scholarly journals Geometric properties of projective manifolds of small degree

2015 ◽  
Vol 160 (2) ◽  
pp. 257-277 ◽  
Author(s):  
SIJONG KWAK ◽  
JINHYUNG PARK
Keyword(s):  
Projective Variety ◽  
Smooth Projective Variety ◽  
Small Degree ◽  
Point Of View ◽  
Geometric Properties ◽  
Projective Varieties ◽  
Cox Rings ◽  
Projective Manifolds ◽  
Simply Connected ◽  
Smooth Projective Varieties

AbstractThe aim of this paper is to study geometric properties of non-degenerate smooth projective varieties of small degree from a birational point of view. First, using the positivity property of double point divisors and the adjunction mappings, we classify smooth projective varieties in $\mathbb{P}$r of degree d ⩽ r + 2, and consequently, we show that such varieties are simply connected and rationally connected except in a few cases. This is a generalisation of P. Ionescu's work. We also show the finite generation of Cox rings of smooth projective varieties in $\mathbb{P}$r of degree d ⩽ r with counterexamples for d = r + 1, r + 2. On the other hand, we prove that a non-uniruled smooth projective variety in $\mathbb{P}$r of dimension n and degree d ⩽ n(r − n) + 2 is Calabi–Yau, and give an example that shows this bound is also sharp.

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 Rational connectivity and analytic contractibility

2019 ◽  
Vol 2019 (747) ◽  
pp. 45-62
Author(s):  
Morgan Brown ◽  
Tyler Foster
Keyword(s):  
Laurent Series ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Homotopy Equivalence ◽  
Formal Laurent Series ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Rationally Connected ◽  
Smooth Projective Varieties

Abstract Let {{k}} be an algebraically closed field of characteristic 0, and let {f:X\to Y} be a morphism of smooth projective varieties over the ring {k((t))} of formal Laurent series. We prove that if a general geometric fiber of f is rationally connected, then the induced map {f^{\mathrm{an}}:X^{\mathrm{an}}\to Y^{\mathrm{an}}} between the Berkovich analytifications of X and Y is a homotopy equivalence. Two important consequences of this result are that the Berkovich analytification of any {\mathbb{P}^{n}} -bundle over a smooth projective {k((t))} -variety is homotopy equivalent to the Berkovich analytification of the base, and that the Berkovich analytification of a rationally connected smooth projective variety over {k((t))} is contractible.

Special varieties in adjunction theory and ample vector bundles

2001 ◽  
Vol 130 (1) ◽  
pp. 61-75 ◽  
Author(s):  
A. LANTERI ◽  
H. MAEDA
Keyword(s):  
Vector Bundles ◽  
Projective Variety ◽  
Global Section ◽  
Smooth Projective Variety ◽  
Ample Vector Bundle ◽  
Projective Varieties ◽  
Adjunction Theory ◽  
Set Up ◽  
Important Position ◽  
Smooth Projective Varieties

In this paper varieties are always assumed to be defined over the field [Copf ] of complex numbers.Given a smooth projective variety Z, the classification of smooth projective varieties X containing Z as an ample divisor occupies an extremely important position in the theory of polarized varieties and it is well-known that the structure of Z imposes severe restrictions on that of X. Inspired by this philosophy, we set up the following condition ([midast ]) in [LM1] in order to generalize several results on ample divisors to ample vector bundles:([midast ]) [Escr ] is an ample vector bundle of rank r [ges ] 2 on a smooth projective variety X of dimension n such that there exists a global section s ∈ Γ([Escr ]) whose zero locus Z = (s)0 is a smooth subvariety of X of dimension n − r [ges ] 1.

scholarly journals Geometric collections and Castelnuovo–Mumford regularity

2007 ◽  
Vol 143 (3) ◽  
pp. 557-578 ◽  
Author(s):  
L. COSTA ◽  
R. M. MIRÓ–ROIG
Keyword(s):  
Projective Variety ◽  
Projective Spaces ◽  
Smooth Projective Variety ◽  
Coherent Sheaf ◽  
Projective Varieties ◽  
Coherent Sheaves ◽  
Quadric Hypersurface ◽  
Basic Facts ◽  
Formal Properties ◽  
Smooth Projective Varieties

