Structure of Kähler cone - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T22:09:06Z http://mathoverflow.net/feeds/question/30926 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/30926/structure-of-kahler-cone Structure of Kähler cone lemega 2010-07-07T17:54:27Z 2011-02-16T00:47:23Z <p>Are there examples of Kähler manifolds whose Kähler cone can be described explicitly, say spanned by certain cohomology classes? As far as I know, Hirzebruch Surface has a complete description for its Kähler cone.</p> http://mathoverflow.net/questions/30926/structure-of-kahler-cone/30936#30936 Answer by Martin Pinsonnault for Structure of Kähler cone Martin Pinsonnault 2010-07-07T18:45:07Z 2010-07-07T18:45:07Z <p>To Hirzebruch surfaces, you can add $\mathrm{CP}^2$, its $k$-folds blow-ups, $1\leq k\leq 8$, and some irrational ruled surfaces.</p> <p>Related to this question is the determination of the symplectic cone. This is now understood for rational $4$-manifolds, ruled $4$-manifolds and their blow-ups, and also for some elliptic fibrations.</p> <p>There is a nice survey by Tian-Jun Li of the relations between symplectic and Kahler cones for $4$-manifolds (and complex surfaces). See arXiv:0805.2931.</p> http://mathoverflow.net/questions/30926/structure-of-kahler-cone/31000#31000 Answer by Allen Knutson for Structure of Kähler cone Allen Knutson 2010-07-08T05:58:56Z 2010-07-08T05:58:56Z <p>Flag manifolds G/B are nice: the K\"ahler cone is the positive Weyl chamber, with edges coming from the Poincar\'e duals of the Schubert divisors.</p> http://mathoverflow.net/questions/30926/structure-of-kahler-cone/32279#32279 Answer by Dmitri for Structure of Kähler cone Dmitri 2010-07-17T11:20:57Z 2010-07-17T11:52:51Z <p>Generalising the case of Hirzebrouch surface, you can say that toric varieties admit explicit description of Kahler cone. Also for each Fano variety its Kahler cone is polyhedral, i.e., it is spanned by a finite number of rays (but this does not mean, that the description is easy). If you leave the class of Fano varieties unpleasant things may start to happen. For example for a generic blow up of $\mathbb CP^2$ in $n\ge 10$ points the structure of Kahler cone it is still unknown (for $n&lt;9$ we get Fano), this is related to Nagata conjecutre <a href="http://en.wikipedia.org/wiki/Nagata" rel="nofollow">http://en.wikipedia.org/wiki/Nagata</a>'s_conjecture_on_curves </p> <p>Morrison's conjecture states that for a Calabi-Yau manifold the quotient of the Kahler cone by the group of isometries of the manifold is polyhedral. The conjecture was proved only for surfaces, there is a recent very nice paper of Burt Totaro on this topic "The cone conjecture for Calabi-Yau pairs in dimension two", <a href="http://arxiv.org/abs/0901.3361" rel="nofollow">http://arxiv.org/abs/0901.3361</a></p> http://mathoverflow.net/questions/30926/structure-of-kahler-cone/55570#55570 Answer by Sándor Kovács for Structure of Kähler cone Sándor Kovács 2011-02-16T00:47:23Z 2011-02-16T00:47:23Z <p>The cone of curves of K3 surfaces is described in <a href="http://www.springerlink.com/content/r654368027w422r2/" rel="nofollow">this paper</a>.</p>