Showing papers in "Mathematische Annalen in 2015"

Acylindrical hyperbolicity of groups acting on trees

Abstract: We provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees. In particular, we consider amalgamated products and HNN-extensions, one-relator groups, automorphism groups of polynomial algebras, $$3$$ -manifold groups and graph products. Acylindrical hyperbolicity is then used to obtain some results about the algebraic structure, analytic properties and measure equivalence rigidity of groups from these classes.

On maximum, typical and generic ranks

Abstract: We show that for several notions of rank including tensor rank, Waring rank, and generalized rank with respect to a projective variety, the maximum value of rank is at most twice the generic rank. We show that over the real numbers, the maximum value of the real rank is at most twice the smallest typical rank, which is equal to the (complex) generic rank.

Linear series on metrized complexes of algebraic curves

Abstract: A metrized complex of algebraic curves over an algebraically closed field $$\kappa $$ is, roughly speaking, a finite metric graph $$\Gamma $$ together with a collection of marked complete nonsingular algebraic curves $$C_v$$ over $$\kappa $$ , one for each vertex $$v$$ of $$\Gamma $$ ; the marked points on $$C_v$$ are in bijection with the edges of $$\Gamma $$ incident to $$v$$ . We define linear equivalence of divisors and establish a Riemann–Roch theorem for metrized complexes of curves which combines the classical Riemann–Roch theorem over $$\kappa $$ with its graph-theoretic and tropical analogues from Amini and Caporaso (Adv Math 240:1–23, 2013); Baker and Norine (Adv Math 215(2):766–788, 2007); Gathmann and Kerber (Math Z 259(1):217–230, 2008) and Mikhalkin and Zharkov (Tropical curves, their Jacobians and Theta functions. Contemporary Mathematics 203–231, 2007), providing a common generalization of all of these results. For a complete nonsingular curve $$X$$ defined over a non-Archimedean field $$\mathbb {K}$$ , together with a strongly semistable model $$\mathfrak {X}$$ for $$X$$ over the valuation ring $$R$$ of $$\mathbb {K}$$ , we define a corresponding metrized complex $$\mathfrak {C}\mathfrak {X}$$ of curves over the residue field $$\kappa $$ of $$\mathbb {K}$$ and a canonical specialization map $$\tau ^{\mathfrak {C}\mathfrak {X}}_*$$ from divisors on $$X$$ to divisors on $$\mathfrak {C}\mathfrak {X}$$ which preserves degrees and linear equivalence. We then establish generalizations of the specialization lemma from Baker (Algebra Number Theory 2(6):613–653, 2008) and its weighted graph analogue from Amini and Caporaso (Adv Math 240:1–23, 2013), showing that the rank of a divisor cannot go down under specialization from $$X$$ to $$\mathfrak {C}\mathfrak {X}$$ . As an application, we establish a concrete link between specialization of divisors from curves to metrized complexes and the theory of limit linear series due to Eisenbud and Harris (Invent Math 85:337–371, 1986). Using this link, we formulate a generalization of the notion of limit linear series to curves which are not necessarily of compact type and prove, among other things, that any degeneration of a $$\mathfrak {g}^r_d$$ in a regular family of semistable curves is a limit $$\mathfrak {g}^r_d$$ on the special fiber.

Center-stable manifold of the ground state in the energy space for the critical wave equation

Abstract: We construct a center-stable manifold of the ground state solitons in the energy space for the critical wave equation without imposing any symmetry, as the dynamical threshold between scattering and blow-up, and also as a collection of solutions which stay close to the ground states. Up to energy slightly above the ground state, this completes the 9-set classification of the global dynamics in our previous paper (DCDS 33:6, 2013). We can also extend the manifold to arbitrary energy size by adding large radiation. The manifold contains all the solutions scattering to the ground state solitons, and also some of those blowing up in finite time by concentration of the ground states.

