Vertical components constitute a new and powerful tool in the study of the local geometry of complex analytic mappings. One can exploit them to establish a certain level of algebraic control over the geometric complexity of analytic morphisms, along the lines of a classical Auslander's freeness criterion. On the other hand, when extended to the category of modules finitely generated over local analytic algebras, vertical components provide a natural setting for the study of homological properties of modules over noetherian rings, allowing for the use of transcendental methods where commutative algebra seemed to fail.

The biggest open problem in the area is desingularization in positive characteristic. Even in characteristic zero, there are several important problems (concerning both algebraic varieties and analytic spaces) that seem to be unsolved, though some are stated as theorems in the classical literature. Questions related to equisingularity in characteristic zero bear on recent programs for desingularization in positive characteristic. I will discuss several problems and their significance.

In the talk I present some general results on local behavior of holomorphic functions along complex submanifolds of C^N. As a corollary, I obtain multi-dimensional generalizations of an important result of Coman and Poletsky on Bernstein type inequalities on transcendental curves in C².

We consider the one-sided holomorphic extension of CR functions defined on non-smooth real analytic hypersurfaces. We show that unlike in the smooth case, the absence of a complex-analytic hypersurface inside the real analytic hypersurface (minimality) is not a sufficient condition for the extension of such functions. We formulate a geometric condition called "two sided support" which is a substitute for minimality in the simplest cases, e.g. for quadratic cones.

A real 4-submanifold in C5 is "CR singular" at a point where the tangent space contains a complex line. The local extrinsic geometry of a real analytic embedding near a CR singularity is studied by finding a normal form for the defining equations under biholomorphic transformations. We also consider one-parameter families of embeddings, and find a normal form for a family exhibiting a cancellation of a pair of CR singularities.

We shall discuss a construction of real hypersurfaces of tube type in complex space with prescribed and constant rank of the Levi form at every point. The construction involves a Cauchy-Kowalevsky type theorem for overdetermined systems (or a nonlinear version of the Poincare lemma).

Let w be a square matrix of (0,1)-forms on a strongly pseudoconvex smooth real hypersurface M in Cⁿ with n ³ 4. Assume that w satisfies the formal integrability condition [`(¶)]<sub>b</sub> w = wÙw. We want to find a non-singular matrix A such that [`(¶)]_b A = -Aw. Assume that the dimension of M is at least seven. We will find local solutions A with sharp regularities in terms of the smoothness of w.

This is joint work with Sidney M. Webster.

This talk is motivated by Grothendieck's theorem, according to which every finite rank vector bundle over P₁ splits into the sum of line bundles. In the 1960s, Gohberg generalized this to a class of Banach bundles. We consider a compact complex manifold X (thus dimX < ¥) and a holomorphic Banach bundle E ® X that is a compact perturbation of a trivial bundle in a sense recently introduced by Lempert. We prove that E splits into the sum of a finite rank bundle and a trivial bundle, provided H¹(X,O)=0.

As it follows from the classical Lindelöf theorem, the boundary values of a bounded holomorphic function defined on the unit disk cannot have discontinuities of the first kind. In our talk we define analogs of almost periodic functions on the unit circle, and show that uniform subalgebras of the algebra H^¥ of bounded holomorphic functions on the unit disk, generated by these functions have, in a sense, the weakest possible discontinuities on the boundary.

Joint work with Alexander Brudnyi.

Let K Ì Pⁿ be a compact subset of complex projective n-space. The projective hull of K is the set [^(K)] of points x Î Pⁿ for which there is a constant C=C(x) with

We prove that if g Ì Pⁿ is a stable real analytic curve (not necessarily connected), then [^(g)] is a 1-dimensional complex analytic subvariety of Pⁿ - g.

Let X be a Stein complex manifold of dimension at least 3. Given a compact codimension 2 real analytic submanifold M of X, that is the boundary of a compact Levi-flat hypersurface H, we study the regularity of H. If M has finitely many CR singularities, which is a generic condition, H must in fact be a real analytic submanifold. If M is real algebraic, it follows that H is real algebraic and in fact extends past M, even near CR singularities. To prove these results we provide two variations on a theorem of Malgrange, one for hypersurfaces with boundary and one for subanalytic sets.

A central result in complex geometry is the finiteness theorem of Cartan and Serre, according to which the cohomology groups of a finite rank holomorphic vector bundle over a compact base are finite dimensional. It is easy to convince oneself that holomorphic Banach bundles over a compact base may very well have infinite dimensional cohomology groups. Nevertheless, first Gohberg in the 1960s, and later Leiterer discovered a class of Banach bundles for which finiteness can be proved.

In the talk I will discuss a theorem on finite dimensionality of cohomology groups of Banach bundles, in a setting that includes Gohberg's and Leiterer's. Although cohomology groups are defined in terms of linear operators, the proof, interestingly, uses a piece of nonlinear analysis.