# Complex Analysis and Complex Geometry (16w5080)

## Organizers

Finnur Larusson (University of Adelaide)

Norman Levenberg (Indiana University)

Rasul Shafikov (University of Western Ontario)

Alexandre Sukhov (Universite des Sciences et Technologies de Lille)

## Objectives

We view complex analysis on the one hand and complex geometry on the other as two aspects of the same subject. The two are inseparable, as most work in the area involves inter-play between analysis and geometry. The fundamental objects of the theory are complex manifolds and, more generally, complex spaces, holomorphic functions on them, and holomorphic maps between them. Holomorphic functions can be defined in three equivalent ways as complex-differentiable functions, as sums of complex power series, and as solutions of the homogeneous Cauchy-Riemann equation. The threefold nature of differentiability over the complex numbers gives complex analysis its distinctive character and is the ultimate reason why it is linked to so many areas of mathematics.

Plurisubharmonic functions are not as well known to nonexperts as holomorphic functions. They were first explicitly defined in the 1940s, but they had already appeared in attempts to geometrically describe domains of holomorphy at the very beginning of several complex variables in the first decade of the 20th century. Since the 1960s, one of their most important roles has been as weights in a priori estimates for solving the Cauchy-Riemann equation. They are intimately related to the complex Monge-Ampère equation, the second partial differential equation of complex analysis. There is also a potential-theoretic aspect to plurisubharmonic functions, which is the subject of pluripotential theory.

In the early decades of the modern era of the subject, from the 1940s into the 1970s, the notion of a complex space took shape and the geometry of analytic varieties and holomorphic maps was developed. Also, three approaches to solving the Cauchy-Riemann equations were discovered and applied. First came a sheaf-theoretic approach in the 1950s, making heavy use of homological algebra. Hilbert space methods appeared in the early 1960s and integral formulas around 1970 through interaction with partial differential equations and harmonic analysis. The complex Monge-Ampère equation came to the fore in the late 1970s with Yau's solution of the Calabi conjectures and Bedford and Taylor's work on the Dirichlet problem.

Today, as before, complex analysis and complex geometry is a highly interdisciplinary field. The foundational work described above has been followed by a broad range of research at the interfaces with numerous other areas, for example algebraic geometry, functional analysis, partial differential equations, and symplectic geometry. Complex analysts and complex geometers share a common toolkit, but find inspiration and open problems in many areas of mathematics.

A single workshop cannot do justice to the breadth and depth of contemporary complex analysis and complex geometry. We have chosen a coherent collection of interrelated topics for the workshop, representing, in our view, some of the most vibrant developments in the subject today. Each of the four topics will be briefly introduced in general terms, followed by a few recent highlights.

**1. Almost complex manifolds and symplectic topology.** In a seminal paper of 1985, M. Gromov introduced almost complex structures and pseudoholomorphic curves into symplectic topology. The key properties of pseudoholomorphic curves---Fredholm theory and compactness---lay the foundation for Gromov's theory and are the reason for their far-reaching applications in symplectic topology (Gromov-Witten invariants, Floer homology, etc.).

The second totally original step was taken by S. Donaldson. In a fundamental paper (*J. Diff. Geom.,* 1996) he introduced a class of functions on an almost complex manifold which are solutions of a $overlinepartial$-inequality that can be viewed as a Beltrami-type condition. The zero sets of such functions are symplectic submanifolds. This observation allowed Donaldson to develop an extremely powerful approach to constructing symplectic objects using tools of complex analysis. The methods of Gromov and Donaldson are dual to each other. Gromov's theory deals with complex curves and is related to one-variable complex analysis, whereas Donaldson's aproach uses methods of several complex variables.

The interaction between complex geometry and symplectic geometry discovered by Gromov and Donaldson have had a deep impact on several branches of mathematics. Our interest is focused on the complex analytic aspects of the theory.

Some of the most important work in this area concerns topological methods in Stein geometry. A fundamental result due to Y. Eliashberg (*Int. J. Math.*, 1990) and R. Gompf (*Ann. Math.*, 1998) gives a remarkable topological characterization of Stein structures in terms of Morse exhaustion functions. This result may be viewed as a soft Oka principle. Related questions on Seiberg-Witten invariants, adjunction inequalities, contact topology, etc., are now under extensive development. Important results are due to F. Forstneriv c, P. Lisca and G. Matic, S. Nemirovski, and others. A related striking result due to J.Duval (*Invent. Math.,* 1990) states that a manifold is rationally convex if and only if it is Lagrangian. A deep generalization is due to Duval-Sibony (1995--1996). These results and related questions touch on some of the fundamental problems of complex analysis.

