Related papers: Rational curves on holomorphic symplectic fourfold…

This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold $X$ in the family has an involution such that the induced involution on the Fano variety…

This is a survey of the geometry of complex cubic fourfolds with a view toward rationality questions. Topics include classical constructions of rational examples, Hodge structures and special cubic fourfolds, associated K3 surfaces and…

In this second paper of a two-part series, we prove that whenever a contact 3-manifold admits a uniform spinal open book decomposition with planar pages, its (weak, strong and/or exact) symplectic and Stein fillings can be classified up to…

Let $M$ be a closed 4-manifold with a free circle action. If the orbit manifold $N^3$ satisfies an appropriate fibering condition, then we show how to represent a cone in $H^2(M;\R)$ by symplectic forms. This generalizes earlier…

Let M be a 2n-dimensional Kahler manifold deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface S. Let Mon be the group of automorphisms of the cohomology ring of M, which are induced by monodromy operators.…

We determine the possible finite groups $G$ of symplectic automorphisms of hyperk\"ahler manifolds which are deformation equivalent to the second Hilbert scheme of a K3 surface. We prove that $G$ has such an action if, and only if, it is…

In this paper, we study the curve cone of an almost complex $4$-manifold which is tamed by a symplectic form. In particular, we prove the cone theorem as in Mori theory for all such manifolds using the Seiberg-Witten theory. For small…

This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing…

Progress on the problem whether the Hilbert schemes of locally Cohen-Macaulay curves in projective 3 space are connected has been hampered by the lack of an answer to a question that was raised by Robin Hartshorne in his paper "On the…

We show that a compact Kahler manifold admitting a nondegenerate holomorphic 2-form valued in a line bundle is a finite cyclic cover of a hyperkahler manifold. With respect to the connection induced by the locally hyperkahler metric, the…

Let $\Sigma$ be a smooth projective surface, let $f' : S' \to \Sigma$ be a double cover of $\Sigma$ and let $\mu : S \to S'$ be the canonical resolution. Put $f = f'\circ\mu$. An irreducible curve $C$ on $\Sigma$ is said to be a splitting…

The main purpose of this paper is to summarize the basic ingredients, illustrated with examples, of a pseudoholomorphic curve theory for symplectic 4-orbifolds. These are extensions of relevant work of Gromov, McDuff and Taubes on…

In this note we prove that the moduli space of torsion-free modules of rank one over an Azumaya algebra on a K3-surface is an irreducible symplectic variety deformation equivalent to a Hilbert scheme of points on the K3-surface.

A self-dual harmonic 2-form on a 4-dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form's zero set, the metric with the 2-form give a compatible almost complex structure and thus…

We extend Donaldson's asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a symplectic orbifold such that the symplectic form defines an integer cohomology class, we prove that there exist sections of large…

Let $M$ be a compact hyperkahler manifold with maximal holonomy (IHS). The group $H^2(M, R)$ is equipped with a quadratic form of signature $(3, b_2-3)$, called Bogomolov-Beauville-Fujiki (BBF) form. This form restricted to the rational…

We construct a uniformly bounded symplectic structure on $S^2 \times \mathbb{R}^4$ admitting embeddings by arbitrarily large balls. This provides a counterexample to a recent conjecture of Savelyev. We then prove the conjecture holds for a…

Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics - conjecturally, the ones that are rational - have specific K3s associated to them geometrically. Hassett has studied cubics with K3s associated to them at the level of…

The Hilbert scheme $X^{[3]}$ of length-$3$ subschemes of a smooth projective variety $X$ is known to be smooth and projective. We investigate whether the property of having a multiplicative Chow-Kuenneth decomposition is stable under taking…

This note (which makes no claim to novelty) presents a proof of the separable rational connectedness of smooth cubic hypersurfaces, in any characteristic, by showing how to explicitly construct very free curves (of degree 3) on them. -----…