scholarly journals Some remarks on regular foliations with numerically trivial canonical class

2017 ◽  
Vol Volume 1 ◽  
Author(s):  
Stéphane Druel
Keyword(s):  
Small Rank ◽  
Codimension Two ◽  
Canonical Class ◽  
Special Cases ◽  
Algebraicity Criterion ◽  
Projective Manifolds ◽  
Complex Projective ◽  
Algebraic Foliations

In this article, we first describe codimension two regular foliations with numerically trivial canonical class on complex projective manifolds whose canonical class is not numerically effective. Building on a recent algebraicity criterion for leaves of algebraic foliations, we then address regular foliations of small rank with numerically trivial canonical class on complex projective manifolds whose canonical class is pseudo-effective. Finally, we confirm the generalized Bondal conjecture formulated by Beauville in some special cases. Comment: 20 pages

Polar Homology

2003 ◽  
Vol 55 (5) ◽  
pp. 1100-1120 ◽  
Author(s):  
Boris Khesin ◽  
Alexei Rosly
Keyword(s):  
Boundary Operator ◽  
Homology Groups ◽  
Complex Dimension ◽  
Linking Numbers ◽  
Complex Submanifolds ◽  
Projective Manifolds ◽  
Complex Projective

AbstractFor complex projective manifolds we introduce polar homology groups, which are holomorphic analogues of the homology groups in topology. The polar k-chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincaré residue on it. One can also define the corresponding analogues for the intersection and linking numbers of complex submanifolds, which have the properties similar to those of the corresponding topological notions.

scholarly journals A note on Lang's conjecture for quotients of bounded domains

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Sébastien Boucksom ◽  
Simone Diverio
Keyword(s):  
Plurisubharmonic Function ◽  
General Type ◽  
Universal Cover ◽  
Projective Manifold ◽  
Bounded Domains ◽  
Final Version ◽  
Lang's Conjecture ◽  
Projective Manifolds ◽  
Lang’S Conjecture ◽  
Complex Projective

It was conjectured by Lang that a complex projective manifold is Kobayashi hyperbolic if and only if it is of general type together with all of its subvarieties. We verify this conjecture for projective manifolds whose universal cover carries a bounded, strictly plurisubharmonic function. This includes in particular compact free quotients of bounded domains. Comment: 10 pages, no figures, comments are welcome. v3: following suggestions made by the referee, the exposition has been improved all along the paper, we added a variant of Theorem A which includes manifolds whose universal cover admits a bounded psh function which is strictly psh just at one point, and we added a section of examples. Final version, to appear on \'Epijournal G\'eom. Alg\'ebrique

scholarly journals Représentations linéaires des groupes kählériens : factorisations et conjecture de Shafarevich linéaire

2014 ◽  
Vol 151 (2) ◽  
pp. 351-376 ◽  
Author(s):  
Fréderic Campana ◽  
Benoît Claudon ◽  
Philippe Eyssidieux
Keyword(s):  
General Type ◽  
Linear Representation ◽  
Kähler Manifolds ◽  
Fundamental Groups ◽  
Kahler Manifolds ◽  
Shafarevich Conjecture ◽  
Classical Work ◽  
Projective Manifolds ◽  
Complex Projective ◽  
Holomorphic Convexity

AbstractWe extend to compact Kähler manifolds some classical results on linear representation of fundamental groups of complex projective manifolds. Our approach, based on an interversion lemma for fibrations with tori versus general type manifolds as fibers, gives a refinement of the classical work of Zuo. We extend to the Kähler case some general results on holomorphic convexity of coverings such as the linear Shafarevich conjecture.

Holomorphic curves in algebraic varieties of maximal albanese dimension

2015 ◽  
Vol 26 (06) ◽  
pp. 1541006 ◽  
Author(s):  
Katsutoshi Yamanoi
Keyword(s):  
Nevanlinna Theory ◽  
Holomorphic Curves ◽  
Algebraic Varieties ◽  
Projective Curve ◽  
Second Main Theorem ◽  
Finite Covering ◽  
Type Estimate ◽  
Geometric Genus ◽  
Projective Manifolds ◽  
Complex Projective

We prove a second main theorem type estimate in Nevanlinna theory for holomorphic curves f : Y → X from finite covering spaces Y → ℂ of the complex plane ℂ into complex projective manifolds X of maximal albanese dimension. If X is moreover of general type, then this implies that the special set of X is a proper subset of X. For a projective curve C in such X, our estimate also yields an upper bound of the ratio of the degree of C to the geometric genus of C, provided that C is not contained in a proper exceptional subset in X.

A FAMILY OF SURFACES CONSTRUCTED FROM GENUS 2 CURVES

2007 ◽  
Vol 18 (05) ◽  
pp. 585-612 ◽  
Author(s):  
CHAD SCHOEN
Keyword(s):  
Riemann Surface ◽  
Two Dimensional ◽  
Hodge Conjecture ◽  
Analytic Variety ◽  
Genus 2 ◽  
Complex Analytic Variety ◽  
Projective Manifolds ◽  
Genus 2 Curves ◽  
Complex Projective

We consider the deformations of the two-dimensional complex analytic variety constructed from a genus 2 Riemann surface by attaching its self-product to its Jacobian in an elementary way. The deformations are shown to be unobstructed, the variety smooths to give complex projective manifolds whose invariants are computed and whose images under Albanese maps (re)verify an instance of the Hodge conjecture for certain abelian fourfolds.

Projective manifolds of sectional genus three as zero loci of sections of ample vector bundles

2008 ◽  
Vol 144 (1) ◽  
pp. 109-118 ◽  
Author(s):  
ANTONIO LANTERI ◽  
HIDETOSHI MAEDA
Keyword(s):  
Vector Bundles ◽  
Projective Variety ◽  
Ample Line Bundle ◽  
Global Section ◽  
Sectional Genus ◽  
Ample Vector Bundle ◽  
Complex Projective Variety ◽  
Projective Manifolds ◽  
Complex Projective ◽  
Zero Locus

AbstractLet ϵ be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of ϵ whose zero locus Z is a smooth subvariety of dimension n-r ≥ 2 of X. Let H be an ample line bundle on X such that the restriction HZ of H to Z is very ample. Triplets (X, ϵ, H) with g(Z, HZ) = 3 are classified, where g(Z, HZ) is the sectional genus of (Z, HZ).

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