Mostrando recursos 1 - 20 de 6.159

  1. On the Tate and Mumford–Tate conjectures in codimension $1$ for varieties with $h^{2,0}=1$

    Moonen, Ben
    We prove the Tate conjecture for divisor classes and the Mumford–Tate conjecture for the cohomology in degree $2$ for varieties with $h^{2,0}=1$ over a finitely generated field of characteristic $0$ , under a mild assumption on their moduli. As an application of this general result, we prove the Tate and Mumford–Tate conjectures for several classes of algebraic surfaces with $p_{g}=1$ .

  2. On the cubical geometry of Higman’s group

    Martin, Alexandre
    We investigate the cocompact action of Higman’s group on a $\operatorname{CAT}(0)$ square complex associated to its standard presentation. We show that this action is in a sense intrinsic, which allows for the use of geometric techniques to study the endomorphisms of the group, and we show striking similarities with mapping class groups of hyperbolic surfaces, outer automorphism groups of free groups, and linear groups over the integers. We compute explicitly the automorphism group and outer automorphism group of Higman’s group and show that the group is both Hopfian and co-Hopfian. We actually prove a stronger rigidity result about the endomorphisms...

  3. Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle

    Filip, Simion
    We describe all the situations in which the Kontsevich–Zorich (KZ) cocycle has zero Lyapunov exponents. Confirming a conjecture of Forni, Matheus, and Zorich, we find this only occurs when the cocycle satisfies additional geometric constraints. We also show that the connected components of the Zariski closure of the monodromy must be from a specific list, and the representations in which they can occur are described. The number of zero exponents of the KZ cocycle is then as small as possible, given its monodromy.

  4. Nonsqueezing property of contact balls

    Chiu, Sheng-Fu
    In this paper we solve a contact nonsqueezing conjecture proposed by Eliashberg, Kim, and Polterovich. Let $B_{R}$ be the open ball of radius $R$ in $\mathbb{R}^{2n}$ , and let $\mathbb{R}^{2n}\times\mathbb{S}^{1}$ be the prequantization space equipped with the standard contact structure. Following Tamarkin’s idea, we apply microlocal category methods to prove that if $R$ and $r$ satisfy $1\leq\pi r^{2}\lt \pi R^{2}$ , then it is impossible to squeeze the contact ball $B_{R}\times\mathbb{S}^{1}$ into $B_{r}\times\mathbb{S}^{1}$ via compactly supported contact isotopies.

  5. Borelian subgroups of simple Lie groups

    de Saxcé, Nicolas
    We prove that in a simple real Lie group, there is no Borel measurable dense subgroup of intermediate Hausdorff dimension.

  6. Algebraic Birkhoff factorization and the Euler–Maclaurin formula on cones

    Guo, Li; Paycha, Sylvie; Zhang, Bin
    We equip the space of lattice cones with a coproduct which makes it a cograded, coaugmented, connnected coalgebra. The exponential generating sum and exponential generating integral on lattice cones can be viewed as linear maps on this space with values in the space of meromorphic germs with linear poles at zero. We investigate the subdivision properties—reminiscent of the inclusion-exclusion principle for the cardinal on finite sets—of such linear maps and show that these properties are compatible with the convolution quotient of maps on the coalgebra. Implementing the algebraic Birkhoff factorization procedure on the linear maps under consideration, we factorize the...

  7. Geometry of webs of algebraic curves

    Hwang, Jun-Muk
    A family of algebraic curves covering a projective variety $X$ is called a web of curves on $X$ if it has only finitely many members through a general point of $X$ . A web of curves on $X$ induces a web-structure (in the sense of local differential geometry) in a neighborhood of a general point of $X$ . We study how the local differential geometry of the web-structure affects the global algebraic geometry of $X$ . Under two geometric assumptions on the web-structure—the pairwise nonintegrability condition and the bracket-generating condition—we prove that the local differential geometry determines the global algebraic...

