scholarly journals A Characterization of Ordinary Abelian Varieties by the Frobenius Push-Forward of the Structure Sheaf II

2018 ◽  
Vol 2019 (19) ◽  
pp. 5975-5988
Author(s):  
Sho Ejiri ◽  
Akiyoshi Sannai
Keyword(s):  
Abelian Variety ◽  
Projective Variety ◽  
Abelian Varieties ◽  
Smooth Projective Variety ◽  
Line Bundles ◽  
Characteristic P ◽  
Structure Sheaf ◽  
Direct Sum ◽  
Push Forward

Abstract In this paper, we prove that a smooth projective variety X of characteristic p > 0 is an ordinary abelian variety if and only if KX is pseudo-effective and $F_{*}^{e}{\mathcal {O}}_{X}$ splits into a direct sum of line bundles for an integer e with pe > 2.

Positivity in varieties of maximal Albanese dimension

2018 ◽  
Vol 2018 (736) ◽  
pp. 225-253 ◽  
Author(s):  
Jungkai Alfred Chen ◽  
Zhi Jiang
Keyword(s):  
Abelian Variety ◽  
General Type ◽  
Projective Variety ◽  
Smooth Projective Variety

AbstractGiven a generically finite morphismffrom a smooth projective varietyXto an abelian varietyA, we show that{f_{*}\omega_{X}}is “sufficiently positive” onA. As an application, we prove that whenXis of general type, the global sections of{\omega_{X}^{2}}define a generically finite map ofX. We also study the structure ofXwhenXis of general type and satisfies{\chi(X,\omega_{X})=0}. We formulate a conjectural characterization of suchXand prove the conjecture whenAhas exactly three simple factors.

scholarly journals Abelian Varieties as Automorphism Groups of Smooth Projective Varieties

2018 ◽  
Vol 2020 (7) ◽  
pp. 1942-1956
Author(s):  
Davide Lombardo ◽  
Andrea Maffei
Keyword(s):  
Automorphism Group ◽  
Projective Variety ◽  
Automorphism Groups ◽  
Abelian Varieties ◽  
Smooth Projective Variety ◽  
Projective Varieties ◽  
Smooth Projective Varieties

Abstract We determine which complex abelian varieties can be realized as the automorphism group of a smooth projective variety.

scholarly journals Seshadri Constants for Curve Classes

2019 ◽  
Author(s):  
Mihai Fulger
Keyword(s):  
Normal Bundle ◽  
Projective Variety ◽  
Smooth Curve ◽  
Smooth Projective Variety ◽  
Projective Varieties ◽  
Seshadri Constants ◽  
Base Locus ◽  
Nef Divisors

Abstract We develop a local positivity theory for movable curves on projective varieties similar to the classical Seshadri constants of nef divisors. We give analogues of the Seshadri ampleness criterion, of a characterization of the augmented base locus of a big and nef divisor, and of the interpretation of Seshadri constants as an asymptotic measure of jet separation. As application, we show in any characteristic that if $C$ is a smooth curve with ample normal bundle in a smooth projective variety then the class of $C$ is in the strict interior of the Mori cone. This was conjectured by Peternell and proved by Ottem and Lau in Characteristic 0.

scholarly journals Index conditions and cup-product maps on Abelian varieties

2014 ◽  
Vol 25 (04) ◽  
pp. 1450036 ◽  
Author(s):  
Nathan Grieve
Keyword(s):  
Asymptotic Behavior ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Abelian Varieties ◽  
General Setting ◽  
Index Theorem ◽  
Line Bundles ◽  
One Dimensional ◽  
Cup Product ◽  
Product Maps

We study questions surrounding cup-product maps which arise from pairs of non-degenerate line bundles on an abelian variety. Important to our work is Mumford's index theorem which we use to prove that non-degenerate line bundles exhibit positivity analogous to that of ample line bundles. As an application we determine the asymptotic behavior of families of cup-product maps and prove that vector bundles associated to these families are asymptotically globally generated. To illustrate our results we provide several examples. For instance, we construct families of cup-product problems which result in a zero map on a one-dimensional locus. We also prove that the hypothesis of our results can be satisfied, in all possible instances, by a particular class of simple abelian varieties. Finally, we discuss the extent to which Mumford's theta groups are applicable in our more general setting.

