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**1. Research topic**

Complex Analysis and Analytic Geometry belong closely together and are one of the few fields in the center of pure mathematics with many applications to other areas of pure mathematics (algebraic geometry, differential geometry, dynamical systems, P.D.E., topology, number theory, etc.) and applied Mathematics (theoritical physics, geophysics, mathematical economy, tomography).

The most deep results in all branches of mathematics use complex variables. There exist many results in applied mathematics which could not be discovered without the utilization of complex analysis.

A first phase of development in this area lasted until about 1965, when for the first
time several powerful quantative theories of the Cauchy-Riemann equations were developed
(L^{2}-theories of Hörmander and J.J. Kohn/L. Nirenberg,
Cauchy-Fantappié kernels of H.Grauert and G. Henkin and others) and the concepts of
currents, plurisubharmonicity, *q*-convexity, etc. had been well established. The
time between 1965 and 1989 was dominated by developing many refinements of these theories
together with the theory of residues, and the foundations of CR-manifolds. Furthermore,
the new methods made it possible to go on studying the strong links between geometry and
analysis. First important areas of this programme were so-to-speak used as testing fields.

Since the early 1990's a new phase in the development of complex analysis has started. The foundations ot the quantative methods have been worked out, such that now the detailed work on the applications could be started. As a new element, the geometry and analysis of CR-manifolds was developed to become a powerful tool. Now, all the different techniques and approaches can be put together in order to tackle new and refined aspects of the link between geometry and analysis in complex analysis and analytic geometry. This will, however, only be possible, if mathematicians with expertise in all the different approaches collaborate on a continuous intensive basis. At the same time, it has to be insured, that a new generation of complex analysts is trained in such a way, that they have a sufficiently broad basis for their own future research. It has become more and more unlikely, that this kind of education can be given to doctoral students and postgraduates at single schools of mathematics.

The goal of the project is to develop research on fundamental problems in complex analysis and partially or totally complex analytic geometry; these problems concern a rather large part of the quoted domains, but are, generally, closely related together.

The different teams have expertise in the following questions which are at the core of complex analysis and geometry: Vector bundles and transcendental methods in algebraic geometry. Cauchy-Riemann theory. Proper holomorphic mappings and convexity theory. Extension and boundary problems. Abel-Radon transform and theory of residues in several variables.

Vector bundles and transcendental methods in algebraic geometry are used in twistor theory, theory of strings and for non linear analysis.

Cauchy-Riemann theory is used for solution of old problems of P.D.E.

Abel-Radon transform is used in tomography and potential theory.

Residues are used in the theory of Feynman integrals and, for example, in geophysics and for sollution of equations of mathematical Physics.

Union of efforts of different teams in a network gives possibility to develop new original ideas, find new methods of solution of main problems and areas for applications.

**2. Project objectives**

Generally speaking, the research objective of the joint programme of work to be undertaken by the participants in the network is the study of many aspects of the link between geometry and analysis in Complex Analysis and Analytic Geometry and the use of recent results in this area in applications in Complex Analysis, Algebraic Geometry and other fields. A synthesis of all modern tools of quantative complex analysis and complex analytic geometry can create a new quality of research in this area of mathematics.

The main areas in which research in this sense will be carried out are complex differential geometry of vector bundles and transcendental methods in algebraic geometry, geometry of CR-manifolds and notions of finite type, Cauchy-Riemann equations on suitable classes of complex manifolds and on CR-manifolds, existence and boundary behaviour of proper holomorphic mappings, refined notions of convexity in complex analysis and corresponding questions of convex hulls and their determination, extension and boundary problems, theory of residues with applications to complex analytic geometry, integral transforms like Bergman and Szegö projections, complex Radon transforms, Hilbert transforms.

More precisely, the project intends to get signifiant progress on the following subjects which are specific aspects of the main theme of the project:

**1. Vector bundles, curvature.**

1.1.Vanishing theorems for ample and numerically effective bundles, criteria for very ample line bundles, classification of the manifolds whose tangent bundle is numerically effective, bundles and jet spaces, manifolds with negative curvature in the sense of jets.

1.2. Complex Monge-Ampère equation and canonical bundle of a complex manifold.

1.3. Quantitative study of the zeroes of holomorphic functions in any codimension.

1.4. Differential geometry of vector bundles on complex manifolds: complex and holomorphic structures, invariants and singularities, gauge theories and moduli, twistor spaces.1.5. Oka-Grauert principle.

1.6. Levi flat surfaces and flat metrics on bundles with Lorentz metrics.

**2. Transcendental methods in algebraic geometry**

2.1. Closed positive currents, approximation of positive currents by algebraic cycles, Lelong numbers and intersection theory, holomorphic Morse inequalities.

