scholarly journals Rationally Connected Varieties and Fundamental Groups

2003 ◽  
pp. 69-92 ◽  
Author(s):  
János Kollár
Keyword(s):  
Fundamental Groups ◽  
Rationally Connected ◽  
Rationally Connected Varieties

scholarly journals KODAIRA DIMENSION OF SUBVARIETIES II

2006 ◽  
Vol 17 (05) ◽  
pp. 619-631 ◽  
Author(s):  
THOMAS PETERNELL
Keyword(s):  
Normal Bundle ◽  
General Type ◽  
Kodaira Dimension ◽  
Projective Manifold ◽  
Fundamental Groups ◽  
Weak Version ◽  
Joint Paper ◽  
Rationally Connected

This paper continues the study of non-general type subvarieties begun in a joint paper with Schneider and Sommese [14]. We prove uniruledness of a projective manifold containing a submanifold not of general type whose normal bundle has positivity properties and study moreover the rational quotient. We also relate the fundamental groups and a prove a cohomological criterion for a manifold to be rationally connected (weak version of a conjecture of Mumford).

scholarly journals Reflexive differential forms on singular spaces. Geometry and cohomology

2014 ◽  
Vol 2014 (697) ◽  
Author(s):  
Daniel Greb ◽  
Stefan Kebekus ◽  
Thomas Peternell
Keyword(s):  
Differential Forms ◽  
Extension Theorem ◽  
Singular Variety ◽  
Singular Spaces ◽  
Complex Spaces ◽  
Rationally Connected ◽  
Smooth Locus ◽  
Recent Extension ◽  
Rationally Connected Varieties

AbstractBased on a recent extension theorem for reflexive differential forms, that is, regular differential forms defined on the smooth locus of a possibly singular variety, we study the geometry and cohomology of sheaves of reflexive differentials.First, we generalise the extension theorem to holomorphic forms on locally algebraic complex spaces. We investigate the (non-)existence of reflexive pluri-differentials on singular rationally connected varieties, using a semistability analysis with respect to movable curve classes. The necessary foundational material concerning this stability notion is developed in an appendix to the paper. Moreover, we prove that Kodaira–Akizuki–Nakano vanishing for sheaves of reflexive differentials holds in certain extreme cases, and that it fails in general. Finally, topological and Hodge-theoretic properties of reflexive differentials are explored.

scholarly journals FANO HYPERSURFACES WITH ARBITRARILY LARGE DEGREES OF IRRATIONALITY

2020 ◽  
Vol 8 ◽  
Author(s):  
NATHAN CHEN ◽  
DAVID STAPLETON
Keyword(s):  
Minimal Degree ◽  
Rational Map ◽  
Rationally Connected ◽  
General Complex ◽  
Special Fibers ◽  
Fano Index ◽  
Low Dimensional ◽  
The Way ◽  
Rationally Connected Varieties

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index  $e$ , then the degree of irrationality of a very general complex Fano hypersurface of index  $e$ and dimension n is bounded from below by a constant times  $\sqrt{n}$ . To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic $p$ argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.

scholarly journals Families of rationally connected varieties

2002 ◽  
Vol 16 (1) ◽  
pp. 57-67 ◽  
Author(s):  
Tom Graber ◽  
Joe Harris ◽  
Jason Starr
Keyword(s):  
Rationally Connected ◽  
Rationally Connected Varieties

scholarly journals One-cycles on rationally connected varieties

2014 ◽  
Vol 150 (3) ◽  
pp. 396-408 ◽  
Author(s):  
Zhiyu Tian ◽  
Hong R. Zong
Keyword(s):  
Complete Intersection ◽  
Finite Fields ◽  
Chow Group ◽  
Rational Curves ◽  
Positive Answer ◽  
Tate Conjecture ◽  
Rationally Connected ◽  
Rationally Connected Varieties

AbstractWe prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. Applying the same technique, we also show that the Chow group of 1-cycles on a separably rationally connected Fano complete intersection of index at least 2 is generated by lines. As a consequence, we give a positive answer to a question of Professor Totaro about integral Hodge classes on rationally connected 3-folds. And by a result of Professor Voisin, the general case is a consequence of the Tate conjecture for surfaces over finite fields.

scholarly journals Rationally Connected Varieties and Loop Spaces

2007 ◽  
Vol 11 (3) ◽  
pp. 485-496 ◽  
Author(s):  
Laszló Lempert ◽  
Endre Szabó
Keyword(s):  
Loop Spaces ◽  
Rationally Connected ◽  
Rationally Connected Varieties

On the vector bundles over rationally connected varieties

2009 ◽  
Vol 347 (19-20) ◽  
pp. 1173-1176 ◽  
Author(s):  
Indranil Biswas ◽  
João Pedro P. dos Santos
Keyword(s):  
Vector Bundles ◽  
Rationally Connected ◽  
Rationally Connected Varieties
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