Geometrie Methods in Complex Analysis

Stefan Nemirovski

Abstract. The talk surveys the applications of geometric topology to complex analysis in several complex variables.

1. Introduction

Complex analysis in several complex variables has always been one of the most omnivorous branches of mathematics. Sheaf theory and Banach algebras, algebraic topology and L2-estimates for PDE's, all these and many other methods have been used to resolve various problems about holomorphic functions.

The purpose of this talk is to discuss complex analytic applications of geo­metric topology. Section 2 presents a rather personal and incomplete overview of complex analysis in C n . Section 3 explains how the topology of Stein manifolds is applied to analytic continuation of holomorphic functions and polynomial approx­imation. Section 4 concerns the connections between the differential topology of four-manifolds and analysis on complex surfaces.

My understanding of the subject has been largely formed at the complex analysis seminar conducted by Anatoliï Georgievich Vitushkin at the mekh-mat of the Moscow State University. I am deeply grateful to the participants of this seminar for many fruitful discussions.

2. Complex Analysis in Cn

2.1. Analytic continuation

Let us consider a holomorphic function / G 0(U) in a domain U C C n . In many situations it is desirable to know what is the maximal analytic extension of / . More precisely, we want to describe the "Riemann surface" of the function / . An important point is that this extension may be multi-valued or, in other words, the Riemann surface of / may be a domain over Cn.

In this setting, for individual functions or specific classes of functions, the problem can be extremely difficult. (Think of L-functions of algebraic varieties over Q.) Therefore, one first looks for extension properties possessed by all holo­morphic functions in a given domain. Technically, this is done by the notion of the envelope of holomorphy.

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Definition 2.1. Let U C Cn be a domain (an open connected subset). The envelope of holomorphy U D U is the maximal domain over C n such that every f G 0(U) extends holomorphically to U.

It is well known that this notion becomes meaningful when n > 2. There are many examples of domains in C2 that are strictly contained in their envelopes of holomorphy. Perhaps the most classical example is given by (a neighbourhood of) the Hartogs figure

H = {\Zl\ = 1, \z2\ < 1} U {\Zl\ < 1, z2 = 0} C C2 .

By the Hartogs Lemma, every function holomorphic in a neighbourhood of H extends to the unit bidisc {\zi\ < 1, \z2\ < ! } •

2.2. Polynomial and rational approximation

Let / E O(K) be a holomorphic function in a neighbourhood of a compact set K <& Cn. The problem is to find conditions (on / and K) under which / can be uniformly approximated by complex polynomials or rational functions. (This may be called "Runge-type" approximation problem. A much more subtle "Mergelyan-type" problem considers functions that are holomorphic in the interior of K only.)

Note that often every sequence of, say, polynomials converging uniformly on K converges on a larger set that does not depend on the sequence. This phe­nomenon can be observed already in C1. For instance, every sequence of polynomi­als converging on the unit circle converges on the entire unit disc by the maximum principle.

Definition 2.2. Let K ê C n be a compact subset. The convex hull of K with respect to a subalgebra of functions V C O(K) is the set

Kv = {zeCn\ \p(z)\ < max |p(£)| for all peV}.

Morally speaking, the larger is the P-convex hull Kp, the more obstructions there are to approximation by functions from V.

2.3. The case of C1

In this case it is relatively easy to determine the envelopes and hulls defined above. Namely,

• the polynomially convex hull of K <ê C1 is obtained by "filling in the holes" in K so that C1 \ K becomes connected (Runge Theorem);

• every compact set K (e C1 is its own rationally convex hull; • every domain [ / c C 1 is its own envelope of holomorphy.

The last two statements follow from the fact that the singularities of rational functions of one complex variable are isolated points. Note also the first appearence of a topological condition in the Runge theorem.

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2.4. Attaching analytic discs

In favourable circumstances, it is possible to approach higher dimensional problems by using essentially one-dimensional objects. The idea is to consider holomorphic maps / : A -> C n from the unit disc A c C 1 such that f(dA) C K where K is some fixed compact subset of C n .

The following properties of analytic discs make them an attractive tool for complex analysis:

• / (A) is contained in the polynomially convex hull of K (maximum principle); • if / (9A) is homologous to zero in K, then / (A) is contained in the rationally

convex hull of K (argument principle) ; • if / can be included in a continuous family ft : A —>• Cn starting from a

disc /o(A) C K, then / (A) lifts to the envelope of holomorphy of K (conti­nuity principle).

A (misleadingly) perfect example is the Hartogs figure H C C2. The obvious family of analytic discs

Dt = {\z1\<l,z2 = t}, \t\<\

satisfies all the above requirements and, in fact, completely exhausts the polyno­mial and rational hulls and the envelope of holomorphy of H.

