scholarly journals On a theorem of Campana and P\u{a}un

2017 ◽  
Vol Volume 1 ◽  
Author(s):  
Christian Schnell
Keyword(s):  
General Type ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Complex Numbers ◽  
Tensor Power

Let $X$ be a smooth projective variety over the complex numbers, and $\Delta \subseteq X$ a reduced divisor with normal crossings. We present a slightly simplified proof for the following theorem of Campana and P\u{a}un: If some tensor power of the bundle $\Omega_X^1(\log \Delta)$ contains a subsheaf with big determinant, then $(X, \Delta)$ is of log general type. This result is a key step in the recent proof of Viehweg's hyperbolicity conjecture. Comment: 9 pages. Formatted using epigamath.sty

A remark on elementary contractions

1995 ◽  
Vol 118 (1) ◽  
pp. 183-188
Author(s):  
Qi Zhang
Keyword(s):  
Projective Variety ◽  
Intersection Number ◽  
Smooth Projective Variety ◽  
Complex Numbers ◽  
Canonical Bundle ◽  
Small Contraction ◽  
Dimension One ◽  
Extremal Ray

Let X be a smooth projective variety of dimension n over the field of complex numbers. We denote by Kx the canonical bundle of X. By Mori's theory, if Kx is not numerically effective (i.e. if there exists a curve on X which has negative intersection number with Kx), then there exists an extremal ray ℝ+[C] on X and an elementary contraction fR: X → Y associated with ℝ+[C].fR is called a small contraction if it is bi-rational and an isomorphism in co-dimension one.

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 A note on cubic equivalences

1986 ◽  
Vol 101 ◽  
pp. 1-26
Author(s):  
Hiroshi Saito
Keyword(s):  
Projective Variety ◽  
Present Note ◽  
Smooth Projective Variety ◽  
Complex Numbers

The present note is intended to be a supplement to [9], in which the following is proven: Let V be a smooth projective variety over the field of complex numbers C, T a smooth quasi-projective variety, Z a cycle in T × V of codimension p. If Z(t) is l-cube equivalent to zero for general t e T, then, setting r = dim V − p,vanishes for l′ < l, where {tZ} is the correspondence defined by Z.

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 Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type

2021 ◽  
Author(s):  
Lie Fu ◽  
Robert Laterveer ◽  
Charles Vial
Keyword(s):  
Projective Variety ◽  
Arbitrary Dimension ◽  
Smooth Projective Variety ◽  
The Other ◽  
Fano Variety ◽  
Fano Varieties ◽  
Intersection Product ◽  
Canonical Class ◽  
Main Input

AbstractGiven a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow–Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Küchle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow–Künneth decomposition, by establishing this for two families of Todorov surfaces.

scholarly journals Vector bundles trivialized by proper morphisms and the fundamental group scheme

2010 ◽  
Vol 10 (2) ◽  
pp. 225-234 ◽  
Author(s):  
Indranil Biswas ◽  
João Pedro P. Dos Santos
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Surjective Morphism ◽  
Finite Vector

AbstractLet X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro-finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.

Nonzero Symmetry Classes of Smallest Dimension

1980 ◽  
Vol 32 (4) ◽  
pp. 957-968 ◽  
Author(s):  
G. H. Chan ◽  
M. H. Lim
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Linear Mapping ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Complex Character ◽  
Tensor Power ◽  
Dimensional Vector Space ◽  
Symmetry Classes ◽  
Image Position

Let U be a k-dimensional vector space over the complex numbers. Let ⊗m U denote the mth tensor power of U where m ≧ 2. For each permutation σ in the symmetric group Sm, there exists a linear mapping P(σ) on ⊗mU such thatfor all x1, …, xm in U.Let G be a subgroup of Sm and λ an irreducible (complex) character on G. The symmetrizeris a projection of ⊗ mU. Its range is denoted by Uλm(G) or simply Uλ(G) and is called the symmetry class of tensors corresponding to G and λ.

scholarly journals FINITENESS OF LOG MINIMAL MODELS AND NEF CURVES ON -FOLDS IN CHARACTERISTIC

2018 ◽  
Vol 239 ◽  
pp. 76-109
Author(s):  
OMPROKASH DAS
Keyword(s):  
Projective Variety ◽  
Smooth Projective Variety ◽  
Minimal Models ◽  
Full Generality ◽  
Canonical Divisor ◽  
Arbitrary Characteristic ◽  
Finiteness Result ◽  
Effective Divisors ◽  
Cone Of Curves ◽  
Nef Cone

In this article, we prove a finiteness result on the number of log minimal models for 3-folds in $\operatorname{char}p>5$. We then use this result to prove a version of Batyrev’s conjecture on the structure of nef cone of curves on 3-folds in characteristic $p>5$. We also give a proof of the same conjecture in full generality in characteristic 0. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor $K_{X}$ of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on $X$.

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.

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