AbstractThe paper begins by overviewing the basic facts on geometric exceptional collections. Then we derive, for any coherent sheaf $\cF$ on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to $\cF$ and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo–Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties X with a geometric collection σ. We define the notion of regularity of a coherent sheaf $\cF$ on X with respect to σ. We show that the basic formal properties of the Castelnuovo–Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we show that in case of coherent sheaves on $\PP^n$ and for a suitable geometric collection of coherent sheaves on $\PP^n$ both notions of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a smooth quadric hypersurface $Q_n \subset \PP^{n+1}$ (n odd) with respect to a suitable geometric collection and we compare it with the Castelnuovo–Mumford regularity of their extension by zero in $\PP^{n+1}$.

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 Rank 3 rigid representations of projective fundamental groups

2018 ◽  
Vol 154 (7) ◽  
pp. 1534-1570 ◽  
Author(s):  
Adrian Langer ◽  
Carlos Simpson
Keyword(s):  
Irreducible Representation ◽  
Projective Variety ◽  
Fundamental Groups ◽  
Projective Varieties ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Geometric Origin ◽  
Rank 2 ◽  
Complex Projective ◽  
Smooth Projective Varieties

Let$X$be a smooth complex projective variety with basepoint$x$. We prove that every rigid integral irreducible representation$\unicode[STIX]{x1D70B}_{1}(X\!,x)\rightarrow \operatorname{SL}(3,\mathbb{C})$is of geometric origin, i.e., it comes from some family of smooth projective varieties. This partially generalizes an earlier result by Corlette and the second author in the rank 2 case and answers one of their questions.

Relations of Normal Bundles and Flips of Small Contractions on Projective Manifolds

2010 ◽  
Vol 17 (01) ◽  
pp. 11-16
Author(s):  
Jihong Su ◽  
Yicai Zhao
Keyword(s):  
Irreducible Component ◽  
Complex Number ◽  
Normal Bundle ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Normal Bundles ◽  
Exceptional Locus ◽  
Small Contraction ◽  
Projective Manifolds ◽  
Complex Number Field

Let X be a smooth projective variety over the complex number field. Let f : X → Y be a small contraction, and suppose that each irreducible component Ei of the exceptional locus E of f is a smooth subvariety. Assume that dim E ≤ ½ ( dim X + 1), and the normal bundle [Formula: see text]. Then each Ei ≅ P dim Ei or Q dim Ei. Moreover, the flip f+ : X+ → Y of f exists.

scholarly journals Demailly’s notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1645-1672 ◽  
Author(s):  
Ariyan Javanpeykar ◽  
Ljudmila Kamenova
Keyword(s):  
Moduli Spaces ◽  
Projective Variety ◽  
Closed Field ◽  
Projective Varieties ◽  
Complex Projective Variety ◽  
Hyperbolic Complex ◽  
Closed Subvariety ◽  
Algebraically Closed Fields ◽  
Complex Projective ◽  
Smooth Projective Varieties

Abstract Demailly’s conjecture, which is a consequence of the Green–Griffiths–Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly’s conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly’s definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a projective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that $${{\,\mathrm{Aut}\,}}(X)$$ Aut ( X ) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore “weaker” notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green–Griffiths–Lang conjecture on hyperbolic projective varieties.

scholarly journals On the closed embedding of the product of theta divisors into product of Jacobians and Chow groups

2018 ◽  
Vol 29 (03) ◽  
pp. 1850021
Author(s):  
Kalyan Banerjee
Keyword(s):  
Smooth Projective Variety ◽  
Chow Groups ◽  
Projective Curve ◽  
Theta Divisor ◽  
Projective Varieties ◽  
Smooth Projective Curve ◽  
Symmetric Powers ◽  
Closed Embedding ◽  
Push Forward ◽  
Smooth Projective Varieties

In this paper, we generalize the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of [Formula: see text] into [Formula: see text] for [Formula: see text], where [Formula: see text] is a smooth projective curve, to symmetric powers of a smooth projective variety of higher dimension. We also prove the analog of this theorem for product of symmetric powers of smooth projective varieties. As an application we prove the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of self-product of theta divisor into the self-product of the Jacobian of a smooth projective curve.

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.

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