Unfortunately, in general it is rather hard to find analytic discs with boundary in a given set. Furthermore, the behaviour of families of analytic discs tends to be very complicated. Nonetheless, various techniques based on families of holomorphic discs (and more general Riemann surfaces with boundary) have found important applications in complex analysis, symplectic and contact geometry.

In this talk, however, we shall pursue a different approach, based on global properties of envelopes and hulls. The results are less explicit but the methods can be applied in a more general setting.

3. Stein Manifolds

3.1. Definition

A complex analytic manifold is Stein if it can be properly holomorphically embed­ded into CN for some N. (This definition is good to work with but not to prove that one manifold or another is Stein.)

The following classical results show the crucial importance of Stein manifolds for several complex variables:

• envelopes of holomorphy are Stein manifolds spread over Cn ; • polynomial and rational hulls have Stein neighbourhoods with the Runge

property, i.e., holomorphic functions therein can be approximated by poly­nomials and rational functions, respectively.

Thus, every statement about Stein manifolds translates into a result on analytic continuation and approximation in Cn .

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3.2. Morse theory on Stein manifolds

Every Stein manifold X of complex dimension n admits a proper Morse function without critical points of index > n. To see this, consider the distance function to a sufficiently generic point in C^ for a proper holomorphic embedding X C CN. Standard Morse theory yields the following topological result:

Theorem 3.1. (Andreotti-Frankel, 1959) A Stein manifold of complex dimension n is diffeomorphic to the interior of a (possibly infinite) handlebody without handles of index > n.

In particular, Stein manifolds have no homology above the middle dimension, HP(X; Z) = 0 for all p > n. This has a strong analytic consequence:

Corollary 3.2. The envelope U D U "fills in" all the cycles of dimension greater than n inU.

For instance, if V C Cn , n > 2, is a bounded domain with connected bound­ary dV and U is a neighbourhood of the boundary, then the envelope of U must contain V. In other words, holomorphic functions defined near dV extend to the entire domain V, which is the classical theorem of Hartogs. Needless to say, there are easier ways to prove this particular result. However, it is much harder to make these more direct methods work in the other situations covered by the above corol­lary.

3.3. Runge property and topology

More can be said about the topology of polynomially convex sets. A polynomially convex subset K <ê C n admits a base of Stein neighbourhoods of the form

Uv = {z£Cn\ ip{z) < 0}

where <p : C n —> R is a proper Morse function without critical points of index > n (see [1]). It follows by Morse theory that

HpÇU^Z) = 0 for a l l p > n .

Therefore, polynomial hulls "fill in" the cycles of dimension n as well. In par­ticular, an n-dimensional closed submanifold in Cn is homologous to zero in its polynomially convex hull.

Somewhat surprisingly it is not clear what might be the analogue of this result for rationally convex sets. Applying Morse theory in the same way as for polynomially convex sets leads to problems about the topology of hypersurface complements in Cn . Homology computations suggest that purely topological ar­guments could be insufficient to settle this question (see [18]).

3.4. Morse theory on Stein manifolds, continued One may ask whether there exist other restrictions on the topology of Stein man­ifolds that are not captured by Morse theory. It turns out that the answer is different for n > 3 and n = 2. More precisely, the difference appears if we are interested in differential topology.

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Let M be an open smooth manifold of real dimension 2n equipped with a Morse function without critical points of index > n and an almost-complex structure. (These assumptions are clearly necessary for M to admit a Stein complex structure.)

Theorem 3.3. (Eliashberg, 1990) If n > 3, then M is diffeomorphic to a Stein manifold.

Hence, the Andreotti-Frankel theorem is sharp in dimensions > 3. In partic­ular, we cannot expect further general results for envelopes of holomorphy in C n , n > 3 .

Theorem 3.4. (Gompf, 1996) Ifn = 2, then M is homeomorphic to a Stein man­ifold.

The assertion is weaker because the differential topology of Stein complex surfaces has some extra rigidity. In many important cases, this statement can be made precise by using gauge theoretic invariants. Consequently, interesting phenomena arise in complex analysis on C2 and other complex surfaces.

Remark 3.5. The results of Eliashberg and Gompf are actually stronger. Namely, there exists a Stein complex structure realizing the homotopy type of the given almost-complex structure on M. In particular, the diffeomorphism (respectively, homeomorphism) to the Stein manifold is orientation preserving.

4. Analytic Continuation and Embedded Real Surfaces in Complex Surfaces

4.1. Envelopes of holomorphy of real surfaces in C2

Before proceeding to the general statements, let us consider two typical analytic corollaries.

