Mostrando recursos 1 - 20 de 6.163

  1. 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...

  2. 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.

  3. 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.

  4. 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.

  5. 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,...

  6. 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.

  7. 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...

  8. 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...

  9. 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.

  10. 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...

  11. 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...

  12. 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...

  13. 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.

  14. 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...

  15. 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...

  16. 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.

  17. 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 $C^{2}$ .

  18. 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.

  19. 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.

  20. 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...

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