Mostrando recursos 1 - 20 de 6.171

  1. Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into $\operatorname{PSL}(3,\mathbb{R})$

    Tholozan, Nicolas
    This article studies the geometry of proper open convex domains in the projective space $\mathbb{R}\mathbf{P}^{n}$ . These domains carry several projective invariant distances, among which are the Hilbert distance $d^{H}$ and the Blaschke distance $d^{B}$ . We prove a thin inequality between those distances: for any two points $x$ and $y$ in such a domain, ¶ \[d^{B}(x,y)\lt d^{H}(x,y)+1.\] ¶ We then give two interesting consequences. The first one answers a conjecture of Colbois and Verovic on the volume entropy of Hilbert geometries: for any proper open convex domain in $\mathbb{R}\mathbf{P}^{n}$ , the volume of a ball of radius $R$ grows at most like $e^{(n-1)R}$...

  2. Infinitesimal Newton–Okounkov bodies and jet separation

    Küronya, Alex; Lozovanu, Victor
    In this article we explore the connection between asymptotic base loci and Newton–Okounkov bodies associated to infinitesimal flags. Analogously to the surface case, we obtain complete characterizations of augmented and restricted base loci. Interestingly enough, an integral part of the argument is a study of the relationship between certain simplices contained in Newton–Okoukov bodies and jet separation; our results also lead to a convex geometric description of moving Seshadri constants.

  3. On the formal degrees of square-integrable representations of odd special orthogonal and metaplectic groups

    Ichino, Atsushi; Lapid, Erez; Mao, Zhengyu
    The formal degree conjecture relates the formal degree of an irreducible square-integrable representation of a reductive group over a local field to the special value of the adjoint $\gamma$ -factor of its $L$ -parameter. In this article, we prove the formal degree conjecture for odd special orthogonal and metaplectic groups in the generic case, which, combined with Arthur’s work on the local Langlands correspondence, implies the conjecture in the nongeneric case.

  4. Involutive Heegaard Floer homology

    Hendricks, Kristen; Manolescu, Ciprian
    Using the conjugation symmetry on Heegaard Floer complexes, we define a $3$ -manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to $\mathbb{Z}_{4}$ -equivariant Seiberg–Witten Floer homology. Further, we obtain two new invariants of homology cobordism, $\underline{d}$ and $\bar{d}$ , and two invariants of smooth knot concordance, $\underline{V}_{0}$ and $\overline{V}_{0}$ . We also develop a formula for the involutive Heegaard Floer homology of large integral surgeries on knots. We give explicit calculations in the case of L-space knots and thin knots. In particular, we show that $\underline{V}_{0}$ detects the nonsliceness of the figure-eight knot. Other applications include...

  5. Modular cocycles and linking numbers

    Duke, W.; Imamoḡlu, Ö.; Tóth, Á.
    It is known that the $3$ -manifold $\operatorname{SL}(2,\mathbb{Z})\backslash\operatorname{SL}(2,\mathbb{R})$ is diffeomorphic to the complement of the trefoil knot in $S^{3}$ . E. Ghys showed that the linking number of this trefoil knot with a modular knot is given by the Rademacher symbol, which is a homogenization of the classical Dedekind symbol. The Dedekind symbol arose historically in the transformation formula of the logarithm of Dedekind’s eta function under $\operatorname{SL}(2,\mathbb{Z})$ . In this paper we give a generalization of the Dedekind symbol associated to a fixed modular knot. This symbol also arises in the transformation formula of a certain modular function. It...

  6. Equivariant indices of $\operatorname{Spin}^{c}$ -Dirac operators for proper moment maps

    Hochs, Peter; Song, Yanli
    We define an equivariant index of $\operatorname{Spin}^{c}$ -Dirac operators on possibly noncompact manifolds, acted on by compact, connected Lie groups. Our main result is that the index decomposes into irreducible representations according to the quantization commutes with reduction principle.