Embedding theorems for Bergman spaces via harmonic analysis

Abstract: Let $$A^p_\omega $$ denote the Bergman space in the unit disc induced by a radial weight $$\omega $$ with the doubling property $$\int _{r}^1\omega (s)\,ds\le C\int _{\frac{1+r}{2}}^1\omega (s)\,ds$$ . The positive Borel measures such that the differentiation operator of order $$n\in \mathbb {N}\cup \{0\}$$ is bounded from $$A^p_\omega $$ into $$L^q(\mu )$$ are characterized in terms of geometric conditions when $$0

Invariant distributions, Beurling transforms and tensor tomography in higher dimensions

Abstract: In the recent articles Paternain et al. (J. Differ Geom, 98:147–181, 2014, Invent Math 193:229–247, 2013), a number of tensor tomography results were proved on two-dimensional manifolds. The purpose of this paper is to extend some of these methods to manifolds of any dimension. A central concept is the surjectivity of the adjoint of the geodesic ray transform, or equivalently the existence of certain distributions that are invariant under the geodesic flow. We prove that on any Anosov manifold, one can find invariant distributions with controlled first Fourier coefficients. The proof is based on subelliptic type estimates and a Pestov identity. We present an alternative construction valid on manifolds with nonpositive curvature, based on the fact that a natural Beurling transform on such manifolds turns out to be essentially a contraction. Finally, we obtain uniqueness results in tensor tomography both on simple and Anosov manifolds that improve earlier results by assuming a condition on the terminator value for a modified Jacobi equation.

Zur Theorie der stochastischen Prozesse

Abstract: Gegenstand der folgenden Untersuchung sind vom Zufall bedingte Prozesse mit einem Freiheitsgrad; genauer handelt es sich um diejenigen Funktionen \(F(t, x; \tau , \xi )\), die hierbei als Ubergangswahrscheinlichkeiten von einem Zustand \(x\) zur Zeit \(t\) in einen Zustand \(\le \xi \) zur Zeit \(\tau > t\) auftreten konnen (eine rein analytische Charakterisierung dieser Funktionen wird in Sect. 1, 1 gegeben).

Clasp technology to knot homology via the affine Grassmannian

Abstract: We categorify all the Reshetikhin–Turaev tangle invariants of type A. Our main tool is a categorification of the generalized Jones–Wenzl projectors (a.k.a. clasps) as infinite twists. Applying this to certain convolution product varieties on the affine Grassmannian we extend our earlier work with Cautis and Kamnitzer (Duke Math J 142:511–588, 2008; Invent Math 174:165–232, 2008) from standard to arbitrary representations.

Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions

Abstract: If stiff inclusions are closely located, then the stress, which is the gradient of the solution, may become arbitrarily large as the distance between two inclusions tends to zero. In this paper we investigate the asymptotic behavior of the stress concentration factor, which is the normalized magnitude of the stress concentration, as the distance between two inclusions tends to zero. For that purpose we show that the gradient of the solution to the case when two inclusions are touching decays exponentially fast near the touching point. We also prove a similar result when two inclusions are closely located and there is no potential difference on boundaries of two inclusions. We then use these facts to show that the stress concentration factor converges to a certain integral of the solution to the touching case as the distance between two inclusions tends to zero. We then present an efficient way to compute this integral.

Collapsing of the Chern–Ricci flow on elliptic surfaces

Abstract: We investigate the Chern–Ricci flow, an evolution equation of Hermitian metrics generalizing the Kahler–Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a Kahler–Einstein metric from the base. Some of our estimates are new even for the Kahler–Ricci flow. A consequence of our result is that, on every minimal non-Kahler surface of Kodaira dimension one, the Chern–Ricci flow converges in the sense of Gromov–Hausdorff to an orbifold Kahler–Einstein metric on a Riemann surface.

The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients

Abstract: The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with \(t\)-independent complex bounded measurable coefficients (\(t\) being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in \(L^{p'}\), subject to the square function and non-tangential maximal function estimates, if and only if the corresponding Regularity problem is solvable in \(L^p\). Moreover, the solutions admit layer potential representations. In particular, we prove that for any elliptic operator with \(t\)-independent real (possibly non-symmetric) coefficients there exists a \(p>1\) such that the Regularity problem is well-posed in \(L^p\).