By now there are a large number of papers concerned with extending concepts and results from complex analysis and complex geometry to the almost complex case. Often the non-integrable case requires new methods that shed light on the integrable case. Notable new work includes a paper by S. Ivashkovich and A. Sukhov (*Ann. Inst. Fourier,* 2010) on the Schwarz reflection principle, boundary regularity and compactness for $J$-complex curves, a paper by A. Sukhov and A. Tumanov on proper $J$-holomorphic discs in Stein domains of dimension two (*Amer. J. Math.*, 2009), a paper by S. Ivashkovich and V. Shevchishin on almost complex structures that are merely Lipschitz (*Math. Z.,* 2011), and many others.

Recently, A. Sukhov and A. Tumanov (`arXiv:1306.5489`) gave a simple proof of Gromov's non-squeezing theorem using an original method of constructing $J$-holomorphic discs based on the Beltrami equation. Using ideas of H. Alexander (*Invent. Math.*, 1998), A. Sukhov and R. Shafikov ({tt arXiv:1302.5613}) gave a simplified proof of Gromov's theorem on the existence of a holomorphic disc attached to a compact Lagrangian *immersed* submanifold of $mathbb C^n$ with transversal self-intersections. Rational convexity of immersed Lagrangian submanifolds was established by J. Duval and D. Gayet (*Math. Ann.*, 2009). Recent work of K. Cielieback and Y. Eliashberg contains a new deep result on the topological characterization of polynomially and rationally convex hulls. This is a remarkable example of the mutually fruitful interaction between the complex and symplectic worlds.

While an almost complex manifold in general admits no $J$-holomorphic functions, plurisubharmonic functions continue to play a fundamental role. F. R. Harvey and H. B. Lawson (`arXiv:1107.2584`) gave an alternative but equivalent definition of subharmonic functions via the viscosity approach to the complex Hessian, and used it to solve the Dirichlet problem for the complex Monge-Ampère equation in both the homogeneous and inhomogeneous forms. J.-P. Rosay (*Math. Z.*, 2010) showed that $J$-holomorphic curves are $infty$-level sets of plurisubharmonic functions.

Other notable results include the proof by V. Shevchishin of the long-standing conjecture of Lagrangian nonimbeddability of the Klein bottle into $mathbb R^4$ and $mathbb{CP}^2$ (*Izv. Ross. Akad. Nauk.,* 2009). This was generalized by S. Nemirovski (*GAFA*, 2009) to higher dimensions.

**2. Elliptic complex geometry and Oka theory.** Elliptic complex geometry is concerned with the flexible analytic geometry of complex affine spaces $mathbb C^n$ and similar mani-folds, opposite to the rigidity that characterizes hyperbolic complex manifolds. Three important classes of similar manifolds, in order of increasing size, are Stein manifolds with the density property, elliptic manifolds as defined by M. Gromov in a seminal paper of 1989, and Oka manifolds that have only recently emerged from the developments inspired by Gromov's paper.

The group of holomorphic automorphisms of $mathbb C^n$, $ngeq 2$, is an infinite-dimensional group with a very rich structure. It has been intensively studied since a groundbreaking paper of E. Anders'en and L. Lempert in 1992. Their work has been extended to Stein manifolds with the density property and applied to a range of embedding problems, in particular to the long-standing conjecture that every open Riemann surface can be embedded into $mathbb C^2$. The flexibility of Oka manifolds is manifested in a tight connection between homotopy theory and complex analysis that has brought D. Quillen's theory of model categories into analysis for the first time in the work of F. L'arusson.

F. Forstneriv c formally introduced Oka manifolds and Oka maps (*C. R. Acad. Sci. Paris,* 2009, 2010), following his proof that more than a dozen Oka properties that had been under investigation for several years are all equivalent. A 500-page monograph of his, published in 2011 in Springer's Ergebnisse series, gives the first systematic exposition of much of the work that has been done in elliptic complex geometry and Oka theory in the past 20 years.

