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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3$\begingroup$ 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). $\endgroup$– Gunnar Þór MagnússonCommented Jul 8, 2010 at 10:52
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5 Answers
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Flag manifolds $G/B$ are nice: the Kähler cone is the positive Weyl chamber, with edges coming from the Poincaré duals of the Schubert divisors.
answered Jul 8, 2010 at 5:58
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$\begingroup$ 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 ... $\endgroup$– NajdorfCommented Jan 22, 2011 at 23:42
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Generalising the case of Hirzebruch surface, you can say that toric varieties admit explicit description of Kähler cone.
Also for each Fano variety its Kähler 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 Kähler cone it is still unknown (for $n<9$ we get Fano), this is related to Nagata's conjecture.
Morrison's conjecture states that for a Calabi-Yau manifold the quotient of the Kähler 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.
answered Jul 17, 2010 at 11:20
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4$\begingroup$ 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. $\endgroup$– user5117Commented Jul 17, 2010 at 11:47
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$\begingroup$ Artie, thanks a lot! You are right, I was thinking about the blow up at 10 points, I corrected the answer. $\endgroup$ Commented Jul 17, 2010 at 11:53
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$\begingroup$ Hi, Dmitri, thanks for your answer. Can you give references on statements in your first paragraph? (Cone of Fano variety is polyhedral) $\endgroup$– lemegaCommented Jul 31, 2010 at 18:48
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2$\begingroup$ 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/… $\endgroup$ Commented Jul 31, 2010 at 22:34
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Explicit description of a Kahler cone for all hyperkahler manifolds is here: https://arxiv.org/abs/1401.0479 (Rational curves on hyperkahler manifolds, Ekaterina Amerik, Misha Verbitsky)
answered Jul 19, 2014 at 9:16
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answered Feb 16, 2011 at 0:47
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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.
answered Jul 7, 2010 at 18:45