The fundamental boundary value problem in the function theory of several complex variables is the ∂-Neumann problem. The L2 existence theory on bounded pseudoconvex domains and the C∞ regularity of solutions up to the boundary on smooth, bounded, strongly pseudoconvex domains were proved in the 1960s. On the other hand, it was discovered quite recently that global regularity up to the boundary fails in some smooth, bounded, weakly pseudoconvex domains. We survey the global regularity theory of the ∂-Neumann problem in the setting of L2 Sobolev spaces on bounded pseudoconvex domains, beginning with the classical results and continuing up to the frontiers of current research. We also briefly discuss the related global regularity theory of the Bergman projection.

AS observed originally by C. Osgood, certain statements in value distribution theory bear a strong resemblance to certain statements in diophantine approximation, and their corollaries for holomorphic curves likewise resemble statements for integral and rational points on algebraic varieties. For example, if X is a compact Riemann surface of genus > 1, then there are no non-constant holomorphic maps f : ℂ → X; on the other hand, if X is a smooth projective curve of genus > 1 over a number field k, then it does not admit an infinite set of /c-rational points. Thus non-constant holomorphic maps correspond to infinite sets of k-rational points.

This article describes the above analogy, and describes the various extensions and generalizations that have been carried out (or at least conjectured) in recent years.

This article is an exposition of our algorithm for canonical resolution of singularities in characteristic zero (Invent. Math. 128 (1997), 207-302), with an essentially complete proof of the theorem in the hypersurface case. We define a local invariant for desingularization whose values are finite sequences that can be compared lexicographically. Our invariant takes only finitely many maximum values (at least locally), and we get an algorithm for canonical desingularization by successively blowing up its maximum loci. The invariant can be described by a local construction that provides equations for the centres of blowing up. Our construction is presented here in parallel with a worked example.

This paper surveys work in partial differential equations and several complex variables that revolves around subelliptic estimates in the ∂-Neumann problem. The paper begins with a discussion of the question of local regularity; one is given a bounded pseudoconvex domain with smooth boundary, and hopes to solve the inhomogeneous system of Cauchy- Riemann equation ∂u = α, where a is a differential form with square integrable coefficients and satisfying necessary compatibility conditions. Can one find a solution u that is smooth wherever α is smooth? According to a fundamental result of Kohn and Nirenberg, the answer is yes when there is a subelliptic estimate. The paper sketches the proof of this result, and goes on to discuss the history of various finite-type conditions on the boundary and their relationships to subelliptic estimates. This includes finite-type conditions involving iterated commutators of vector fields, subelliptic multipliers, finite type conditions measuring the order of contact of complex analytic varieties with the boundary, and Catlin's multitype. The paper also discusses additional topics such as nonpseudoconvex domains, Holder and Lp estimates for ∂, and finite-type conditions that arise when studying holomorphic extension, convexity, and the Bergman kernel function. The paper contains a few new examples and some new calculations on CR manifolds. The paper ends with a list of nine open problems.

On a polarized uniruled projective manifold we pick an irreducible component X of the Chow space whose generic members are free rational curves of minimal degree. The normalized Chow space of minimal rational curves marked at a generic point is nonsingular, and its strict transform under the tangent map gives a variety of minimal rational tangents, or VMRT. In this survey we present a systematic study of VMRT by means of techniques from differential geometry (distributions, G-structures), projective geometry (the Gauss map, tangency theorems), the deformation theory of (rational) curves, and complex analysis (Hartogs phenomenon, analytic continuation). We give applications to a variety of problems on uniruled projective manifolds, especially on irreducible Hermitian symmetric manifolds S of the compact type and more generally on rational homogeneous manifolds G/P of Picard number 1, including the deformation rigidity of S and the same for homogeneous contact manifolds of Picard number 1, the characterization of S of rank at least 2 among projective uniruled manifolds in terms of G-structures, solution of Lazarsfeld's Problem for finite holomorphic maps from G/P of Picard number 1 onto projective manifolds, local rigidity of finite holomorphic maps from a fixed projective manifold onto G/P of Picard number 1 other than ℙn , and a proof of the stability of tangent bundles of certain Fano manifolds.

We give a systematic treatment of the quotient theory for a holomorphic action of a reductive group G = Kℂ on a not necessarily compact Kählerian space X. This is carried out via the complex geometry of Hamiltonian actions and in particular uses strong exhaustion properties of K-invariant plurisubharmonic potential functions.

In this article, written at the end of 1996, we survey some of the most important results in Seiberg-Witten Theory which are directly related to Algebraic or Kählerian Geometry. We begin with an introduction to abelian Seiberg-Witten Theory, with special emphasis on the generalized Seiberg-Witten invariants, which take also into account 1-homology classes of the base manifold. The more delicate case of manifolds with b+ = 1 is discussed in detail; we present our universal wall-crossing formula which shows that, crossing a wall in the parameter space, produces jumps of the invariants which are of a purely topological nature.

Next we introduce nonabelian Seiberg-Witten equations associated with very general compact Lie groups, and we describe in detail some of the properties of the moduli spaces of PU(2)-monopoles. The latter play an important role in our approach to prove Witten's conjecture. Then we specialize to the case where the base manifold is a Kahler surface, and we present the complex geometric interpretation of the corresponding moduli spaces of monopoles. This interpretation is another instance of a Kobayashi-Hitchin correspondence, which is based on the analysis of various types of vortex equations. Finally we explain our strategy for a proof of Witten's conjecture in an abstract setting, using the algebraic geometric "coupling principle" and "master spaces" to relate the relevant correlation functions.

We investigate attractors for holomorphic maps from ℙ k→ ℙ k, emphasizing the case k = 2. The interest in attractors stems from the fact that when a map is subject to small random perturbations, the long-term dynamics of the resulting system live near the map's attractors. In the case k = 1, that is, the case of rational functions on the Riemann sphere, the attractors are either periodic orbits or the whole sphere. In higher dimensions, however, there are other possibilities, which we call nontrivial. In addition to giving some examples of nontrivial attractors, we prove some general results about such attractors inℙ 2, among them that a given map can have at most one nontrivial attractor K, that K is then connected, has pseudoconvex complement, and contains a nonconstant entire image of ℂ, and that an attractor for a map f is also an attractor for any iterate fn.