By Morse theory, Stein domains in C2 can have non-trivial homology up to real dimension 2. So, let E C C2 be a smoothly embedded closed real surface of genus g (a sphere with g handles) and let us try to use (a deformation of) S as the two-skeleton of a Stein domain. This turns out to be possible only if g > 1.

Theorem 4.1. (Forstneric, 1992) If g > I, then there exists a C°-close embedding with a base of Stein neighbourhoods.

For instance, the Clifford torus T — {\zi\ = \z2\ = 1} C C2 has a fundamental system of Stein neighbourhoods

U£ = {l-e< |* i | , |* 2 | < l + e } .

Moreover, T is rationally convex. This is perhaps the simplest example to the Duval-Sibony theorem [5] on the rational convexity of Lagrangian submanifolds.

On the other hand, embedded two-spheres never have "small" Stein neigh­bourhoods:

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Theorem 4.2. (Nemirovski, 1998) A smoothly embedded 2-sphere E C C2 is ho­mologous to zero in every Stein domain over C2 containing it

In particular, E is homologous to zero in the envelope of holomorphy of any neighbourhood U D E. In other words, envelopes of holomorphy of domains in C2 "fill in" two-dimensional cycles represented by smooth embeddings of the two-sphere.

For instance, the envelope of holomorphy of the standard two-sphere

5 = {|z!|2 + \z2\2 = 1, lmz2 = 0} C C2

is the Levi-flat three-ball

B = {\z1\2 + \z2\

2 <l, lmz2 = 0} .

This follows from the continuity principle applied to the family of analytic discs Dt = {\zl\

2 + t*<i,z2 = t}, t e [-1,1].

The last example was generalized by several authors (see [3,13]). It was shown that the envelope of holomorphy of a generic two-sphere is given by a family of analytic discs provided that the sphere is contained in the boundary of a strictly pseudoconvex domain. Without the latter assumption, however, this geometric result is false [7].

4.2. Adjunction inequalities for real surfaces in Stein surfaces

The theorems stated in the previous subsection are a particular instance of general results on real surfaces in Stein complex surfaces [17].

Theorem 4.3. (Adjunction inequality) Let E C X be a smoothly embedded closed oriented real surface in a Stein complex surface. Then

[£]-[E] + | C l ( X ) . [ £ ] | < 2 9 ( £ ) - 2 (*)

if only E is not a null-homologous 2-sphere.

Here " • " denotes the intersection pairing, [E] G H2{X\ Z) is the cohomology class Poincaré dual to E, and c\(X) G H2(X; Z) is the first Chern class of X.

Inequality (*) is called the "adjunction inequality" because it resembles the classical adjunction formula for complex curves. Inequalities of this type were proved for embedded real surfaces in compact complex surfaces by using the Seiberg-Witten invariants (see [12, 15], and [19]). The result for Stein surfaces can be deduced from this and the algebraic approximation theorem of Stout [21].

In the compact case, adjunction inequalities yield a proof of the so-called "generalized Thorn conjecture." The latter asserts that a non-singular complex curve in a compact Kahler surface attains the minimal genus among all embedded real surfaces realizing the same homology class.

Stein manifolds do not contain compact complex curves, and hence this in­terpretation of the adjunction inequality does not make much sense for them. However, there is a different geometric way to see that inequality (*) is sharp in the Stein case.

Geometrie Methods in Complex Analysis 61

Theorem 4.4. (Existence of Stein neighbourhoods) / / a smoothly embedded real surface E c Y in an arbitrary complex surface Y satisfies inequality (*), then there exists a C°-close embedded surface with a base of Stein neighbourhoods.

This result is essentially a consequence of Gromov's /i-principle for totally real embeddings. More precisely, a generic embedded real surface S CY admits a Stein neighbourhood base if and only if it has only hyperbolic complex tangencies. (A complex tangency is a point p G S such that the tangent plane TPS is a complex line in TpY.) The /i-principle tells us that the surface E is isotopie to such a surface if there are no topological obstructions. However, by Lai's formulas for the number of complex tangencies of different types, the obstructions take on the form of inequality (*).

4.3. Legendrian surgery and Bennequin's inequality

To emphasize the role of the adjunction inequality as an obstruction to an /i-prin-ciple, we return briefly to the results of Eliashberg and Gompf on the existence of Stein manifolds with prescribed topology.

Let us restrict ourselves to the simplest situation. Suppose that we wish to build up a Stein complex surface diffeomorphic to the 4-ball with a two-handle attached to it along an unknotted circle in the 3-sphere. Note that to glue a two-handle, we must first choose a framing on this circle. In our case, the framing is determined by the integer n equal to the self-intersection index of the two-sphere formed by the core of the handle and a disc bounded by our circle in the ball.