  7. The Prym–Green conjecture for torsion line bundles of high order

    Farkas, Gavril; Kemeny, Michael
    Using a construction of Barth and Verra that realizes torsion bundles on sections of special K3 surfaces, we prove that the minimal resolution of a general paracanonical curve $C$ of odd genus $g$ and order $\ell\geq\sqrt{\frac{g+2}{2}}$ is natural, thus proving the Prym–Green conjecture. In the process, we confirm the expectation of Barth and Verra concerning the number of curves with $\ell$ -torsion line bundle in a linear system on a special K3 surface.

  8. On the center of quiver Hecke algebras

    Shan, P.; Varagnolo, M.; Vasserot, E.
    We compute the equivariant cohomology ring of the moduli space of framed instantons over the affine plane. It is a Rees algebra associated with the center of cyclotomic degenerate affine Hecke algebras of type $A$ . We also give some related results on the center of quiver Hecke algebras and the cohomology of quiver varieties.

  9. Chern slopes of surfaces of general type in positive characteristic

    Urzúa, Giancarlo
    Let $\mathbf{{k}}$ be an algebraically closed field of characteristic $p\gt 0$ , and let $C$ be a nonsingular projective curve over $\mathbf{{k}}$ . We prove that for any real number $x\geq2$ , there are minimal surfaces of general type $X$ over $\mathbf{{k}}$ such that (a) $c_{1}^{2}(X)\gt 0$ , $c_{2}(X)\gt 0$ , (b) $\pi_{1}^{\acute{e}t}(X)\simeq\pi_{1}^{\acute{e}t}(C)$ , and (c) $c_{1}^{2}(X)/c_{2}(X)$ is arbitrarily close to $x$ . In particular, we show the density of Chern slopes in the pathological Bogomolov–Miyaoka–Yau interval $(3,\infty)$ for any given $p$ . Moreover, we prove that for $C=\mathbb{P}^{1}$ there exist surfaces $X$ as above with $H^{1}(X,\mathcal{O}_{X})=0$ , that is,...

  10. Functional calculus for generators of symmetric contraction semigroups

    Carbonaro, Andrea; Dragičević, Oliver
    We prove that every generator of a symmetric contraction semigroup on a $\sigma$ -finite measure space admits, for $1\lt p\lt \infty$ , a Hörmander-type holomorphic functional calculus on $L^{p}$ in the sector of angle $\phi^{*}_{p}=\operatorname{arcsin}\vert1-2/p\vert$ . The obtained angle is optimal.

  11. Linear differential equations on the Riemann sphere and representations of quivers

    Hiroe, Kazuki
    Our interest in this article is a generalization of the additive Deligne–Simpson problem, which was originally defined for Fuchsian differential equations on the Riemann sphere. We extend this problem to differential equations having an arbitrary number of unramified irregular singular points, and we determine the existence of solutions of the generalized additive Deligne–Simpson problems. Moreover, we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the Riemann sphere (namely, open embedding of the moduli spaces into quiver varieties and the nonemptiness condition of the moduli spaces). Furthermore, we detail the connectedness...

  12. The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra

    Morton, Hugh; Samuelson, Peter
    We give a generators and relations presentation of the HOMFLYPT skein algebra $H$ of the torus $T^{2}$ , and we give an explicit description of the module corresponding to the solid torus. Using this presentation, we show that $H$ is isomorphic to the $\sigma=\bar{\sigma}^{-1}$ specialization of the elliptic Hall algebra of Burban and Schiffmann. ¶ As an application, for an iterated cable $K$ of the unknot, we use the elliptic Hall algebra to construct a 3-variable polynomial that specializes to the $\lambda$ -colored HOMFLYPT polynomial of $K$ . We show that this polynomial also specializes to one constructed by Cherednik and Danilenko...

  13. 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 $p_{g}=1$ .

  14. 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...

  15. 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.

  16. 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.

  17. 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.

  18. 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...

  19. 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...

  20. 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...

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