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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.

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Tori admit very explicit Kahler cones. Their cohomology is generated by forms with constant coefficients, so the Kahler cone of an n-dimensional complex torus identifies with the set of positive definite hermitian nxn matrices (maybe modulo some relation). –  Gunnar Þór Magnússon Jul 8 '10 at 10:52

5 Answers 5

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.

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Any indication if anything similar to this happens in infinite dimensional case of Kac-Moody algebras? You do have a flag manifold and you can define a positive Weyl chamber ... –  Najdorf Jan 22 '11 at 23:42

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 $n<9$ we get Fano), this is related to Nagata conjecutre http://en.wikipedia.org/wiki/Nagata's_conjecture_on_curves

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", http://arxiv.org/abs/0901.3361

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In fact for the blowup of P^2 in 9 points the answer was found by Borcea: "On desingularized Horrocks-Mumford quintics", J. Reine Angew. Math. 421 (1991), 23--41 (most of the paper is about threefolds, but this result is in the last section). For some reason this result seems to be not at all well-known: my supervisor gave it to me as a problem at the start of my Ph.D. –  Artie Prendergast-Smith Jul 17 '10 at 11:47
    
Artie, thanks a lot! You are right, I was thinking about the blow up at 10 points, I corrected the answer. –  Dmitri Jul 17 '10 at 11:53
    
Hi, Dmitri, thanks for your answer. Can you give references on statements in your first paragraph? (Cone of Fano variety is polyhedral) –  lemega Jul 31 '10 at 18:48
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Lemega, after googling a bit I found, there was a discussion of this question on matheoverflow previously, and there is a "refference" there in the answer of Michael Thaddeus: mathoverflow.net/questions/27249/… –  Dmitri Jul 31 '10 at 22:34

To Hirzebruch surfaces, you can add $\mathrm{CP}^2$, its $k$-folds blow-ups, $1\leq k\leq 8$, and some irrational ruled surfaces.

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.

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.

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The cone of curves of K3 surfaces is described in this paper.

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Explicit description of a Kahler cone for all hyperkahler manifolds is here: http://arxiv.org/abs/1401.0479 (Rational curves on hyperkahler manifolds, Ekaterina Amerik, Misha Verbitsky)

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