An integrable system of K3-Fano flags

Given a K3 surface S, we show that the relative intermediate Jacobian of the universal family of Fano 3-folds V containing S as an anticanonical divisor is a Lagrangian fibration. The proof uses variations of the mixed Hodge structure of the pair (V,S).Comment: 10 pages; final version as published in Math. Annale

Uhlenbeck-Donaldson compactification for framed sheaves on projective surfaces

We construct a compactification $M^{\mu ss}$ of the Uhlenbeck-Donaldson type for the moduli space of slope stable framed bundles. This is a kind of a moduli space of slope semistable framed sheaves. We show that there exists a projective morphism $\gamma \colon M^{ss} \to M^{\mu ss}$, where $M^{ss}$ is the moduli space of S-equivalence classes of Gieseker-semistable framed sheaves. The space $M^{\mu ss}$ has a natural set-theoretic stratification which allows one, via a Hitchin-Kobayashi correspondence, to compare it with the moduli spaces of framed ideal instantons.Comment: 18 pages. v2: a few very minor changes. v3: 27 pages. Several proofs have been considerably expanded, and more explanations have been added. v4: 28 pages. A few minor changes. Final version accepted for publication in Math.

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 65 2 and second Chern class n 65 r, n 61 r(mod2). We give an explicit construction of an irreducible component In 17,r of this space for each such value of n and show that In 17,r has the expected dimension 4n(r + 1) 12 r(2r + 1). \ua9 2012 Versita Warsaw and Springer-Verlag Wien

Minimal Discrepancy for a Terminal cDV Singularity Is 1

An answer to a question raised by Shokurov on the minimal discrepancy of a terminal singularity of index 1 is given. It is proved that the minimal discrepancy is 1 (it is 2 for a non-singular point and 0 for all other canonical singularities of index 1). A rough classification of terminal singularities of index 1 based on finding certain low degree monomials in their equations, and the toric techniques of weighted blow ups are used. This result has been generalized to terminal singularities of index $r>1$ by Y.Kawamata; his theorem states that the minimal discrepancy is $1/r$