  8. Proof of linear instability of the Reissner–Nordström Cauchy horizon under scalar perturbations

    Luk, Jonathan; Oh, Sung-Jin
    It has long been suggested that solutions to the linear scalar wave equation ¶ \[\Box_{g}\phi=0\] on a fixed subextremal Reissner–Nordström spacetime with nonvanishing charge are generically singular at the Cauchy horizon. We prove that generic smooth and compactly supported initial data on a Cauchy hypersurface indeed give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. In particular, the solution generically does not belong to $W^{1,2}_{\mathrm{loc}}$ . This instability is related to the celebrated blue-shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture...

  9. A topological property of asymptotically conical self-shrinkers of small entropy

    Bernstein, Jacob; Wang, Lu
    For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder we show that the link of the asymptotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the self-shrinker. Combining this with recent work of Brendle, we conclude that the round sphere uniquely minimizes the entropy among all nonflat two-dimensional self-shrinkers. This confirms a conjecture of Colding, Ilmanen, Minicozzi, and White in dimension two.

  10. Level-raising and symmetric power functoriality, III

    Clozel, Laurent; Thorne, Jack A.
    The simplest case of the Langlands functoriality principle asserts the existence of the symmetric powers $\operatorname{Sym}^{n}$ of a cuspidal representation of $\operatorname{GL}(2)$ over the adèles of $F$ , where $F$ is a number field. In 1978, Gelbart and Jacquet proved the existence of $\operatorname{Sym}^{2}$ . After this, progress was slow, eventually leading, through the work of Kim and Shahidi, to the existence of $\operatorname{Sym}^{3}$ and $\operatorname{Sym}^{4}$ . In this series of articles we revisit this problem using recent progress in the deformation theory of modular Galois representations. As a consequence, our methods apply only to classical modular forms on a...

  11. Transition asymptotics for the Painlevé II transcendent

    Bothner, Thomas
    We consider real-valued solutions $u=u(x|s)$ , $x\in\mathbb{R}$ , of the second Painlevé equation $u_{xx}=xu+2u^{3}$ which are parameterized in terms of the monodromy data $s\equiv(s_{1},s_{2},s_{3})\subset\mathbb{C}^{3}$ of the associated Flaschka–Newell system of rational differential equations. Our analysis describes the transition, as $x\rightarrow-\infty$ , between the oscillatory power-like decay asymptotics for $|s_{1}|\lt 1$ (Ablowitz–Segur) to the power-like growth behavior for $|s_{1}|=1$ (Hastings–McLeod) and from the latter to the singular oscillatory power-like growth for $|s_{1}|\gt 1$ (Kapaev). It is shown that the transition asymptotics are of Boutroux type; that is, they are expressed in terms of Jacobi elliptic functions. As applications of our results...

  12. K-stability for Fano manifolds with torus action of complexity $1$

    Ilten, Nathan; Süß, Hendrik
    We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension $1$ . Using a recent result of Datar and Székelyhidi, we effectively determine the existence of Kähler–Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kähler–Einstein Fano threefolds and Fano threefolds admitting a nontrivial Kähler–Ricci soliton.

  13. The Cauchy–Szegő projection for domains in $\mathbb{C}^{n}$ with minimal smoothness

    Lanzani, Loredana; Stein, Elias M.
    We prove the $L^{p}(bD)$ -regularity of the Cauchy–Szegő projection (also known as the Szegő projection) for bounded domains $D\subset\mathbb{C}^{n}$ which are strongly pseudoconvex and whose boundary satisfies the minimal regularity condition of class $C^{2}$ .

  14. Derived automorphism groups of K3 surfaces of Picard rank $1$

    Bayer, Arend; Bridgeland, Tom
    We give a complete description of the group of exact autoequivalences of the bounded derived category of coherent sheaves on a K3 surface of Picard rank $1$ . We do this by proving that a distinguished connected component of the space of stability conditions is preserved by all autoequivalences and is contractible.

  15. Derived equivalences for rational Cherednik algebras

    Losev, Ivan
    Let $W$ be a complex reflection group, and let $H_{c}(W)$ be the rational Cherednik algebra for $W$ depending on a parameter $c$ . One can consider the category $\mathcal{O}$ for $H_{c}(W)$ . We prove a conjecture of Rouquier that the categories $\mathcal{O}$ for $H_{c}(W)$ and $H_{c'}(W)$ are derived-equivalent, provided that the parameters $c,c'$ have integral difference. Two main ingredients of the proof are a connection between the Ringel duality and Harish-Chandra bimodules and an analogue of a deformation technique developed by the author and Bezrukavnikov. We also show that some of the derived equivalences we construct are perverse.