scholarly journals Formality of -objects

2019 ◽  
Vol 155 (5) ◽  
pp. 973-994
Author(s):  
Andreas Hochenegger ◽  
Andreas Krug
Keyword(s):  
Projective Variety ◽  
Smooth Projective Variety ◽  
Complex Numbers ◽  
Cohomology Algebra ◽  
Triangulated Category ◽  
Endomorphism Algebra ◽  
Structure Sheaf

We show that a$\mathbb{P}$-object and simple configurations of$\mathbb{P}$-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient triangulated category. We also observe that the category generated by the structure sheaf of a smooth projective variety over the complex numbers only depends on its graded cohomology algebra.

scholarly journals Higher-dimensional Clifford–Severi equalities

2019 ◽  
Vol 22 (08) ◽  
pp. 1950079 ◽  
Author(s):  
Miguel Ángel Barja ◽  
Rita Pardini ◽  
Lidia Stoppino
Keyword(s):  
Lower Bound ◽  
Line Bundle ◽  
Abelian Variety ◽  
Projective Variety ◽  
Line Bundles ◽  
Smooth Complex ◽  
Higher Dimensional ◽  
Complex Projective Variety ◽  
Irregular Varieties ◽  
Complex Projective

Let [Formula: see text] be a smooth complex projective variety, [Formula: see text] a morphism to an abelian variety such that [Formula: see text] injects into [Formula: see text] and let [Formula: see text] be a line bundle on [Formula: see text]; denote by [Formula: see text] the minimum of [Formula: see text] for [Formula: see text]. The so-called Clifford–Severi inequalities have been proven in [M. A. Barja, Generalized Clifford–Severi inequality and the volume of irregular varieties, Duke Math. J. 164(3) (2015) 541–568; M. A. Barja, R. Pardini and L. Stoppino, Linear systems on irregular varieties, J. Inst. Math. Jussieu (2019) 1–39; doi:10.1017/S1474748019000069]; in particular, for any [Formula: see text] there is a lower bound for the volume given by: [Formula: see text] and, if [Formula: see text] is pseudoeffective, [Formula: see text] In this paper, we characterize varieties and line bundles for which the above Clifford–Severi inequalities are equalities.

On the Chow Groups of Supersingular Varieties

2002 ◽  
Vol 45 (2) ◽  
pp. 204-212 ◽  
Author(s):  
Najmuddin Fakhruddin
Keyword(s):  
Projective Variety ◽  
Abelian Varieties ◽  
Smooth Projective Variety ◽  
Chow Groups ◽  
Coniveau Filtration

AbstractWe compute the rational Chow groups of supersingular abelian varieties and some other related varieties, such as supersingular Fermat varieties and supersingular K3 surfaces. These computations are concordant with the conjectural relationship, for a smooth projective variety, between the structure of Chow groups and the coniveau filtration on the cohomology.

scholarly journals THE MOTIVE OF THE HILBERT CUBE

2016 ◽  
Vol 4 ◽  
Author(s):  
MINGMIN SHEN ◽  
CHARLES VIAL
Keyword(s):  
Recent Work ◽  
Hilbert Scheme ◽  
Projective Variety ◽  
Abelian Varieties ◽  
Smooth Projective Variety ◽  
Hilbert Cube ◽  
Rational Map ◽  
Chow Rings ◽  
Hyperkähler Varieties

The Hilbert scheme $X^{[3]}$ of length-3 subschemes of a smooth projective variety $X$ is known to be smooth and projective. We investigate whether the property of having a multiplicative Chow–Künneth decomposition is stable under taking the Hilbert cube. This is achieved by considering an explicit resolution of the rational map $X^{3}{\dashrightarrow}X^{[3]}$. The case of the Hilbert square was taken care of in Shen and Vial [Mem. Amer. Math. Soc.240(1139) (2016), vii+163 pp]. The archetypical examples of varieties endowed with a multiplicative Chow–Künneth decomposition is given by abelian varieties. Recent work seems to suggest that hyperKähler varieties share the same property. Roughly, if a smooth projective variety $X$ has a multiplicative Chow–Künneth decomposition, then the Chow rings of its powers $X^{n}$ have a filtration, which is the expected Bloch–Beilinson filtration, that is split.

Sign in / Sign up

Export Citation Format

Share Document