Intersection Numbers of Extremal Rays on Holomorphic Symplectic Varieties

By Brendan Hassett, Yuri Tschinkel. 2009

51 citations recorded by the source corpus.

Suppose X is a smooth projective complex variety. Let N1(X, Z) ⊂ H2(X, Z) and N 1 (X, Z) ⊂ H 2 (X, Z) denote the group of curve classes modulo homological equivalence and the Neron-Severi group respec- tively. The monoids of effective classes in each group generate cones NE1(X) ⊂ N1(X, R) and NE 1 (X) ⊂ N 1 (X, R) with closures NE1(X) and NE 1 (X), the pseudoeffective cones. These play a central role in the birational geometry of X. By Kleiman's criterion (16), a divisor D is ample if and only if DC > 0 for each C ∈ NE1(X) \ {0}. More generally, a divisor D is nef if DC ≥ 0 for each such C. Divisors in the interior of NE 1 (X) are big and induce birational maps from X. In general, very little is known about the structure of these cones be- yond Mori's Cone Theorem (5, §3.3) and more general finiteness results arising from the Minimal Model Program (2, 1.1.9). Let X be an irreducible holomorphic symplectic variety, i.e., a smooth projective simply-connected manifold admitting a unique nondegener- ate holomorphic two-form. Let (,) denote the Beauville-Bogomolov form on the cohomology group H 2 (X, Z), normalized so that it is in- tegral and primitive. Duality gives a Q-valued form on H2(X, Z), also denoted (,). When X is a K3 surface these coincide with the intersec- tion form. For a polarized K3 surface (S, g), we can read off the ample cone from the Hodge structure on the middle cohomology and the polarizing class. Precisely, the cone of effective curves can be expressed (17) NE1(S) = h C ∈ N1(S, Z) : (C, C) ≥ −2, Cg > 0i .

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/08c24b86effd1b0aa43a25849c8f3c1a40d2ca06

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Citations in the corpus

MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations (2013, background): “ in thiscase, it is easy to see that θ∨ induces an isomorphism H∗ alg(X,Z)/v→ N1(M) of lattices, respecting the integral structures. Given a class R∈ N1(M) satisfying the inequality (R,R) ≥ −n+3 2 of [HT10], let a0 ∈ v⊥ Q be the (rational) class with θ ∨(a0) = R. Let kbe any integer satisfying k≤ n−”
Hodge theory and Lagrangian planes on generalized Kummer fourfolds (2010, background): “We expect only the primitive class occurs because • we know of no examples where the minimal effective generator l of an extremal ray associated to a birational contraction of holomorphic symplectic manifolds fails to be primitive; • we conjecture [14] that classes of type e arise from divisorial co”
A survey of Torelli and monodromy results for holomorphic-symplectic varieties (2011, background): “The homomorphism f takes the K¨ ahler-type chamber in Problem 10.2 to KY . Hassett and Tschinkel formulated a precise conjectural solution to problem 10.2 [ HT4 ].”
A note on the Kähler and Mori cones of hyperkähler manifolds (2015, background): “A nice geometrical interpretation of some walls of the Kähler cone can be found in [7], where Hasset and Tschinkel analyzed contractions of some extremal rays of the”
Projectivity and birational geometry of Bridgeland moduli spaces (2012, unknown): “Our result shows, in particular, that [29, Conjecture 1.2] will need to be modiﬁed (see Remark 10.4).”