On the Hodge conjecture and the Tate conjecture for the Hilbert schemes of an abelian surface

2006 ◽  
Vol 279 (1-2) ◽  
pp. 217-231
Author(s):  
Ken-ichi Sugiyama
Keyword(s):  
Abelian Surface ◽  
Hilbert Schemes ◽  
Hodge Conjecture ◽  
Tate Conjecture

scholarly journals Higher Chow cycles on Abelian surfaces and a non-Archimedean analogue of the Hodge--conjecture

2014 ◽  
Vol 150 (4) ◽  
pp. 691-711
Author(s):  
Ramesh Sreekantan
Keyword(s):  
Local Field ◽  
Chow Group ◽  
Abelian Surface ◽  
Hodge Conjecture ◽  
Boundary Map ◽  
Localization Sequence ◽  
Hyperelliptic Jacobians ◽  
Higher Chow Group ◽  
Classical Construction ◽  
Ordinary Reduction

AbstractWe construct new indecomposable elements in the higher Chow group $CH^2(A,1)$ of a principally polarized Abelian surface over a $p$-adic local field, which generalize an element constructed by Collino [Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J. Algebraic Geom. 6 (1997), 393–415]. These elements are constructed using a generalization, due to Birkenhake and Wilhelm [Humbert surfaces and the Kummer plane, Trans. Amer. Math. Soc. 355 (2003), 1819–1841 (electronic)], of a classical construction of Humbert. They can be used to prove a non-Archimedean analogue of the Hodge-${\mathcal{D}}$-conjecture – namely, the surjectivity of the boundary map in the localization sequence – in the case where the Abelian surface has good and ordinary reduction.

scholarly journals THE INTEGRAL HODGE CONJECTURE FOR 3-FOLDS OF KODAIRA DIMENSION ZERO

2020 ◽  
pp. 1-21
Author(s):  
Burt Totaro
Keyword(s):  
Kodaira Dimension ◽  
Hodge Conjecture ◽  
Dimension Zero ◽  
Tate Conjecture

We prove the integral Hodge conjecture for all 3-folds $X$ of Kodaira dimension zero with $H^{0}(X,K_{X})$ not zero. This generalizes earlier results of Voisin and Grabowski. The assumption is sharp, in view of counterexamples by Benoist and Ottem. We also prove similar results on the integral Tate conjecture. For example, the integral Tate conjecture holds for abelian 3-folds in any characteristic.

scholarly journals Universal functors on symmetric quotient stacks of Abelian varieties

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Andreas Krug ◽  
Ciaran Meachan
Keyword(s):  
Elliptic Curve ◽  
Abelian Varieties ◽  
Abelian Surface ◽  
Hilbert Schemes ◽  
Abelian Surfaces ◽  
The One ◽  
Dimension One

AbstractWe consider certain universal functors on symmetric quotient stacks of Abelian varieties. In dimension two, we discover a family of $${{\mathbb {P}}}$$ P -functors which induce new derived autoequivalences of Hilbert schemes of points on Abelian surfaces; a set of braid relations on a holomorphic symplectic sixfold; and a pair of spherical functors on the Hilbert square of an Abelian surface, whose twists are related to the well-known Horja twist. In dimension one, our universal functors are fully faithful, giving rise to a semiorthogonal decomposition for the symmetric quotient stack of an elliptic curve (which we compare to the one discovered by Polishchuk–Van den Bergh), and they lift to spherical functors on the canonical cover, inducing twists which descend to give new derived autoequivalences here as well.

scholarly journals On the Hodge, Tate and Mumford-Tate Conjectures for Fibre Products of Families of Regular Surfaces with Geometric Genus 1

2018 ◽  
Vol 25 (3) ◽  
pp. 312-322
Author(s):  
Olga V. Oreshkina (Nikol’skaya)
Keyword(s):  
Real Field ◽  
Hodge Conjecture ◽  
Geometric Genus ◽  
Smooth Projective Curve ◽  
Tate Conjecture ◽  
Smooth Part ◽  
Fibre Products ◽  
Fibre Product ◽  
Totally Real ◽  
Hodge Groups

