Lagrangian fibrations of holomorphic-symplectic varieties of K3^[n]-type

Let X be a compact Kahler holomorphic-symplectic manifold, which is deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface. Let L be a nef line-bundle on X, such that the 2n-th power of c_1(L) vanishes and c_1(L) is primitive. Assume that the two dimensional subspace H^{2,0}(X) + H^{0,2}(X), of the second cohomology of X with complex coefficients, intersects trivially the integral cohomology. We prove that the linear system of L is base point free and it induces a Lagrangian fibration on X. In particular, the line-bundle L is effective. A determination of the semi-group of effective divisor classes on X follows, when X is projective. For a generic such pair (X,L), not necessarily projective, we show that X is bimeromorphic to a Tate-Shafarevich twist of a moduli space of stable torsion sheaves, each with pure one dimensional support, on a projective K3 surface.Comment: 34 pages. v3: Reference [Mat5] and Remark 1.8 added. Incorporated improvement to the exposition and corrected typos according to the referees suggestions. To appear in the proceedings of the conference Algebraic and Complex Geometry, Hannover 201

Integral constraints on the monodromy group of the hyperkahler resolution of a symmetric product of a K3 surface

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. The second integral cohomology of M is endowed with the Beauville-Bogomolov bilinear form. We prove that the restriction homomorphism from Mon to the isometry group O[H^2(M)] is injective, for infinitely many n, and its kernel has order at most 2, in the remaining cases. For all n, the image of Mon in O[H^2(M)] is the subgroup generated by reflections with respect to +2 and -2 classes. As a consequence, we get counter examples to a version of the weight 2 Torelli question, when n-1 is not a prime power.Comment: Version 3: Latex, 54 pages. Expository change

Multi-Hamiltonian structures for r-matrix systems

For the rational, elliptic and trigonometric r-matrices, we exhibit the links between three "levels" of Poisson spaces: (a) Some finite-dimensional spaces of matrix-valued holomorphic functions on the complex line; (b) Spaces of spectral curves and sheaves supported on them; (c) Symmetric products of a surface. We have, at each level, a linear space of compatible Poisson structures, and the maps relating the levels are Poisson. This leads in a natural way to Nijenhuis coordinates for these spaces. At level (b), there are Hamiltonian systems on these spaces which are integrable for each Poisson structure in the family, and which are such that the Lagrangian leaves are the intersections of the symplective leaves over the Poisson structures in the family. Specific examples include many of the well-known integrable systems.Comment: 26 pages, Plain Te

Curve classes on irreducible holomorphic symplectic varieties

We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of K3 type and of Generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds.Comment: 15 page

Loss of testosterone impairs anti-tumor neutrophil function.

In men, the incidence of melanoma rises rapidly after age 50, and nearly two thirds of melanoma deaths are male. The immune system is known to play a key role in controlling the growth and spread of malignancies, but whether age- and sex-dependent changes in immune cell function account for this effect remains unknown. Here, we show that in castrated male mice, neutrophil maturation and function are impaired, leading to elevated metastatic burden in two models of melanoma. Replacement of testosterone effectively normalized the tumor burden in castrated male mice. Further, the aberrant neutrophil phenotype was also observed in prostate cancer patients receiving androgen deprivation therapy, highlighting the evolutionary conservation and clinical relevance of the phenotype. Taken together, these results provide a better understanding of the role of androgen signaling in neutrophil function and the impact of this biology on immune control of malignancies

On Integrable Systems and Supersymmetric Gauge Theories

The properties of the N=2 SUSY gauge theories underlying the Seiberg-Witten hypothesis are discussed. The main ingredients of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential are presented, the invariant sense of these definitions is illustrated. Recently found exact nonperturbative solutions to N=2 SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to integrable systems are discussed.Comment: LaTeX, 38 pages, no figures; based on the lecture given at INTAS School on Advances in Quantum Field Theory and Statistical Mechanics, Como, Italy, 1996; minor changes, few references adde

Using serum CA125 to assess the activity of potential cytostatic agents in ovarian cancer

Objective: New strategies are required to rapidly identify novel cytostatic agents before embarking on large randomized trials. This study investigates whether a change in rate of rise (slope) of serum CA125 from before to after starting a novel agent could be used to identify cytostatic agents. Tamoxifen was used to validate this hypothesis. Methods: Asymptomatic patients with relapsed ovarian cancer who had responded to chemotherapy were enrolled and had CA125 measurements taken every 4 weeks, then more frequently when rising. Once levels reached 4 times the upper limit of normal or nadir, they started continuous tamoxifen 20 mg daily, as well as fortnightly CA125 measurements until symptomatic progression. Because of the potentially nonlinear relationship of CA125 over time, it was felt that to enable normal approximations to be utilized a natural logarithmic standard transformation [ln(CA125)] was the most suitable to improve linearity above the common logarithmic transformation to base 10. Results: From 235 recruited patients, 81 started tamoxifen and had at least 4 CA125 measurements taken before and 4 CA125 measurements taken after starting tamoxifen, respectively. The mean regression slopes from using at least 4 1n(CA125) measurements immediately before and after starting tamoxifen were 0I0149 and 0I0093 [ln(CA125)/d], respectively. This difference is statistically significant, P = 0I001. Therefore, in a future trial with a novel agent, at least as effective as tamoxifen, using this effect size, the number of evaluable patients needed, at significance level of 5% and power of 80%, is 56. Conclusions: Further validation of this methodology is required, but there is potential to use comparison of mean regression slopes of ln(CA125) as an interim analysis measure of efficacy for novel cytostatic agents in relapsed ovarian cancer.Peer reviewedFinal Accepted Versio

Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant.Comment: v2: added due credits to the work of Burger, Iozzi and Wienhard. v3: corrected count of connected components for G=SU(p,q) (p \neq q); added due credits to the work of Xia and Markman-Xia; minor corrections and clarifications. 31 page

Dualities in integrable systems and N=2 theories

We discuss dualities of the integrable dynamics behind the exact solution to the N=2 SUSY YM theory. It is shown that T duality in the string theory is related to the separation of variables procedure in dynamical system. We argue that there are analogues of S duality as well as 3d mirror symmetry in the many-body systems of Hitchin type governing low-energy effective actions.Comment: 16 pages, Latex, Talk given at QFTHEP-99, Moscow, May 27-June