scholarly journals MINIMAL CLASSES ON THE INTERMEDIATE JACOBIAN OF A GENERIC CUBIC THREEFOLD

2010 ◽  
Vol 12 (01) ◽  
pp. 55-70 ◽  
Author(s):  
ANDREAS HÖRING
Keyword(s):  
Cohomology Class ◽  
Cubic Threefold ◽  
Intermediate Jacobian

Let X be a smooth cubic threefold. We can associate two objects to X: the intermediate Jacobian J and the Fano surface F parametrizing lines on X. By a theorem of Clemens and Griffiths, the Fano surface can be embedded in the intermediate Jacobian and the cohomology class of its image is minimal. In this paper, we show that if X is generic, the Fano surface is the only surface of minimal class in J.

scholarly journals The Intermediate Jacobian of the Cubic Threefold

1972 ◽  
Vol 95 (2) ◽  
pp. 281 ◽  
Author(s):  
C. Herbert Clemens ◽  
Phillip A. Griffiths
Keyword(s):  
Cubic Threefold ◽  
Intermediate Jacobian

Irrationality of generic cubic threefold via Weil's conjectures

2018 ◽  
Vol 20 (07) ◽  
pp. 1750078
Author(s):  
Dimitri Markushevich ◽  
Xavier Roulleau
Keyword(s):  
Upper Bounds ◽  
Prym Varieties ◽  
Reduction Modulo ◽  
Cubic Threefold ◽  
Arithmetic Method ◽  
Intermediate Jacobian ◽  
Counting Points

An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo [Formula: see text]. It is illustrated by an application to a cubic threefold, for which the hypothesis that its intermediate Jacobian is isomorphic to the Jacobian of a curve is contradicted by reducing modulo 3 and counting points over appropriate extensions of [Formula: see text]. As a spin-off, it is shown that the 5-dimensional Prym varieties arising as intermediate Jacobians of certain cubic 3-folds have the maximal number of points over [Formula: see text] which attains Perret's and Weil's upper bounds.

scholarly journals Folding of Hitchin Systems and Crepant Resolutions

2021 ◽  
Author(s):  
Florian Beck ◽  
Ron Donagi ◽  
Katrin Wendland
Keyword(s):  
Fixed Point ◽  
Lie Algebras ◽  
Integrable Systems ◽  
Trace Back ◽  
Dynkin Diagrams ◽  
Or Groups ◽  
Hitchin Systems ◽  
Singular Fibers ◽  
Crepant Resolutions ◽  
Intermediate Jacobian

Abstract Folding of ADE-Dynkin diagrams according to graph automorphisms yields irreducible Dynkin diagrams of $\textrm{ABCDEFG}$-types. This folding procedure allows to trace back the properties of the corresponding simple Lie algebras or groups to those of $\textrm{ADE}$-type. In this article, we implement the techniques of folding by graph automorphisms for Hitchin integrable systems. We show that the fixed point loci of these automorphisms are isomorphic as algebraic integrable systems to the Hitchin systems of the folded groups away from singular fibers. The latter Hitchin systems are isomorphic to the intermediate Jacobian fibrations of Calabi–Yau orbifold stacks constructed by the 1st author. We construct simultaneous crepant resolutions of the associated singular quasi-projective Calabi–Yau three-folds and compare the resulting intermediate Jacobian fibrations to the corresponding Hitchin systems.

scholarly journals The Walker Abel–Jacobi map descends

2021 ◽  
Author(s):  
Jeffrey D. Achter ◽  
Sebastian Casalaina-Martin ◽  
Charles Vial
Keyword(s):  
Abelian Variety ◽  
Projective Manifold ◽  
Field Of Definition ◽  
Definition Of ◽  
Complex Projective ◽  
Intermediate Jacobian ◽  
Cycle Classes ◽  
Coniveau Filtration

AbstractFor a complex projective manifold, Walker has defined a regular homomorphism lifting Griffiths’ Abel–Jacobi map on algebraically trivial cycle classes to a complex abelian variety, which admits a finite homomorphism to the Griffiths intermediate Jacobian. Recently Suzuki gave an alternate, Hodge-theoretic, construction of this Walker Abel–Jacobi map. We provide a third construction based on a general lifting property for surjective regular homomorphisms, and prove that the Walker Abel–Jacobi map descends canonically to any field of definition of the complex projective manifold. In addition, we determine the image of the l-adic Bloch map restricted to algebraically trivial cycle classes in terms of the coniveau filtration.

scholarly journals Troisi\`eme groupe de cohomologie non ramifi\'ee d'un solide cubique sur un corps de fonctions d'une variable

2018 ◽  
Vol Volume 2 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Alena Pirutka
Keyword(s):  
Function Field ◽  
Cohomology Group ◽  
Chow Group ◽  
Hodge Conjecture ◽  
Complex Curve ◽  
The Third ◽  
Generic Fibre ◽  
Nous Montrons ◽  
Unramified Cohomology ◽  
Cubic Threefold

En combinant une m\'ethode de C. Voisin avec la descente galoisienne sur le groupe de Chow en codimension $2$, nous montrons que le troisi\`eme groupe de cohomologie non ramifi\'ee d'un solide cubique lisse d\'efini sur le corps des fonctions d'une courbe complexe est nul. Ceci implique que la conjecture de Hodge enti\`ere pour les classes de degr\'e 4 vaut pour les vari\'et\'es projectives et lisses de dimension 4 fibr\'ees en solides cubiques au-dessus d'une courbe, sans restriction sur les fibres singuli\`eres. --------------- We prove that the third unramified cohomology group of a smooth cubic threefold over the function field of a complex curve vanishes. For this, we combine a method of C. Voisin with Galois descent on the codimension $2$ Chow group. As a corollary, we show that the integral Hodge conjecture holds for degree $4$ classes on smooth projective fourfolds equipped with a fibration over a curve, the generic fibre of which is a smooth cubic threefold, with arbitrary singularities on the special fibres. Comment: in French

THE MINIMAL GENUS PROBLEM IN RULED MANIFOLDS

2008 ◽  
Vol 17 (04) ◽  
pp. 471-482
Author(s):  
XU-AN ZHAO ◽  
HONGZHU GAO
Keyword(s):  
Cohomology Class ◽  
Standard Form ◽  
Geometric Construction ◽  
Diffeomorphism Group ◽  
Minimal Genus ◽  
Adjunction Formula ◽  
Embedded Surfaces

In this paper, we consider the minimal genus problem in a ruled 4-manifold M. There are three key ingredients in the studying, the action of diffeomorphism group of M on H2(M,Z), the geometric construction of surfaces representing a cohomology class and the generalized adjunction formula. At first, we discuss the standard form (see Definition 1.1) of a class under the action of diffeomorphism group on H2(M,Z), we prove the uniqueness of the standard form. Then we construct some embedded surfaces representing the standard forms of some positive classes, the generalized adjunction formula is used to show that these surfaces realize the minimal genera.

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