2.2. Applications to number theory; Hodge conjecture.

2.3. Currents and calibrated geometry: special metrics of type (n-1)-Kähler and existence of positive forms on complex manifolds.

**3. Cauchy-Riemann theory.**

3.1. Differential geometry of CR structures. Foliations with complex leaves:

1) to investigate foliations with complex leaves as ringed spaces (local rings of CR functions) and to develop a "q-pseudoconvexity theory" as well as for complex spaces. Crucial problems: vanishing theorems, Oka-Grauert principle, topology of such foliations.

2) Study foliations with complex leaves in

CP^{n}Notions and properties of CR-manifolds of finite type, their geometry and applications to analytic problems.

3.2. Microlocal analysis of CR functions, Cauchy-Riemann equations in p-convex domains. Analytic discs attached to submanifolds of

C^{n}.3.3. Quantitative theory for -equation and applications, in particular to the corona problem the Bergman and Szegö theory, extension problems. Resolution with L

^{2}methods and integral representation formulas of tangential Cauchy-Riemann equations in CR manifolds.Construction and estimation of Cauchy-Fantappié kernels on suitable classes of domains of finite type (convex domains, domains of finite regular type) and applications to the quantative theory of the CR-equations.

3.4. Integral transforms: L

^{2}-estimates for integral transforms in complex analysis. Relation to Fourier integral operators. Sampling and frames in spaces of analytic functions.3.5. -cohomology and global cohomology of CR manifolds;

3.6. Geometric and analytic invariants for CR structures: -Neuman problem on manifolds with non smooth boundary, Bergman kernel, standard invariant metrics and the Monge-Ampère equation.

3.7. Cauchy-Riemann equation and extension of CR objects: removable singularities of holomorphic and pluriharmonic functions.

**4. Proper holomorphic mappings.**

4.1. Problems of existence and classification.

4.2. Behaviour at the boundary.

4.3. Invariant metrics.

4.3. Holomorphic mappings between bounded domains; infinite dimensional polynomials.

**5. Convexity in complex analysis.**

5.1. Levi problem.

5.2. Serre problem.

5.3. Theory of

q-convex spaces.5.4. Boundary behaviour of holomorphic functions.

5.5. Closed positive currents and representing measures; relations to holomorphic and rational convexity.

**6. Extension problems, boundary problems.**

6.1. Extension of CR functions and differential forms: Hartogs-Bochner phenomenon for CR functions and differential forms.

6.2. Quantitative study of the extension and restriction of holomorphc functions and applications to complex differential geometry.

6.3. Extension of pluriharmonic functions; removable singularities of CR functions.

6.4. Boundary problem for holomorphic chains in a complex manifold, in particular, in q-concave open submanifold of a complex projective space.

6.5. Hull of holomorphy of submanifolds and subsets in

C^{n}.6.6. Boundary problem for CR chains in a CR manifold.

6.7. Problems of embedding of CR structures and boundary problems.

6.8. The Plateau problem for Levi flat surfaces

**7. Abel-Radon-Penrose transform and Theory of
residues.**

7.1. Integrals of meromorphic forms on Riemann surfaces and Abel-Radon transform.Applications to algebraic geometry.

7.2. Radon transform of analytic currents and of -cohomology on projective varieties. Application to P.D.E.; relations to web geometry.

7.3. Deformations of complex structures of concave manifolds and Einstein equation.

7.4. Residue morphisms; structure of residual currents; characterization of residual currents as a generalization of the characterization of holomorphic chains. Application to holomorphic differential forms on complex spaces. Analytic approach to the Grothendieck residue and duality theory.

**3. Tasks of the teams.**

The teams are designated by their number in the list of Participants

The tasks are designated by their number in the list of Objectives of the Project
(section **2**)

Team 1: 1 ,2, 3, 4, 5, 6 ,7.

Team 2: 1, 2, 3, 6.

Team 3: 3, 4, 5, 6.

Team 4: 1, 3, 6.

Team 5: 1, 2, 3, 6.

Team 6: 1, 2, 3, 5, 6, 7.

**ANNUAL PROGRESS REPORT** (1. October 1998- 30. September 1999)

Progress

(1) Scientific results achieved compared to the objectives set out in the work programme.

We refer to the sections of the Work Programme (Annex I) and denote the teams by their number in boldface. Joint work between two different teams are denoted ( )-( ).