Subadditivity of syzygies of Koszul algebras

Abstract: Estimates are obtained for the degrees of minimal syzygies of quotient algebras of polynomial rings. For a class that includes Koszul algebras in almost all characteristics, these degrees are shown to increase by at most two from one syzygy module to the next one. Even slower growth is proved if, in addition, the algebra satisfies Green and Lazarsfeld’s condition \(N_q\) with \(q\ge 2\).

Exponential estimate for the asymptotics of Bergman kernels

Abstract: We prove an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. Further, we give Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line bundles.

Maximal regularity for non-autonomous evolution equations

Abstract: We consider the maximal regularity problem for non-autonomous evolution equations Open image in new window (1) Each operator \(A(t)\) is associated with a sesquilinear form \({\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )}\) on a Hilbert space \(H\). We assume that these forms all have the same domain and satisfy some regularity assumption with respect to \(t\) (e.g., piecewise \(\alpha \)-Holder continuous for some Open image in new window). We prove maximal \(L_p\)-regularity for all \(u_0 \) in the real-interpolation space Open image in new window. The particular case where \(p = 2\) improves previously known results and gives a positive answer to a question of Lions (Equations differentielles operationnelles et problemes aux limites, Springer, Berlin, 1961) on the set of allowed initial data \(u_0\).

Filtrations and test-configurations

Abstract: We introduce a strengthening of K-stability, based on filtrations of the homogeneous coordinate ring. This allows for considering certain limits of families of test-configurations, which arise naturally in several settings. We prove that if a manifold with no automorphisms admits a cscK metric, then it satisfies this stronger stability notion. We also discuss the relation with the birational transformations in the definition of $$b$$ -stability.

Geometric models for higher Grothendieck–Witt groups in \(\mathbb {A}^1\)-homotopy theory

Abstract: We show that the higher Grothendieck–Witt groups, a.k.a. algebraic hermitian \(K\)-groups, are represented by an infinite orthogonal Grassmannian in the \(\mathbb {A}^1\)-homotopy category of smooth schemes over a regular base for which \(2\) is a unit in the ring of regular functions. We also give geometric models for various \(\mathbb {P}^1\)- and \(S^1\)-loop spaces of hermitian \(K\)-theory.

Orbifold quasimap theory

Abstract: We extend to orbifolds the quasimap theory of Ciocan-Fontanine and Kim (Adv Math 225(6):3022–3051, 2010; J Geom Phys 75:17–47, 2014) as well as the genus zero wall-crossing results from (Algebr Geom 1(4):400–448, 2014; Proceedings of the Conference on the Occasion of Mukai’s 60th Birthday, 2015). As a consequence, we obtain generalizations of orbifold mirror theorems, in particular, of the mirror theorem for toric orbifolds recently proved independently by Coates et al. (A mirror theorem for toric stacks).

Viscosity solutions of general viscous Hamilton–Jacobi equations

Abstract: We present comparison principles, Lipschitz estimates and study state constraints problems for degenerate, second-order Hamilton–Jacobi equations.

Vanishing sequences and Okounkov bodies

Abstract: We define and study the vanishing sequence along a real valuation of sections of a line bundle on a normal projective variety. Building on previous work of the first author with Huayi Chen, we prove an equidistribution result for vanishing sequences of large powers of a big line bundle, and study the limit measure; in particular, the latter is described in terms of restricted volumes for divisorial valuations. We also show on an example that the associated concave function on the Okounkov body can be discontinuous at boundary points.

A combinatorial Li–Yau inequality and rational points on curves

Abstract: We present a method to control gonality of nonarchimedean curves based on graph theory. Let $$k$$ denote a complete nonarchimedean valued field. We first prove a lower bound for the gonality of a curve over the algebraic closure of $$k$$ in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some “volume” of the original graph; this can be seen as a substitute for graphs of the Li–Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs. Finally, we apply the results to give a lower bound for the gonality of arbitrary Drinfeld modular curves over finite fields and for general congruence subgroups $$\varGamma $$ of $$\varGamma (1)$$ that is linear in the index $$[\varGamma (1):\varGamma ]$$ , with a constant that only depends on the residue field degree and the degree of the chosen “infinite” place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian.

