which induces an isomorphism $Ch_{\mathrm{orb}}:K_{\mathrm{orb}}(\mathcal{X})\rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})$ when restricted to the sub-algebra $K_{\mathrm{orb}}(\mathcal{X})$ . Here $H_{\mathrm{orb}}^\bullet(\mathcal{X})$ is the Chen–Ruan orbifold cohomology. We further show that $\mathcal{C}\mathbf{h}$ and $\mathfrak{C}\mathfrak{h}_\mathrm{orb}$ preserve many properties of these algebras and satisfy the Grothendieck–Riemann–Roch theorem with respect to etale maps. All of these results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds. We further prove that $\mathcal{H}(X,G)$ is isomorphic to Fantechi and Gottsche’s construction [FG, JKK]. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results greatly simplify the definitions of the Fantechi–Gottsche ring, Chen–Ruan orbifold cohomology, and the Abramovich–Graber–Vistoli orbifold Chow ring. We conclude by showing that a K-theoretic version of Ruan’s Hyper-Kahler Resolution Conjecture holds for the symmetric product of a complex projective surface with trivial first Chern class.

Hida families and rational points on elliptic curves

Abstract: Let E be an elliptic curve over Q of conductor N = Mp, having a prime p‖N of multiplicative reduction. Because E is modular, it corresponds to a normalised weight two eigenform on Γ0(N), whose q-expansion is denoted f = ∑n anq . LetX := hom(Z×p ,Z×p ) Z/(p−1)Z×Zp, which contains Z as a dense subset by associating to k ∈ Z the character x → xk−2. Denote by A(U) the ring of Cp-valued p-adic analytic functions on a compact open subset U of X. Hida’s theory associates to f a neighborhood U of 2 ∈ X (which can be assumed, for simplicity, to be contained in the residue class of 2 modulo p − 1) and a formal q-expansion

The Bogomolov conjecture for totally degenerate abelian varieties

Abstract: We prove the Bogomolov conjecture for a totally degenerate abelian variety A over a function field. We adapt Zhang’s proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A key step is the tropical equidistribution theorem for A at the totally degenerate place v. As a corollary, we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v.

Symmetric groups and expander graphs

Abstract: We construct explicit generating sets Sn and \(\tilde{S}_{n}\) of the alternating and the symmetric groups, which turn the Cayley graphs \(\mathcal{C}(\text{Alt}(n), S_{n})\) and \(\mathcal{C}(\text{Sym}(n), \tilde{S}_{n})\) into a family of bounded degree expanders for all n.