**1. Vector bundles, curvature.**

1.1. For an ample vector bundle E over a smooth projective variety, the symmetric power
of E tensor determinant of E are positive in the sense of Nakano. (1)

Pseudo-effective bundles and hard Lefschetz theorem. (2)

1.4. Special Lagrangian submanifolds, Frobenius manifolds and Mirror symmetry. (5)

*1.7. LS(2 )coherent cohomology on etal coverings of a complex space. (2)

The space of LS(2) holomorphic sections of invariant line bundles over Galois coverings of
Zariski open sets in Moishezon manifolds has the following property: under
some curvcature conditions, its von Neuman dimension is bounded from below. (4)

**2. Transcendental methods in algebraic geometry.**

2.1. Extension of positive currents through small obstacles. Slicing and tangent cone of locally normal currents.(1) Improvements of bounds for the degree of generic hyperbolic hypersurfaces. (2)

*3. Cauchy-Riemann theory.*

3.1. Exension of CR functions from a non generic submanifold of CS(n) contained in a generic one.(1) Homogeneous CR-manifolds . (5) Complex structures on homogeneous spaces which are left-invariant by the action of a solvable Lie group. (5) Peter Ebenfelt works on questions related to the geometry of real submanifolds in complex space. His cooperation with Baouendi and Rothschild has resulted in numerous new theorems, several of which have been collected in the recent monograph ``Real submanifolds in complex space and their mappings" (6)

3.3. Powers of the Szegö kernel and Hankel operators in Hardy spaces.

Regularity CS(k) up to the boundary for on q-convex or q-concave corners and for
tangential on q-concave CR manifolds. (1) Bo Berndtsson has been working mainly with
various

problems concerning -estimates for the -operator. One
result, related to Sobolev regularity of the Bergman projector, has been published jointly
with Ph Charpentier (Bordeaux). Another result, on weighted estimates on the boundary of
pseudoconvex domains, (6)

3.3 and 3.6. Based on several important conversations with G. Henkin (1), G.Herbort and Klas Diederich gave new fine estimates on the boundary behavior of the pluricomplex Green function on regular domains and applied them to questions about the growth of the Bergman metric at the boundary. Furthermore, they succeeded to show that their new Green function with a given divisor as pole set has a series of interesting properties which make it possible to use it for showing that proper holomorphic maps between a large class of pseudoconvex domains cannot "blow down" complex analytic sets in the boundary. These results are contained in two preprints (1999). (3)

3.4. Mats Andersson has studied extensions of Taylors functional calculus that allow
non-holomorphic functions acting on operators. This has made necessary a reformulation of
Taylors theory in terms of differential forms and a precise study of asymptotically
holomorphic functions in several variables. Berndtsson has also collaborated with Mats
Andersson on non-holomorphic functional calculus in several variables. (6) Sandberg has in
particular obtained a generalization of the one-variable resolvent identity to several
variables, thus giving a new proof of the multiplicative property of taylors functional
calculus. (6) Problems of sampling in weighted spaces of entire functions. (6)

3.5. Generalization of the Andreotti-Grauert theory to real hypersusfaces; separation theorem for q-convex-concave real hypersurfaces.Dolbeault isomorphism on q-concave generic CR manifolds (for convenient dimensions); it coincides with the natural map (2)-(4)

3.7. New results on removable singularities on CR-manifolds. (4) Jointly with B Joricke N.
Shcherbina has also been working on non-removable singularities for CR-functions in higher
dimensions. A counterexample to the removability of generic balls has been given in a
paper to be published in Duke Math J. (6)

*3.8. Holomorphic dynamics and iteration theory. (5)

*3.9. Schwarz reflection in CS() and classification of germs of diffeomorphisms. (1)

**4. Proper holomorphic mappings.**

4.2. Proper holomorphic maps. Automorphisms group of domains with singular boundary. (5)

4.3. Characterization of analytic isomorphisms by infinitesimal metrics and properties of the Carathéodory pseudodistance on an infinite product. (1)

*5. Convexity in complex analysis.*

5.3. Complete characterization of Hankel operators for convex bounded domains of finite
type of CS(n). (1)

Serre's duality theorrem for cohomology with compact support of non compact complex
manifolds. Applications: separation theorem of Henkin-Leiterer for q-concave-(n-q)-convex
manifolds ; separation theorem of Andreotti-Vesentini for increasing union of q-concave
manifolds. (2)-(4) Interplay between real convexity and pseudoconvexity. (5)

5.4. Solution of the Dirichlet problem for for (1,1)-forms, with boundary value in the sense of currents, in a completely strictly pseudoconvex domain of a complex manifold. (2) Evolution by Levi form and pseudoconvexity in complex analysis. (5)

*5.6. Pseudoconvexe-concave duality. (2). Extension problems, boundary problems.

