Moving and Ample Cones of Holomorphic Symplectic Fourfolds

By Brendan Hassett, Yuri Tschinkel. 2007

107 citations recorded by the source corpus.

We analyze the ample and moving cones of holomorphic symplectic manifolds, in light of recent advances in the minimal model program. As an application, we establish a numerical criterion for ampleness of divisors on fourfolds deformation-equivalent to punctual Hilbert schemes of K3 surfaces.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/df3bda6942f757d8e8c692dd6338effc04485cd9

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Citations in the corpus

Rational curves on hyperkahler manifolds (2014, background): “asses do not, apriori, have to be minimal, since extremal rays of the cone of curves could accumulate. One can conjecture that in real life it never happens (as this is true in the projective case by [HT]). 5 Twistor lines inthe Teichmu¨ller space 5.1 Twistorlines and 3-dimensional planes inH2(M,R).”
Projectivity and birational geometry of Bridgeland moduli spaces (2012, unknown): “…the movable cone for moduli spaces of stable sheaves (for general results and conjectures on the movable cone for an Hyperk¨ahler manifold, see [28,29]); in particular, it would likely imply the above-mentioned conjecture on Lagrangian ﬁbrations for any moduli space of Gieseker-stable sheaves…”
MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations (2013, background): “mation equivalent to Hilbert schemes of points on K3 surfaces. The cone theorem and the minimal model program (MMP) induce a locally polyhedral chamber decomposition of the movable cone of MH(v) (see [HT09]): • chambers correspond one-to-one to smoothK-trivial birational modelsMf 99K MH(v) of the mo”
Intersection Numbers of Extremal Rays on Holomorphic Symplectic Varieties (2009, background): “This generalizes to higher dimensions conjectures of [8] in the special case n = 2; see [9] for a proof in this case that the divisors predicted to be nef are in fact nef.”
Automorphism groups of Calabi-Yau manifolds of Picard number two (2012, background): “u99], [Ve09], [Hu11]). Theorem (1.3) (3) is proved in Section 5 by using the surjectivity of the period map for hyperk¨ahler manifold due to Huybrechts ([Hu99]) and a result of Hassett and Tschinkel ([HT09]). See also Proposition (5.1) for Hilbert schemes of points on K3 surfaces and generalized Kum”
A note on the Kähler and Mori cones of hyperkähler manifolds (2015, methodology): “Using minimal model program, they proved in [6] one direction of the conjecture in dimension four.”
Moduli of K3 Surfaces and Irreducible Symplectic Manifolds (2010, unknown): “So far, no complete analogue is known for the higher dimensional case, but Hassett and Tschinkel have proved partial results for n = 2 in [58], and more precise results in a special case in [59].”
Prime exceptional divisors on holomorphic symplectic varieties and monodromy reflections (2009, background): “class of an irreducible divisor E. Then (e,e) = −2 or (e,e) = 2−2n. If (e,e) = 2−2n, then the class (e,•) in H2(X,Z)∗ is divisible by n− 1. The Theorem is related to Theorem 22 in the beautiful paper [HT2]. In particular, the case of fourfolds is settled in that paper. The hypothesis that the diviso”
Hodge theory and Lagrangian planes on generalized Kummer fourfolds (2010, background): “(2) If X is deformation equivalent to the Hilbert scheme of length two subschemes of a K3 surface then (l, l) = −5/2 [15].”