One of the most striking applications of the modern development of Oka theory is the solution of Gromov's Vaserstein problem by B. Ivarsson and F. Kutzschebauch (*Ann. Math.,* 2012). Kutzschebauch and S. Kaliman have a long-standing collaboration that straddles elliptic complex geometry and affine algebraic geometry. They have introduced an algebraic volume density property and used it to produce new examples of manifolds with the volume density property (*Invent. Math.,* 2008, 2010).

In papers that appeared in 2005--2008, E. F. Wold introduced powerful new techniques for constructing Fatou-Bieberbach domains and embeddings of open Riemann surfaces into $mathbb C^2$. He has continued this work in collaboration with Forstneriv c. They have solved the embedding problem for many infinitely connected planar domains (*Anal. PDE,* 2013). Wold, Kutzschebauch and S. Baader used Fatou-Bieberbach domains to show that holomorphic embeddings of the disc into $mathbb C^2$ can be topologically knotted (*J. reine angew. Math.,* 2010).

Among other new developments in this area is J. Prezelj's work extending the Oka principle from Stein spaces to $1$-convex spaces (*Trans. A.M.S.,* 2010), and its continuation with M. Slapar (*Michigan Math. J.,* 2011). Forstneriv c and Slapar have brought differential topo-logy into Oka theory and shown that the Oka principle holds for arbitrary target manifolds if one is allowed to deform the complex structure of the source manifold, and in real dimension 4 also the smooth structure (*Math. Res. Lett.,* 2007). T. Ritter has proved that every open Riemann surface embeds acyclically and holomorphically into an elliptic manifold (*Proc. Amer. Math. Soc.,* 2013). Kutzschebauch, L'arusson, and G. W. Schwarz have applied the Oka principle in geometric invariant theory (*J. reine angew. Math.,* to appear). Forstneriv c and L'arusson have clarified the place of Oka manifolds in the classification of compact complex surfaces (*Int. Math. R es. Not.*, to appear).

**3. Holomorphic foliations.** In recent years the theory of holomorphic foliations has become a central topic in geometric complex analysis and complex geometry. It is a very natural area of application of a wide range of techniques, and also a source of deep motivating questions.

An important problem which has attracted the attention of several researchers concerns the existence of Levi-flat hypersurfaces in complex manifolds. D. Burns and X. Gong (*Amer. J. Math.,* 1999) obtained a local classification of Levi-flat real analytic hypersurfaces near a Morse-type singularity. Many interesting results and questions concerning the structure of Levi-degeneracy sets of CR manifolds are contained in the work of D. Cerveau and A. Lins Neto (*Amer. J. Math.,* 2011) and of Cerveau and P. Sad (*Ann. Sc. Norm. Sup. Pisa,* 2004). One of the most challenging problems in this area is the conjecture that there does not exist a real analytic compact Levi-flat hypersurface in the complex projective plane. In higher dimensions this is a well-known result of Lins Neto (*Ann. Inst. Fourier,* 1999). Developing ideas of T. Ohsawa, S. Ivashkovich recently proposed a promising approach to this problem, based on his Bochner-Hartogs-type theorems for roots of holomorphic line bundles (`arXiv:1004.2618`).

An important result of Ivashkovich concerns an analogue of Novikov's vanishing cycle theory for holomorphic foliations (*Geom. Funct. Anal.,* 2011). He proves that a vanishing cycle only appears when a much richer complex-geometric object, called a foliated shell, is present. There has been remarkable progress on Hilbert's sixteenth problem on an upper bound for the number of limit cycles in a complex polynomial dynamical system due to G. Binyamini, D. Novikov, and S. Yakovenko (*Invent. Math.,* 2010).

Another aspect of the theory involves the relationship between holomorphic foliations and the complex Monge-Ampère equation. The classical work of E. Bedford and M. Kalka in the late 1970s has had important applications in the recent work of Lempert and L. Vivas (`arXiv:1105.2188`), and of Y. Rubinstein and S. Zelditch (*Adv. Math.,* 2011, and {tt arXiv:1205.4793}). Lempert's approach (*Bull. Soc. Math. France,* 1981) to the construction of solutions of the complex Monge-Ampère equation was further developed by S. Donaldson (*J. Sympl. Geom.,* 2002), and more recently by G. Patrizio and A. Spiro (*Adv. Math.,* 2010), and in the almost complex setting by H. Gaussier and J. Joo (*Ann. Sc. Norm. Super. Pisa,* 2010). More advanced analytic tools are used by V. Tosatti and B. Weinkove (* J. Amer. Math. Soc.,* 2010). A series of papers by H. B. Lawson and F. R. Harvey (starting with *Amer. J. Math.,* 2009) is devoted to a geome tric approach to the fundamental notion of plurisubharmonicity; they prove deep results on the Monge-Ampère equation on almost complex manifolds.