The crucial observation of Eliashberg is that the resulting four-manifold can be endowed with a Stein complex structure if the following assumptions are satis­fied:

(i) the gluing circle is Legendrian with respect to the standard contact structure on the 3-sphere,

(ii) the framing is the canonical Legendrian framing minus 1.

However, the framings realizable by Legendrian embeddings of the unknot are negative by the famous Bennequin inequality [2]. Therefore, the two-sphere obtained as the "core" of our Stein surface can have self-intersection at most —2, which is precisely the bound given by the adjunction inequality.

To overcome this difficulty Gompf uses infinite constructions of four-manifold topology (Casson handles). This leads to Stein manifolds that are only homeomor-phic to the initial handlebody. On the other hand, there is no "Bennequin inequal­ity" in higher dimensions, and the method works within the smooth category.

Let us also note that, in the opposite direction, the adjunction inequality for real surfaces in Stein surfaces was used by Lisca and Matic [14] to obtain generalizations of the Bennequin inequality.

4.4. More applications Adjunction inequality (*) has several other applications to analytic continuation. Sometimes it is necessary to leave C2 and consider holomorphic functions over

62 S. Nemirovski

other complex surfaces. We shall state two related results whose proofs use en­velopes of holomorphy over C x CP 1 and CP 2 .

The first theorem generalizes the Hartogs Lemma in C2 in the following way. The disc {\zi\ < 1, z2 = 0} in the Hartogs figure is replaced by a graph of a continuous function bounded by 1.

Theorem 4.5. (Generalized Hartogs Lemma) Let f : A —> A be a continuous self-map of the unit disc and let Vf C A x A c C2 be its graph. Then every holomorphic function in a neighbourhood of the generalized Hartogs figure

ff/ = {|^| = i, \z2\<i}urfcc2

extends to the unit bidisc.

This theorem was proved by Chirka [4] in 1997. A proof based on inequal­ity (*) is explained in [17]. However, the original argument was inspired by the approach developed by Ivashkovich-Shevchishin [11] using Gromov's theory of pseudoholomorphic curves.

The classical Hartogs Lemma holds true in Cn for all n > 2. On the other hand, Rosay showed in [20] that the generalization is false when n > 3. The counterexample is obtained by an "embedded version" of Eliashberg's Legendrian surgery. This shows once again the difference between complex dimension 2 and higher dimensions.

The next theorem (proved in [16]) was probably the first application of ad­junction inequalities to complex analysis. This result was conjectured by Vitushkin in the 1980s as a by-product of his studies on the geometric aspects of the Jacobian conjecture in C2 (cf., for instance, [22]).

Theorem 4.6. (Vitushkin's Conjecture) Let S C CP 2 be an embedded 2-sphere that is not homologous to zero. Then every holomorphic function in a neighbourhood of S is constant.

The proof uses R. Fujita's theorem [9] that the envelope of holomorphy of a domain in the complex projective space C P n is either Stein or coincides with the entire C P n . If a domain in CP 2 contains a homologically non-trivial two-sphere (which must have positive self-intersection), the adjunction inequality excludes the first possibility. Hence, every holomorphic function extends from this domain onto the entire CP 2 and so is constant by the maximum principle.

The idea to regard Vitushkin's conjecture as an extension theorem is due to Sergei Ivashkovich. A similar argument shows that every meromorphic function in a neighbourhood of S is rational, that is, extends meromorphically onto CP 2 .

On the other hand, it follows from Theorem 4.4 that there exists an embedded real surface S C CP 2 of genus 3 homologous to the projective line and having a base of Stein neighbourhoods.

4.5. Selected problems: Fake Stein R4s In [10], Gompf proves that there exist uncountably many exotic Stein R4s (i.e., Stein complex surfaces homeomorphic but not diffeomorphic to R4). However, he

Geometrie Methods in Complex Analysis 63

observes that the construction works only for some classes of exotic R4s. In partic­ular, all Stein examples are actually diffeomorphic to domains in the standard E4 .

It will be interesting to know which exotic R4s are Stein. For instance, it does not seem plausible that the "universal" exotic M4 of Freedman and Taylor admits a Stein complex structure.

Somewhat more precisely one may ask the following:

Question. Are all exotic Stein R4s diffeomorphic to domains in M4?

Question. Are any (or all) of them biholomorphic to domains in C2 ?

The last problem is related to the old question about the existence of Stein manifolds (of arbitrary dimension) that are homeomorphic to the 2n-dimensional ball but cannot be realized as domains in Cn .

In any case, a Stein domain in C2 that is homeomorphic to E4 but not diffeomorphic to it would be an interesting object for "hard" analysis.

References

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Steklov Mathematical Insti tute Russian Academy of Sciences Gubkina str. 8 117966 Moscow GSP-1, Russia E-mail address: stefan@mccme.ru