64 lines on smooth quartic surfaces

Abstract: Let $$k$$ be a field of characteristic $$p\ge 0$$ with $$p e 2,3$$ . We prove that there are no geometrically smooth quartic surfaces $$S \subset \mathbb {P}^3_k$$ with more than 64 lines. As a key step, we derive the sharp bound that any line meets at most 20 other lines on $$S$$ .

Multiscale analysis of 1-rectifiable measures: necessary conditions

Abstract: We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $$\mathbb {R}^n$$ , $$n\ge 2$$ . To each locally finite Borel measure $$\mu $$ , we associate a function $$\widetilde{J}_2(\mu ,x)$$ which uses a weighted sum to record how closely the mass of $$\mu $$ is concentrated near a line in the triples of dyadic cubes containing $$x$$ . We show that $$\widetilde{J}_2(\mu ,\cdot )<\infty \ \mu $$ -a.e. is a necessary condition for $$\mu $$ to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from 2000. A novelty of this result is that no assumption is made on the upper Hausdorff density of the measure. Thus we are able to analyze general 1-rectifiable measures, including measures which are singular with respect to 1-dimensional Hausdorff measure.

W^*-Superrigidity for arbitrary actions of central quotients of braid groups

Abstract: For any $$n\geqslant 4$$ let $$\tilde{B}_n=B_n/Z(B_n)$$ be the quotient of the braid group $$B_n$$ through its center. We prove that any free ergodic probability measure preserving (pmp) action $$\tilde{B}_n\curvearrowright (X,\mu )$$ is virtually $$\hbox {W}^*$$ -superrigid in the following sense: if $$L^{\infty }(X)\rtimes \tilde{B}_n\cong L^{\infty }(Y)\rtimes \Lambda $$ , for an arbitrary free ergodic pmp action $$\Lambda \curvearrowright (Y, u )$$ , then the actions $$\tilde{B}_n\curvearrowright X,\Lambda \curvearrowright Y$$ are virtually conjugate. Moreover, we prove that the same holds if $$\tilde{B}_n$$ is replaced with a finite index subgroup of the direct product $$\tilde{B}_{n_1}\times \cdots \times \tilde{B}_{n_k}$$ , for some $$n_1,\ldots ,n_k\geqslant 4$$ . The proof uses the dichotomy theorem for normalizers inside crossed products by free groups from Popa and Vaes (212, 141–198, 2014) in combination with the OE superrigidity theorem for actions of mapping class groups from Kida (131, 99–109, 2008). Similar techniques allow us to prove that if a group $$\Gamma $$ is hyperbolic relative to a finite family of proper, finitely generated, residually finite, infinite subgroups, then the $$\hbox {II}_1$$ factor $$L^{\infty }(X)\rtimes \Gamma $$ has a unique Cartan subalgebra, up to unitary conjugacy, for any free ergodic pmp action $$\Gamma \curvearrowright (X,\mu )$$ .

The obstacle problem for the porous medium equation

Abstract: We prove existence results for the obstacle problem related to the porous medium equation. For sufficiently regular obstacles, we find continuous solutions whose time derivative belongs to the dual of a parabolic Sobolev space. We also employ the notion of weak solutions and show that for more general obstacles, such a weak solution exists. The latter result is a consequence of a stability property of weak solutions with respect to the obstacle.

Weighted Calderón–Zygmund and Rellich inequalities in L^p

Abstract: We find necessary and sufficient conditions for the validity of weighted Rellich and Calderon–Zygmund inequalities with respect to \(L^p\)-norm, \(1\le p \le \infty \), for functions in the whole space and in the half-space with Dirichlet boundary conditions. General operators like \(L=\varDelta +c\frac{x}{|x|^2}\cdot abla -\frac{b}{|x|^2}\) are considered. We compute best constants in some situations.

