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Rational curves on holomorphic symplectic fourfolds. (English) Zbl 1081.14515

From the introduction: One of the main problems in the theory of irreducible holomorphic symplectic manifolds is the description of the ample cone in the Picard group. The goal of this paper is to formulate explicit Hodge-theoretic criteria for the ampleness of line bundles on certain irreducible holomorphic symplectic manifolds. It is well known that for \(K3\) surfaces the ample cone is governed by \((-2)\)-curves. More generally, we expect that certain distinguished two-dimensional homology classes of the symplectic manifold should correspond to explicit families of rational curves, and that these govern its ample cone.
The program for analyzing the ample cone of a symplectic manifold divides naturally into three parts. First, for each deformation type of irreducible holomorphic symplectic manifolds we identify distinguished Hodge classes in \(H_2(\mathbb{Z})\) that should be represented by rational curves. Second, one shows that rational curves with a special geometry govern the ample cone of \(F\). This entails classifying possible contractions of symplectic manifolds – a very active topic of current research (see the discussion of the literature below) – and interpretting this classification in terms of the numerical properties of the contracted curves. It also involves the classification of base loci for sections of line bundles on a symplectic manifold. The third part of the program is to show that a divisor class satisfying certain numerical conditions arises from a big line bundle and thus yields a birational transformation of thesymplectic manifold.
We are mainly concerned with the first step of this program in a specific case. Let \(F\) be an irreducible holomorphic symplectic fourfold deformation equivalent to the punctual Hilbert scheme \(S^{[2]}\) for some \(K3\) surface \(S\). Given the Hodge structure on \(H^2(F)\), we describe explicitly (but conjecturally) the cone of effective curves on \(F\) and, by duality, the ample cone of \(F\). As in the case of \(K3\) surfaces, each divisor class of square \(-2\) induces a reflection preserving the Hodge structure. The ‘birational ample cone’ is conjectured to be the interior of a fundamental domain for this reflection group. However, the ample cone may be strictly smaller than the birational ample cone, owing to the existence of elementary transformations along \(\mathbb{P}^2\)’s in \(F\). The corresponding classes have square \(-10\) with respect to the Beauville form.
This paper is organized as follows. In section 2 we recall basic results and conjectures for irreducible holomorphic symplectic manifolds. In the next section, we introduce the notion of nodal classes and state our main conjectures. In section 4 we give some deformation-theoretic evidence for our conjectures. The rest of the paper is devoted to examples supporting the conjectures. Section 5 is devoted to Hilbert schemes \(S^{[2]}\) for \(K3\) surfaces of small degree. We describe examples of nonnodal rational curves and certain codimension-two behavior in section 6. We turn to the projective geometry of cubic fourfolds in the last section. Questions of rationality and unirationality are addressed in section 7.2.

MSC:

14J35 \(4\)-folds
14C20 Divisors, linear systems, invertible sheaves
32J18 Compact complex \(n\)-folds
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