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

Author(s):  
Daniel Greb ◽  
Stefan Kebekus ◽  
Thomas Peternell

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.

2020 ◽  
Vol 126 (2) ◽  
pp. 221-228
Author(s):  
Håkan Samuelsson Kalm ◽  
Martin Sera

For a reduced pure dimensional complex space $X$, we show that if Barlet's recently introduced sheaf $\alpha _X^1$ of holomorphic $1$-forms or the sheaf of germs of weakly holomorphic $1$-forms is locally free, then $X$ is smooth. Moreover, we discuss the connection to Barlet's well-known sheaf $\omega _X^1$.


2020 ◽  
Vol 8 ◽  
Author(s):  
NATHAN CHEN ◽  
DAVID STAPLETON

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.


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

2009 ◽  
Vol 146 (1) ◽  
pp. 193-219 ◽  
Author(s):  
Daniel Greb ◽  
Stefan Kebekus ◽  
Sándor J. Kovács

AbstractGiven a normal variety Z, a p-form σ defined on the smooth locus of Z and a resolution of singularities $\pi : \widetilde {Z} \to Z$, we study the problem of extending the pull-back π*(σ) over the π-exceptional set $E \subset \widetilde {Z}$. For log canonical pairs and for certain values of p, we show that an extension always exists, possibly with logarithmic poles along E. As a corollary, it is shown that sheaves of reflexive differentials enjoy good pull-back properties. A natural generalization of the well-known Bogomolov–Sommese vanishing theorem to log canonical threefold pairs follows.


2014 ◽  
Vol 150 (3) ◽  
pp. 396-408 ◽  
Author(s):  
Zhiyu Tian ◽  
Hong R. Zong

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.


2007 ◽  
Vol 11 (3) ◽  
pp. 485-496 ◽  
Author(s):  
Laszló Lempert ◽  
Endre Szabó

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