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We obtain results on approximation of holomorphic maps by algebraic maps, the jet transversality theorem for holomorphic and algebraic maps between certain classes of manifolds, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds. [ABSTRACT FROM AUTHOR]

We study the Gauss-Martin connection for the meduli space of an arrangement of complex hyperplanes in the cohomology of a nontesonant complex rank one local system. Aomoto and Kita determined this Gauss-Martin connection for arrangements in general position. We use their results and an algorithm constructed in this paper to determine this Gauss-Martin connection for all arrangements. [ABSTRACT FROM AUTHOR]

This paper addresses conjectures of E. Bombieri and E Vojta in the special case of ruled surfaces not birational to p². Apart from this implicit restriction to p¹ bundles S over an elliptic curve, the ultimate question of the arithmetic of pairs (S,D) for a divisor D requires further restrictions on D which turn the proposed conjectures into the study of Roth's theorem on approximation of algebraic numbers α, but for α now parametrized by an elliptic curve. With these restrictions, best possible answers are obtained. The same study may also be carried out for holomorphic maps, and this is done simultaneously. [ABSTRACT FROM AUTHOR]

Suppose that a sequence of Riemannian manifolds with Ricci curvature ≥ -k2 converges to a Gromov-Hausdorff limit X. We show that if the amount of sectional curvature below K of the limiting manifolds approaches 0 in a suitable L1-sense, then X is an Alexandrov space of curvature ≥ K. As applications we present several generalizations of classical theorems. [ABSTRACT FROM AUTHOR]

We consider the wave maps equation with values into a Riemannian manifold which is isometrically embedded in Rm. Our main result asserts that the Cauchy problem is globally well- posed for initial data which is small in the critical Sobolev spaces. This extends and completes recent work of Tao and other authors. [ABSTRACT FROM AUTHOR]

Let f : X → S be a Lagrangian fibration between projective varieties. We prove that Rif*OX ≈ ΩiS if S is smooth. Suppose that X is an irreducible symplectic manifold or a certain moduli space of semistable torsion free sheaves on a K3 surface, the Hodge numbers satisfy hp,q(S) = hp,q(Pn), where n = dim S. If S ≈ Pn and X is an irreducible symplectic manifold, there exists a hypersurface Mf of the Kuranishi space of X such that every member of the Kuranishi family over Mf admits a Lagrangian fibration over Pn. [ABSTRACT FROM AUTHOR]

We consider compact connected six dimensional symplectic manifolds with Hamilto- nian SU (2) or SO (3) actions with cyclic principal stabilizers. We classify such manifolds up to equivariant symplectomorphisms. [ABSTRACT FROM AUTHOR]

We propose a new approach to solve the inverse boundary value problems in the Euclidean space. The idea consists of embedding the problem into hyperbolic manifolds and using their spectral properties. As a by-product, one can discuss the reconstruction of local conformal deformation of the metric of hyperbolic manifolds from the spectral data at infinity. We also propose a new spectral data observed from the cusp neighborhood at infinity. [ABSTRACT FROM AUTHOR]

A four-dimensional integrable rigid-body system is considered and it is shown that it represents two twisted three-dimensional Lagrange tops. A polynomial Lax representation, which doesn't fit neither in Dubrovin's nor in Adler-van Moerbeke's picture is presented. The algebro-geometric integration procedure is based on deep facts from the geometry of the Prym varieties of double coverings of hyperelliptic curves: Mumford's relation and Mumford-Dalalyan theory. The correspondence between all such coverings with Prym varieties splitted as a sum of two varieties of the same dimension and the integrable hierarchy associated to the initial system is established. [ABSTRACT FROM AUTHOR]

We analyze the integral points on varieties defined by one equation of the form f1 … fr = g, where the fi , g are polynomials in n variables with algebraic coefficients, and g has "small" degree; we shall use a method that we recently introduced in the context of Siegel's Theorem for integral points on curves. Classical, very particular, instances of our equations arise, e.g., in a well-known corollary of Roth's Theorem (the case n = 2, fi linear forms, deg g < r - 2) and with the norm-form equations, treated by W. M. Schmidt. Here we shall prove (Thm. 1) that the integral points are not Zariski-dense, provided &Sum; deg fi > n · max ( degfi) + deg g and provided the fi, g, satisfy certain (mild) assumptions which are "generically" verified. Our conclusions also cover certain complete- intersection subvarieties of our hypersurface (Thm. 2). Finally, we shall prove (mm. 3) an analogue of the Schmidt's Subspace Theorem for arbitrary polynomials in place of linear forms. [ABSTRACT FROM AUTHOR]

We study special Lagrangian cones in &doubleC;n with isolated singularities especially the case n = 3. Our main result constructs an infinite family of special Lagrangian cones in &doubleC;n each of which has a toroidal link. We obtain a detailed geometric description of these toil. We prove a regularity result for special Lagrangian cones in &doubleC;n with a spherical link—any such cone must be a plane. We also construct a one-parameter family of asymptotically conical special Lagrangian submanifolds from any special Lagrangian cone. [ABSTRACT FROM AUTHOR]

In this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the Laplacian on symmetric spaces of rank greater than one and their geometric perturbations. In our previous work we described the resolvent, and specifically the asymptotic behavior of the Green's function, on SL (3)/SO (3) using methods from three-particle scattering. Here we extend the technique of complex scaling to symmetric spaces to show that the resolvent continues analytically across the spectrum. [ABSTRACT FROM AUTHOR]