  16. The dynamical André–Oort conjecture: Unicritical polynomials

    Ghioca, D.; Krieger, H.; Nguyen, K. D.; Ye, H.
    We establish equidistribution with respect to the bifurcation measure of postcritically finite (PCF) maps in any one-dimensional algebraic family of unicritical polynomials. Using this equidistribution result, together with a combinatorial analysis of certain algebraic correspondences on the complement of the Mandelbrot set $\mathcal{M}_{2}$ (or generalized Mandelbrot set $\mathcal{M}_{d}$ for degree $d\gt 2$ ), we classify all curves $C\subset{\mathbb{A}}^{2}$ defined over ${\mathbb{C}}$ with Zariski-dense subsets of points $(a,b)\in C$ , such that both $z^{d}+a$ and $z^{d}+b$ are simultaneously PCF for a fixed degree $d\geq2$ . Our result is analogous to the famous result of André regarding plane curves which contain infinitely...

  17. Multivariable $(\varphi,\Gamma)$ -modules and locally analytic vectors

    Berger, Laurent
    Let $K$ be a finite extension of $\mathbf{Q}_{p}$ , and let $G_{K}=\mathrm{Gal}(\overline{\mathbf{Q}}_{p}/K)$ . There is a very useful classification of $p$ -adic representations of $G_{K}$ in terms of cyclotomic $(\varphi,\Gamma)$ -modules (cyclotomic means that $\Gamma=\mathrm{Gal}(K_{\infty}/K)$ where $K_{\infty}$ is the cyclotomic extension of $K$ ). One particularly convenient feature of the cyclotomic theory is the fact that the $(\varphi,\Gamma)$ -module attached to any $p$ -adic representation is overconvergent. ¶ Questions pertaining to the $p$ -adic local Langlands correspondence lead us to ask for a generalization of the theory of $(\varphi,\Gamma)$ -modules, with the cyclotomic extension replaced by an infinitely ramified $p$ -adic Lie...

  18. Polynomials vanishing on Cartesian products: The Elekes–Szabó theorem revisited

    Raz, Orit E.; Sharir, Micha; De Zeeuw, Frank
    Let $F\in{\mathbb{C}}[x,y,z]$ be a constant-degree polynomial, and let $A,B,C\subset{\mathbb{C}}$ be finite sets of size $n$ . We show that $F$ vanishes on at most $O(n^{11/6})$ points of the Cartesian product $A\times B\times C$ , unless $F$ has a special group-related form. This improves a theorem of Elekes and Szabó and generalizes a result of Raz, Sharir, and Solymosi. The same statement holds over ${\mathbb{R}}$ , and a similar statement holds when $A,B,C$ have different sizes (with a more involved bound replacing $O(n^{11/6})$ ). This result provides a unified tool for improving bounds in various Erdős-type problems in combinatorial geometry, and...

  19. Global well-posedness and scattering for the defocusing, $L^{2}$ -critical, nonlinear Schrödinger equation when $d=2$

    Dodson, Benjamin
    In this article we prove that the defocusing, cubic nonlinear Schrödinger initial value problem is globally well posed and scattering for $u_{0}\in L^{2}(\mathbf{R}^{2})$ . The proof uses the bilinear estimates of Planchon and Vega and a frequency-localized interaction Morawetz estimate similar to the high-frequency estimate of Colliander, Keel, Staffilani, Takaoka, and Tao and especially the low-frequency estimate of Dodson.

  20. Equations of tropical varieties

    Giansiracusa, Jeffrey; Giansiracusa, Noah
    We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T}=(\mathbb{R}\cup\{-\infty\},\mathrm{max},+)$ by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring $R$ with non-Archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of $\mathbb{T}$ -points this reduces to Kajiwara–Payne’s extended tropicalization, and in the case of a...

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