Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections

Let X be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of X is a lattice with respect to the Beauville-Bogomolov pairing. A divisor E on X is called a prime exceptional divisor, if E is reduced and irreducible and of negative Beauville-Bogomolov degree. Let E be a prime exceptional divisor on X. We first observe that associated to E is a monodromy involution of the integral cohomology of X, which acts on the second cohomology lattice as the reflection by the cohomology class of E (Theorem 1.1). We then specialize to the case that X is deformation equivalent to the Hilbert scheme of length n zero-dimensional subschemes of a K3 surface. We determine the set of classes of exceptional divisors on X (Theorem 1.11). This leads to a determination of the closure of the movable cone of X.Comment: v2: 53 pages, Latex. The main Conjecture 1.11 is now Theorem 1.11. Final version. To appear in KJM, Maruyama memorial volum

Classification of involutions on Enriques surfaces

We present the classification of involutions on Enriques surfaces. We classify those into 18 types with the help of the lattice theory due to Nikulin. We also give all examples of the classification.Comment: 25 pages, 42 figure

Open problems, questions, and challenges in finite-dimensional integrable systems

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference “Finite-dimensional Integrable Systems, FDIS 2017” held at CRM, Barcelona in July 2017.Postprint (updated version

A brief introduction to Enriques surfaces

This is a brief introduction to the theory of Enriques surfaces over arbitrary algebraically closed fields. Some new results about automorphism groups of Enriques surfaces are also included.Comment: Minor corrections, to appear in "Development of Moduli Theory---Kyoto 2013