**6.
Extension problems, boundary problems.**

6.1. Hartogs-type extension theorem for meromorphic mappings into q-complete analytic spaces.

Boundary regularity of and Hartogs-Bochner Theorem in CPS(n). Hartogs phenomenon on real analytic, generic, non signed CR-manifolds. (1) LS(2) extension theorem for cohomology classes of positive degree (problems of regularity). (2)

6.3. Solution of the boundary problem for holomorphic chains on the product of a connected complex manifold and a disk-convex Kähler manifold (conteining the known results on CS(n) and CPS(n)). (1)

6.6. Stability of embeddings for pseudoconcave surfaces and their boundaries.

Embeddings for 3-dimensional CR-submanifolds. (1)

6.7. Preliminary results on Levi-flat hypersurfaces with given boundary in CS(n), using
currents method. (2),(1), (1)-(5)

Nikolay Shcherbina has in collaboration with G Tomassini been studying the Dirichlet
problem for Leviflat surfaces, resulting in a joint publication in Int Math. Res. Notes.
(5)-(6) Monge-Ampère equation on Grauert tubes. (5)

*7. Abel-Radon transform and Theory of residues.*

7.1. New inversions of Abel's theorem. Application: existence of Stein domains in CPS(n) which meet every algebraic hypersurface.(1)

7.2. Mikael Passare has worked on basic questions related to Laurent series inseveral complex variables. This led to a partial answer to a question posed by Gelfand, Kapranov and Zelevinsky regarding the number of possible expansions for a given rational function. In particular an upper estimate was found in terms of the integer points of the Newton polytope of the denominator of the rational function. (6)

**Advancement of the different tasks:**

Significant results have obtained in the different tasks. Published, prepublished papers,
or works in preparation, in collaboration betwwen two teams, have been obtained,
concerning the tasks 5 (2)-(4), 6 (2)-(4), (1)-(5), (5)-(6), 7 (1)-(6).

The team (6) has been involved in more sub-tasks than foreseen.

Results have been obtained in new sub-tasks 1.7, 3.8, 3.9, 5.6, noted with *.

What appears to be a "final" result in 1.5 (extension to Gromov theory) has been
obained by Forstneric under the influence of (1)-(4),

**Networking: **

Short (about one week) working visits to another team of the network:

From (2) to (3):1 , to (4): 2, to (5): 1;

(3) to (6): 1;

(4) to (2): 3;

(5) to (1): 1;

(6) to (1): 2.

Number of participations to meetings on the specility of the Network, in the E. U.:

(1): 5; (2): 4; (4): 5; (5): 3; (6): 5.

Several meetings, not financed by the Network, in the speciality of the Network, have been
partially organized, and partially attended by members of the Network. (we indicate the
number of the team of the organizers): March 99: Oberwolfach (3), Poitiers (1); April 99:
Stockholm (6); May 99: CIRM at Luminy (1), Pisa (5); June 99: CIRM at Trento (5), Cortona
(5). Common works have been elaborated at these occasions.

Appointement of young researchers: (in man-month)

2 postdocs have been appointed by the Network; one for 9 months in Gôteborg (6), the
other for 3 months in Wuppertal (3); both got new results to be published.

Publicity for vacant positions has been made inside the Network and by private conversations during participation in meetings. Public announcements have also been made during meetings.

The 2 postdocs from France to Göteborg and Wuppertal have been integrated quite easily and produced papers to be published.

**Joint publications**: we denote the involved teams by their number in
boldface.

Henkin G., Passare M., *Abelian differentials on singular varieties and variations
on a theorem of Lie-Griffiths*, Inv. Math. 135, n. 2, 1999, 297-328 (refereed
journal).**(1)-(6) **

Ch. Laurent-Thiébaut (Grenoble), J. Leiterer (Berlin)** (2)-(4),**

*Isomorphisme de Dolbeault dans les variétés CR,* Comptes Rend. Ac. Sci. Paris,
328, Série I, 1999, 469-472 (refereed journal).

*Some applications of Serre duality in CR manifolds*, Nagoya math. Journal, 154,
1999, 141-156 (refereed journal).

On Serre duality, à paraître au Bulletin des Sciences Mathématiques (refereed journal).

*Andreotti-Vesentini separation theorem on real hypersurfaces*, Prépublication de
l'Institut Fourier, 428, 1998, 1-12.

N. Shcherbina and G. Tomassini , *The Dirichlet problem for Leviflat graphs over
unbounded domains*, Int. Math. Res. Notes 3 1999 (refereed journal).** (5)-(6)**