Hodge Classes and Abelian Varieties of Quaternionic Type

Giuseppe Lombardo

Displaying similar documents to “Hodge Classes and Abelian Varieties of Quaternionic Type”

Variations of mixed Hodge structure attached to the deformation theory of a complex variation of Hodge structures

Philippe Eyssidieux, Carlos Simpson (2011)

Journal of the European Mathematical Society

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Let $X$ be a compact Kähler manifold, $x \in X$ be a base point and $ρ : π_{1} (X, x) \to G L_{N} (C)$ be the monodromy representation of a $𝒞$ -VHS. Building on Goldman–Millson’s classical work, we construct a mixed Hodge structure on the complete local ring of the representation variety at $ρ$ and a variation of mixed Hodge structures whose monodromy is the universal deformation of $ρ$ .

Naive boundary strata and nilpotent orbits

Matt Kerr, Gregory Pearlstein (2014)

Annales de l’institut Fourier

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We give a Hodge-theoretic parametrization of certain real Lie group orbits in the compact dual of a Mumford-Tate domain, and characterize the orbits which contain a naive limit Hodge filtration. A series of examples are worked out for the groups $S U (2, 1)$ , $S p_{4}$ , and $G_{2}$ .

Derived invariance of the number of holomorphic $1$ -forms and vector fields

Mihnea Popa, Christian Schnell (2011)

Annales scientifiques de l'École Normale Supérieure

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We prove that smooth projective varieties with equivalent derived categories have isogenous Picard varieties. In particular their irregularity and number of independent vector fields are the same. This implies that all Hodge numbers are the same for arbitrary derived equivalent threefolds, as well as other consequences of derived equivalence based on numerical criteria.

Theta height and Faltings height

Fabien Pazuki (2012)

Bulletin de la Société Mathématique de France

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Using original ideas from J.-B. Bost and S. David, we provide an explicit comparison between the Theta height and the stable Faltings height of a principally polarized Abelian variety. We also give as an application an explicit upper bound on the number of $K$ -rational points of a curve of genus $g \geq 2$ under a conjecture of S. Lang and J. Silverman. We complete the study with a comparison between differential lattice structures.

Lagrangian fibrations on generalized Kummer varieties

Martin G. Gulbrandsen (2007)

Bulletin de la Société Mathématique de France

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We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface $A$ of Picard number one we find the following: The Kummer variety $K^{n} A$ is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if $n$ is a perfect square. And this is the case if and only if $K^{n} A$ carries a divisor with vanishing Beauville-Bogomolov square.

Projective normality of abelian varieties with a line bundle of type $(2, \dots)$

Elena Rubei (1998)

Bollettino dell'Unione Matematica Italiana

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Sia $X$ una varietà abeliana e $L$ un fibrato in rette ampio di tipo $(2, 2 d_{2}, \dots, 2 d_{g})$ su $X$ ; sia $φ_{L}$ l'applicazione associata a $L$ . In questo lavoro si dimostra il seguente fatto: se $d_{i} \leq 2$ per qualsiasi $i$ , $L$ non è mai normalmente generato (quindi, se $φ_{L}$ è un embedding, $φ_{L} (X)$ non è proiettivamente normale); negli altri casi invece $L$ è normalmente generato per $(X, c_{1} (L))$ generico nello spazio dei moduli delle varietà abeliane polarizzate di tipo $(2, 2 d_{2}, \dots, 2 d_{g})$ .

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Claire Voisin (2013)

Annales scientifiques de l'École Normale Supérieure

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We prove that Bloch’s conjecture is true for surfaces with $p_{g} = 0$ obtained as $0$ -sets $X_{σ}$ of a section $σ$ of a very ample vector bundle on a variety $X$ with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as $0$ on holomorphic $2$ -forms of $X_{σ}$ , then it acts as $0$ on $0$ -cycles of degree $0$ of $X_{σ}$ . In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general $X_{σ}$ ...

A new proof of the $L^{p}$ -conjecture for locally compact abelian groups

David L. Johnson (1982)

Colloquium Mathematicae

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Number of solutions in a box of a linear equation in an Abelian group

Maciej Zakarczemny (2016)

Colloquium Mathematicae

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For every finite Abelian group Γ and for all $g, a ₁, . . ., a_{k} \in Γ$ , if there exists a solution of the equation $\sum_{i = 1}^{k} a_{i} x_{i} = g$ in non-negative integers $x_{i} \leq b_{i}$ , where $b_{i}$ are positive integers, then the number of such solutions is estimated from below in the best possible way.

On a generalization of Abelian sequential groups

Saak S. Gabriyelyan (2013)

Fundamenta Mathematicae

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Let (G,τ) be a Hausdorff Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdorff (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group ${(G, τ)}^{\land}$ is a dense -closed subgroup of the compact group ${(G_{d})}^{\land}$ , where $G_{d}$ is the group G with...

Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn)

David Masser, Umberto Zannier (2015)

Journal of the European Mathematical Society

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In recent papers we proved a special case of a variant of Pink’s Conjecture for a variety inside a semiabelian scheme: namely for any curve inside anything isogenous to a product of two elliptic schemes. Here we go beyond the elliptic situation by settling the crucial case of any simple abelian surface scheme defined over the field of algebraic numbers, thus confirming an earlier conjecture of Shou-Wu Zhang. This is of particular relevance in the topic, also in view of very recent counterexamples...

New estimates for elliptic equations and Hodge type systems

Jean Bourgain, Haïm Brezis (2007)

Journal of the European Mathematical Society

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We establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in arbitrary dimension $n$ , with data in $L^{1}$ . We also present related results concerning differential forms with coefficients in the limiting Sobolev space $W^{1, n}$ .

Asymptotics of eigensections on toric varieties

A. Huckleberry, H. Sebert (2013)

Annales de l’institut Fourier

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Using exhaustion properties of invariant plurisubharmonic functions along with basic combinatorial information on toric varieties, we prove convergence results for sequences of densities $| ϕ_{n} |^{2} = | s_{N} |^{2} / | | s_{N} {| |}_{L^{2}}^{2}$ for eigensections $s_{N} \in Γ (X, L^{N})$ approaching a semiclassical ray. Here $X$ is a normal compact toric variety and $L$ is an ample line bundle equipped with an arbitrary positive bundle metric which is invariant with respect to the compact form of the torus. Our work was motivated by and extends that of Shiffman, Tate...

Overconvergent de Rham-Witt cohomology

Christopher Davis, Andreas Langer, Thomas Zink (2011)

Annales scientifiques de l'École Normale Supérieure

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The goal of this work is to construct, for a smooth variety $X$ over a perfect field k of finite characteristic $p > 0$ , an overconvergent de Rham-Witt complex $W^{†} Ω_{X / k}$ as a suitable subcomplex of the de Rham-Witt complex of Deligne-Illusie. This complex, which is functorial in $X$ , is a complex of étale sheaves and a differential graded algebra over the ring $W^{†} (𝒪_{X})$ of overconvergent Witt-vectors. If $X$ is affine one proves that there is an isomorphism between Monsky-Washnitzer cohomology and (rational) overconvergent...

Singularities of theta divisors and the geometry of $𝒜_{5}$

Gavril Farkas, Samuele Grushevsky, Salvati R. Manni, Alessandro Verra (2014)

Journal of the European Mathematical Society

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We study the codimension two locus $H$ in $𝒜_{g}$ consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class $[H] \in C H^{2} (𝒜_{g})$ for every $g$ . For $g = 4$ , this turns out to be the locus of Jacobians with a vanishing theta-null. For $g = 5$ , via the Prym map we show that $H \subset 𝒜_{5}$ has two components, both unirational, which we describe completely. We then determine the slope of the effective cone of $\bar{𝒜_{5}}$ and show that the component $\bar{N_{0}^{'}}$ of the Andreotti-Mayer...

$L^{2}$ -cohomology

J. Eichhorn (1986)

Banach Center Publications

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Hodge type decomposition

Wojciech Kozłowski (2007)

Annales Polonici Mathematici

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In the space $Λ^{p}$ of polynomial p-forms in ℝⁿ we introduce some special inner product. Let $H^{p}$ be the space of polynomial p-forms which are both closed and co-closed. We prove in a purely algebraic way that $Λ^{p}$ splits as the direct sum $d * (Λ^{p + 1}) \oplus δ * (Λ^{p - 1}) \oplus H^{p}$ , where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.

Polarizations of Prym varieties for Weyl groups via abelianization

Herbert Lange, Christian Pauly (2009)

Journal of the European Mathematical Society

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Let $π : Z \to X$ be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply connected Lie group $G$ . For any dominant weight $λ$ consider the curve $Y = Z / Stab (λ)$ . The Kanev correspondence defines an abelian subvariety $P_{λ}$ of the Jacobian of $Y$ . We compute the type of the polarization of the restriction of the canonical principal polarization of $Jac (Y)$ to $P_{λ}$ in some cases. In particular, in the case of the group $E_{8}$ we obtain families of Prym-Tyurin varieties. The main idea is...

The finiteness of compact varieties in $C^{n}$

Walter Rudin (1981)

Annales Polonici Mathematici

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Some Remarks on Prym-Tyurin Varieties

Giuliano Parigi (2007)

Bollettino dell'Unione Matematica Italiana

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The aims of the present paper can be described as follows: a) In [2] Beauville showed that if some endomorphism $u$ a Jacobian $J(C)$ has connected kernel, the principal polarization on $J(C)$ induces a multiple of the principal polarization on the image of $u$ . We reformulate and complete this theorem proving "constructively" the following: Theorem. Let $Z\subset J(C)$ be an abelian subvariety and $Y$ its complementary variety. $Z$ is a Prym-Tyurin variety with respect to $J(C)$ if and only if the following sequence...