Rational curves, Dynkin diagrams and Fano manifolds with nef tangent bundle

Abstract: A Fano manifold $$X$$ with nef tangent bundle is of Flag-Type if it has the same kind of elementary contractions as a complete flag manifold. In this paper we present a method to associate a Dynkin diagram $$\mathcal {D}(X)$$ with any such $$X$$ , based on the numerical properties of its contractions. We then show that $$\mathcal {D}(X)$$ is the Dynkin diagram of a semisimple Lie group. As an application we prove that Campana–Peternell conjecture holds when $$X$$ is a Flag-Type manifold whose Dynkin diagram is $$A_n$$ , i.e. we show that $$X$$ is the variety of complete flags of linear subspaces in $$\mathbb {P}^n$$ .

Equivalence of ELSV and Bouchard-Mariño conjectures for r-spin Hurwitz numbers

Abstract: We propose two conjectures on Hurwitz numbers with completed \((r+1)\)-cycles, or, equivalently, on certain relative Gromov-Witten invariants of the projective line. The conjectures are analogs of the ELSV formula and of the Bouchard–Marino conjecture for ordinary Hurwitz numbers. Our \(r\)-ELSV formula is an equality between a Hurwitz number and an integral over the space of \(r\)-spin structures, that is, the space of stable curves with an \(r\)th root of the canonical bundle. Our \(r\)-BM conjecture is the statement that \(n\)-point functions for Hurwitz numbers satisfy the topological recursion associated with the spectral curve \(x = -y^r + \log y\) in the sense of Chekhov, Eynard, and Orantin. We show that the \(r\)-ELSV formula and the \(r\)-BM conjecture are equivalent to each other and provide some evidence for both.

Universal torsors and values of quadratic polynomials represented by norms

Abstract: Let \(K/k\) be an extension of number fields, and let \(P(t)\) be a quadratic polynomial over \(k\). Let \(X\) be the affine variety defined by \(P(t) = N_{K/k}(\mathbf {z})\). We study the Hasse principle and weak approximation for \(X\) in three cases. For \([K:k]=4\) and \(P(t)\) irreducible over \(k\) and split in \(K\), we prove the Hasse principle and weak approximation. For \(k=\mathbb {Q}\) with arbitrary \(K\), we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. For \([K:k]=4\) and \(P(t)\) irreducible over \(k\), we determine the Brauer group of smooth proper models of \(X\). In a case where it is non-trivial, we exhibit a counterexample to weak approximation.

The scale and tidy subgroups for endomorphisms of totally disconnected locally compact groups

Abstract: The scale of an endomorphism, $$\alpha $$ , of a totally disconnected, locally compact group $$G$$ is the minimum index $$[\alpha (U) : \alpha (U)\cap U]$$ , for $$U$$ a compact, open subgroup of $$G$$ . A structural characterization of subgroups at which the minimum is attained is established. This characterization extends the notion of subgroup tidy for $$\alpha $$ from the previously understood case when $$\alpha $$ is an automorphism to the case when $$\alpha $$ is merely an endomorphism.

The Radon transform between monogenic and generalized slice monogenic functions

Abstract: In Bures et al. (Elements of quaternionic analysis and Radon transform, 2009), the authors describe a link between holomorphic functions depending on a parameter and monogenic functions defined on \({\mathbb {R}}^{n+1}\) using the Radon and dual Radon transforms. The main aim of this paper is to further develop this approach. In fact, the Radon transform for functions with values in the Clifford algebra \({\mathbb {R}}_n\) is mapping solutions of the generalized Cauchy–Riemann equation, i.e., monogenic functions, to a parametric family of holomorphic functions with values in \({\mathbb {R}}_n\) and, analogously, the dual Radon transform is mapping parametric families of holomorphic functions as above to monogenic functions. The parametric families of holomorphic functions considered in the paper can be viewed as a generalization of the so-called slice monogenic functions. An important part of the problem solved in the paper is to find a suitable definition of the function spaces serving as the domain and the target of both integral transforms.