The Hodge, Tate and Mumford-Tate conjectures are proved for the fibre product of two non-isotrivial 1-parameter families of regular surfaces with geometric genus 1 under some conditions on degenerated fibres, the ranks of the N\'eron - Severi groups of generic geometric fibres and representations of Hodge groups in transcendental parts of rational cohomology.Let \(\pi_i:X_i\to C\quad (i = 1, 2)\) be a projective non-isotrivial family (possibly with degeneracies) over a smooth projective curve \(C\). Assume that the discriminant loci \(\Delta_i=\{\delta\in C\,\,\vert\,\, Sing(X_{i\delta})\neq\varnothing\} \quad (i = 1, 2)\) are disjoint, \(h^{2,0}(X_{ks})=1,\quad h^{1,0}(X_{ks}) = 0\) for any smooth fibre \(X_{ks}\), and the following conditions hold:\((i)\) for any point \(\delta \in \Delta_i\) and the Picard-Lefschetz transformation \( \gamma \in GL(H^2 (X_{is}, Q)) \), associated with a smooth part \(\pi'_i: X'_i\to C\setminus\Delta_i\) of the morphism \(\pi_i\) and with a loop around the point \(\delta \in C\), we have \((\log(\gamma))^2\neq0\);\((ii)\) the variety \(X_i \, (i = 1, 2)\), the curve \(C\) and the structure morphisms \(\pi_i:X_i\to C\) are defined over a finitely generated subfield \(k \hookrightarrow C\).If for generic geometric fibres \(X_{1s}\) \, and \, \(X_{2s}\) at least one of the following conditions holds: \((a)\) \(b_2(X_{1s})- rank NS(X_{1s})\) is an odd prime number, \(\quad\,\,\) \(b_2(X_{1s})- rank NS(X_{1s})\neq b_2(X_{2s})- rank NS(X_{2s})\); \((b)\) the ring \(End_{ Hg(X_{1s})} NS_ Q(X_{1s})^\perp\) is an imaginary quadratic field, \(\quad\,\, b_2(X_{1s})- rank NS(X_{1s})\neq 4,\) \(\quad\,\, End_{ Hg(X_{2s})} NS_ Q(X_{2s})^\perp\) is a totally real field or \(\,\, b_2(X_{1s})- rank NS(X_{1s})\,>\, b_2(X_{2s})- rank NS(X_{2s})\) ; \((c)\) \([b_2(X_{1s})- rank NS(X_{1s})\neq 4, \, End_{ Hg(X_{1s})} NS_ Q(X_{1s})^\perp= Q\); \(\quad\,\,\) \(b_2(X_{1s})- rank NS(X_{1s})\neq b_2(X_{2s})- rank NS(X_{2s})\),then for the fibre product \(X_1 \times_C X_2\) the Hodge conjecture is true, for any smooth projective \(k\)-variety \(X_0\) with the condition \(X_1 \times_C X_2\) \(\widetilde{\rightarrow}\) \(X_0 \otimes_k C\) the Tate conjecture on algebraic cycles and the Mumford-Tate conjecture for cohomology of even degree are true.

scholarly journals Families of abelian surfaces with real multiplication over Hilbert modular surfaces

1982 ◽  
Vol 88 ◽  
pp. 17-53 ◽  
Author(s):  
G. van der Geer ◽  
K. Ueno
Keyword(s):  
Endomorphism Ring ◽  
Symplectic Group ◽  
Abelian Surface ◽  
Quadratic Relation ◽  
Holomorphic Map ◽  
Real Multiplication ◽  
Compact Complex Surfaces ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Abelian Functions

Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.

scholarly journals Curved Rickard complexes and link homologies

2020 ◽  
Vol 2020 (769) ◽  
pp. 87-119
Author(s):  
Sabin Cautis ◽  
Aaron D. Lauda ◽  
Joshua Sussan
Keyword(s):  
Spectral Sequence ◽  
Quantum Groups ◽  
Braid Group ◽  
Derived Categories ◽  
Duke Math ◽  
Khovanov Homology ◽  
Hilbert Schemes ◽  
Coherent Sheaves ◽  
Knot Homology ◽  
Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

scholarly journals On the Hodge conjecture for quasi-smooth intersections in toric varieties

2021 ◽  
Author(s):  
Ugo Bruzzo ◽  
William Montoya
Keyword(s):  
Complete Intersection ◽  
Toric Variety ◽  
Toric Varieties ◽  
Picard Group ◽  
Hodge Conjecture ◽  
Smooth Hypersurface ◽  
Suitable Notion

AbstractWe establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in Bruzzo and Grassi (Commun Anal Geom 28: 1773–1786, 2020). We also show that the Hodge Conjecture holds asymptotically for suitable quasi-smooth hypersurface in the Noether–Lefschetz locus, where “asymptotically” means that the degree of the hypersurface is big enough, under the assumption that the ambient variety $${{\mathbb {P}}}_\Sigma ^{2k+1}$$ P Σ 2 k + 1 has Picard group $${\mathbb {Z}}$$ Z . This extends to a class of toric varieties Otwinowska’s result in Otwinowska (J Alg Geom 12: 307–320, 2003).

scholarly journals The close relation between border and Pommaret marked bases

2021 ◽  
Author(s):  
Cristina Bertone ◽  
Francesca Cioffi
Keyword(s):  
Finite Order ◽  
Polynomial Ring ◽  
Group Actions ◽  
Close Relation ◽  
Open Covering ◽  
Order Ideal ◽  
Lower Dimension ◽  
Hilbert Schemes ◽  
Affine Spaces ◽  
The Ideal

AbstractGiven a finite order ideal $${\mathcal {O}}$$ O in the polynomial ring $$K[x_1,\ldots , x_n]$$ K [ x 1 , … , x n ] over a field K, let $$\partial {\mathcal {O}}$$ ∂ O be the border of $${\mathcal {O}}$$ O and $${\mathcal {P}}_{\mathcal {O}}$$ P O the Pommaret basis of the ideal generated by the terms outside $${\mathcal {O}}$$ O . In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $$\partial {\mathcal {O}}$$ ∂ O -marked sets (resp. bases) and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked sets (resp. bases). We prove that a $$\partial {\mathcal {O}}$$ ∂ O -marked set B is a marked basis if and only if the $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked set P contained in B is a marked basis and generates the same ideal as B. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $$\partial {\mathcal {O}}$$ ∂ O -marked bases and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gröbner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in affine spaces of lower dimension. Furthermore, we observe that Pommaret marked schemes give an open covering of Hilbert schemes parameterizing 0-dimensional schemes without any group actions. Several examples are given throughout the paper.

scholarly journals Derived Hilbert schemes

2002 ◽  
Vol 15 (4) ◽  
pp. 787-815 ◽  
Author(s):  
Ionuţ Ciocan-Fontanine ◽  
Mikhail M. Kapranov
Keyword(s):  
Hilbert Schemes
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