Dimitri Markushevich

Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces

A rational Lagrangian fibration f on an irreducible symplectic variety V is a rational map which is birationally equivalent to a regular surjective morphism with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect that a rational Lagrangian fibration exists if and only if V has a divisor D with Bogomolov–Beauville square 0. This conjecture is proved in the case when V is the Hilbert scheme of d points on a generic K3 surface S of genus g under the hypothesis that its degree 2g−2 is a square times 2d−2. The construction of f uses a twisted Fourier–Mukai transform which induces a birational isomorphism of V with a certain moduli space of twisted sheaves on another K3 surface M, obtained from S as its Fourier–Mukai partner.

Kowalevski top and genus-2 curves

Kowalevskis curve of genus 2 is related to two other curves arising from the solution of the Kowalevski top by the method of spectral curves in the case when the angular momentum of the top is orthogonal to the gravity vector. One is the Bobenko-Reyman-Semenov-Tian-Shansky curve of genus 2, the other is the spectral curve of the Kuznetsov-Tsiganov Lax matrix, of genus 3. The relations between the curves are given by correspondences, that is, multivalued maps, inducing isogenies of the corresponding Jacobian or Prym varieties.

Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ3

Symplectic instanton vector bundles on the projective space ℙ3 constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space In;r of rank-2r symplectic instanton vector bundles on ℙ3 with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus In;r* of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).

Two infinite series of moduli spaces of rank 2 sheaves on \({\mathbb {P}}^3\)

We describe new components of the Gieseker–Maruyama moduli scheme

An integrable system of K3-Fano flags

{\mathcal {M}}(n)

SOME EXAMPLES OF INSTANTONS IN SIGMA MODELS II: K3 MANIFOLDS

M(n) of semistable rank 2 sheaves E on

Editors’ preface for the topical issue “Instantons, coherent sheaves and their moduli spaces”

{\mathbb {P}^{3}}