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.

Analytical method for the determination of trichlorobenzenes in marine biota (poster)

Trichlorobenzenes (TCBs) were intensively used in the last decades as essential components of dielectric fluids, intermediates in chemical synthesis, solvents, coolants, lubricants, heat-transfer medium; insecticide, additive in polyester dyeing and components of termite-control preparations (1, 2). Due to their widespread occurrence in the various environmental compartments they have been classified by OSPARCOM (Oslo and Paris Commissions) (3) as chemicals for priority action and have been proposed by the Marine Chemistry Working Group (MCWG) as chemical parameters in the Water Framework Directive (4). Based on their octanol-water partitioning coefficients (log Kow = 4.02-4.49) (5) and bioconcentration factors in fish (ranging from 182 to 3200, depending on the lipid content) (6), these chemicals are expected to bioaccumulate in aquatic organisms.Against their potential significance in the marine environment there is relatively little information available concerning the actual concentration levels and distribution of trichlorobenzenes in marine organisms (7, 8).The aim of this work was to develop an analytical method appropriate for the determination of TCBs in marine biota.The analytical method consists of saponification of the fish tissue with methanolic potassium hydroxide, liquid-liquid extraction of the solution with pentane, clean up of the concentrated extract on alumina column and analysis of the extract with gas chromatograph equipped with electron capture detector (ECD). The method proved to be appropriate for the detection of concentration levels typical of the organic contaminants in biota (7) (~1 ng /g wet weight of tissue). The relative standard deviation of the analysis of 1,3,5-, 1,2,4- and 1,2,3-trichlorobenzene was 8, 6 and 18% (n=4) respectively. Higher recoveries of the analytes were obtained with spiked fish samples than with standard solutions (88, 96 and 78 instead of 53, 50 and 32% of 1,3,5-, 1,2,4- and 1,2,3-trichlorobenzene respectively). One plausible explanation of the difference is that the proteins and glycerides of the fish tissue compete effectively with trichlorobenzenes for the base and their presence decrease their decomposition rate

Webs of Lagrangian Tori in Projective Symplectic Manifolds

For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperk\"ahler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauville's. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandt's theory of subnormal subgroups.Comment: 18 pages, minor latex problem fixe

Derived categories of cubic fourfolds

We discuss the structure of the derived category of coherent sheaves on cubic fourfolds of three types: Pfaffian cubics, cubics containing a plane and singular cubics, and discuss its relation to the rationality of these cubics.Comment: 18 page

Holomorphic symplectic geometry: a problem list

A list of open problems on holomorphic symplectic, contact and Poisson manifolds

On the Heterotic World-sheet Instanton Superpotential and its individual Contributions

For supersymmetric heterotic string compactifications on a Calabi-Yau threefold $X$ endowed with a vector bundle $V$ the world-sheet superpotential $W$ is a sum of contributions from isolated rational curves \C in $X$; the individual contribution is given by an exponential in the K\"ahler class of the curve times a prefactor given essentially by the Pfaffian which depends on the moduli of $V$ and the complex structure moduli of $X$. Solutions of $DW=0$ (or even of $DW=W=0$) can arise either by nontrivial cancellations between the individual terms in the summation over all contributing curves or because each of these terms is zero already individually. Concerning the latter case conditions on the moduli making a single Pfaffian vanish (for special moduli values) have been investigated. However, even if corresponding moduli - fulfilling these constraints - for the individual contribution of one curve are known it is not at all clear whether {\em one} choice of moduli exists which fulfills the corresponding constraints {\em for all contributing curves simultaneously}. Clearly this will in general happen only if the conditions on the 'individual zeroes' had already a conceptual origin which allows them to fit together consistently. We show that this happens for a class of cases. In the special case of spectral cover bundles we show that a relevant solution set has an interesting location in moduli space and is related to transitions which change the generation number.Comment: 47 page

Improved convergence and stability properties in a three-dimensional higher-order ice sheet model

We present a finite difference implementation of a three-dimensional higher-order ice sheet model. In comparison to a conventional centred difference discretisation it enhances both numerical stability and convergence. In order to achieve these benefits the discretisation of the governing force balance equation makes extensive use of information on staggered grid points. Using the same iterative solver, a centred difference discretisation that operates exclusively on the regular grid serves as a reference. The reprise of the ISMIP-HOM experiments indicates that both discretisations are capable of reproducing the higher-order model inter-comparison results. This setup allows a direct comparison of the two numerical implementations also with respect to their convergence behaviour. First and foremost, the new finite difference scheme facilitates convergence by a factor of up to 7 and 2.6 in average. In addition to this decrease in computational costs, the accuracy for the resultant velocity field can be chosen higher in the novel finite difference implementation. Changing the discretisation also prevents build-up of local field irregularites that occasionally cause divergence of the solution for the reference discretisation. <br><br> The improved behaviour makes the new discretisation more reliable for extensive application to real ice geometries. Higher accuracy and robust numerics are crucial in time dependent applications since numerical oscillations in the velocity field of subsequent time steps are attenuated and divergence of the solution is prevented