A very recent and highly active direction of research concerns the ergodic theory of holomorphic foliations, developed by T.-C. Dinh and N. Sibony (`arXiv:1004.3931}), and by C. Dupont and B. Deroin ({tt arXiv:1203.6244`). They have produced new and original constructions of invariant measures and Green currents associated to holomorphic foliations by analogy with discrete dynamical systems. This promising direction contains various natural and attractive questions. The interaction between holomorphic foliations and pluripotential theory is fundamental and will be a focus of activity in the coming years.

**4. Pluripotential theory and its applications.** Pluripotential theory involves the study of plurisubharmonic functions on complex spaces. The theory of the complex Monge-Ampère operator on classes of plurisubharmonic functions on domains in $mathbb C^n$ initiated by Bedford and A. Taylor during the years 1976--1989 has been further developed by Z. Bl ocki, U. Cegrell, S. Kol odziej, and others. The maximal domain of definition of the Monge-Ampère operator in this setting was determined by Bl ocki (*Amer. J. Math.,* 2006). Extremal plurisubharmonic functions, a priori defined as upper envelopes of certain families of plurisubharmonic functions, play an essential role in applications of pluripotential theory to approximation theory, complex dynamics, and arithmetic geometry.

The work of S. Boucksom and R. Berman (*Invent. Math.,* 2010) in the setting of compact complex manifolds has opened up new avenues of research, even in the setting of $mathbb C^n$. These include the solution by Berman, Boucksom, and D. Nyström of problems on asymptotics of Fekete points and normalized Bergman kernels (*Acta Math.,* 2011). These works have applications to K"ahler-Einstein metrics and Arakelov geometry, and have led to new probabilistic results, for example by O. Zeitouni and S. Zelditch (*Int. Math. Res. Not.,* 2010). It is hoped that these techniques can be used to study zero currents associated to random polynomials and random polynomial mappings in several complex variables and, more generally, random sections of holomorphic line bundles.

Pluripotential-theoretic methods continue to be fruitful in higher-dimensional complex dynamics. The Green current and the equilibrium measure for endomorphisms of $mathbb P^n$ and polynomial-like maps of $mathbb P^n$ have been studied by a host of researchers, such as Dinh and Sibony (see for example Springer *Lecture Notes in Math.,* vol. 1998, 2010). In particular, differences of $omega$-plurisubharmonic functions and superpotentials provide tools for a calculus of positive closed $(p,p)$-currents on a compact K"ahler manifold. Dinh and Sibony (*Comm. Math. Helv.,* 2006) also introduced a $dd^c$-method in dynamics that can possibly be applied to study random polynomials. This technique was successfully used by Dinh, Marinescu, and Schmidt (*J. Stat. Phys.,* 2012) to study equidistribution of zeros of holomorphic sections in a non-compact setting.

In other applications, D. Coman and E. Poletsky have derived new Bernstein, Bezout, and Markov-type inqualities for restrictions of holomorphic polynomials to certain transcendental curves using pluripotential theory, and applied them to transcendental number theory (*Invent. Math.,* 2007). A. Brudnyi has generalized these results to higher-dimensional complex submanifolds (*Invent. Math.,* 2008). This work has applications in many other areas of mathematics, such as convex geometry, subelliptic differential equations, and even distribution of zeros of families of random analytic functions.

**Timeliness and community interest.** Why is this a timely proposal for 2015? Most of the work we have cited is brand new, some of it not yet in print. We want to hold an international workshop on these developments as soon as possible, and the earliest it can be, at least if we want to avail ourselves of the excellent facilities and inspiring natural setting of BIRS, is 2015.

There is strong community interest in the workshop proposed here. Most of the 36 people on our list of possible participants (not counting the 4 of us) have expressed their interest in the workshop. We are reserving $42-(36+4)=2$ places for new PhDs and advanced graduate students, whom we will identify later.