Geometry of analytic continuation on complex manifolds – history, survey, and report

1 Introduction

In the theory of several complex variables, questions of extending analytic objects keeping their analyticity arise in many situations. The maximal domain of definition for such extensions has geometric properties similar to the convexity. At this point, basic questions arise characterizing the realm of several complex variables.

The study of domains of holomorphy was the starting point of general theory of several complex variables. Recall that a domain Ω over C n is said to be a domain of holomorphy if Ω ∈ π 0 ( O C n ) ^[1], i.e., if Ω is biholomorphically equivalent to a connected component of the structure sheaf O C n of C n , which is, by definition, the sheaf π : O ( = O C n ) → C n , of the germs of holomorphic functions. That C 2 ⧹ { ( 0 , 0 ) } is not a domain of holomorphy was first explicitly stated in the international congress of mathematics talk of Hurwitz in 1897. There exist various geometric characterizations of domains of holomorphy. A successful example is the following equivalences that originated in the works of Hartogs, Levi, Cartan, and Thullen and finalized by Oka in 1953 as follows:

Here δ Ω ( x ) ≔ sup { r ; ∃ ι : B ( π ( x ) , r ) ⟶ holomorphic Ω s.t. π ∘ ι = I d } ,

B ( w , r ) ≔ { z ∈ C n ; ‖ z − w ‖ < r } and PSH = plurisubharmonic .

The equivalence between − log δ Ω ∈ PSH ( Ω ) and Ω ∈ π 0 ( O ) is the solution of the Levi problem^[2] on C n . Recall that a complex manifold M is said to be holomorphically convex if

or equivalently,

A striking result in the early stage of several complex variables (SCV) was that a finitely sheeted domain over C n is holomorphically convex if and only if it is a domain of holomorphy (cf. [12]). We shall call a domain Ω → π M locally pseudoconvex if every point x ∈ M has a neighborhood U such that π − 1 ( U ) is holomorphically convex^[3]. It is known from the works of Fujita [25] and Takeuchi [72] that one can replace C n by CP n in Oka’s theorem by interpreting δ as “the distance to the boundary of Ω ” with respect to the Fubini-Study metric or replacing B ( π ( x ) , r ) by the corresponding geodesic ball. The result was further generalized by Ueda [73] for the domains over Grassmannian manifolds.

Note that C n is the unique simply connected complete Kähler manifold whose sectional curvature is zero (cf. [68]) and CP n is characterized as a compact Kähler manifold whose biholomorphic sectional curvature is positive (cf. [45] and [69]). For any noncompact locally pseudoconvex domain π : Ω → CP n , the curvature property can be used to show that, for any fixed C ∞ real-valued function ψ on CP n , − log δ Ω + ε π * ψ ∈ PSH ( Ω ) for sufficiently small ε > 0 . Recall that a holomorphically convex manifold M is called a Stein manifold if O ( M ) ↠ C { x , y } ^[4], i.e., the natural restriction map O ( M ) → C { x , y } is surjective, for all x , y ∈ M .

According to what J.-P. Serre told S. Hitotumatu, it had not yet been proven in 1954 that locally pseudoconvex domains over Stein manifolds are Stein (cf. [38])^[5]. As is well known, this question became an exercise because of the following characterization of Steinness due to Grauert [27] (see also [35, pp. 280–284]).

Since every locally pseudoconvex domain Ω over CP n can be shown to have strictly PSH exhaustion functions using the aforementioned property of − log δ Ω , one obtains Oka’s theorem and its generalizations by Fujita and Takeuchi from Theorem 1. Roughly speaking, the curvature positivity implies strict pseudoconvexity and the latter yields holomorphic functions.

Grauert’s method of showing “(2) ⇒ (1)” in Theorem 1 is basically a generalization of Oka’s method in the sense that it is by solving Cousin’s problem. However, it is not so constructive as Oka’s since it is based on the finite dimensionality of sheaf cohomology groups rather than their vanishing. Nevertheless, a great advantage of Grauert’s method is that it is available to produce holomorphic functions under weaker assumptions as in the following.

M is called strongly pseudoconvex if it satisfies (1’) or (2’).

Strongly pseudoconvex manifolds arise naturally as neighborhoods of compact analytic subsets that are holomorphically contractible to points. A decisive result in this context is that strongly pseudoconvex manifolds are nothing but the nonsingular models of Stein spaces with finitely many singular points (cf. Grauert [28], Hironaka [36], and Artin [2]).

On the other hand, the picture of O M changes quite a lot for other complex manifolds M . For instance, there exist a complex torus T and a domain Ω ⊂ T with ∂ Ω ≠ ϕ and Ω ∉ π 0 ( O T ) such that − log δ Ω ∈ PSH with respect to a flat metric (cf. [48]).

This change enlarges the range of questions on the geometry of analytic continuation on complex manifolds. Grauert [29] suggested one direction by introducing the notion of bundle convexity. The idea is to find a geometric condition for a holomorphic vector bundle π : E → M so that the sheaf of germs of its holomorphic sections has properties similar to O C n and O CP n .

Note that

so that Grauert’s E -convexity is more restrictive than the mere E -convexity. To find a reasonable class of ( M , E ) for which M is E -convex or so in Grauert’s sense, basic things to be studied are consequences of the curvature properties of M and E . In order to describe a reasonable statement in terms of curvature properties, we shall fix a Hermitian metric along the fibers of E (a fiber metric of E in short) and restrict ourselves to the following weaker convexity notion.

Let L → M be a holomorphic line bundle and let K M → M be the canonical line bundle of M . L is said to be positive (denoted L > 0 ) if it admits a fiber metric whose curvature form is everywhere positive. If the dual bundle L * is positive, L is said to be negative (denoted L < 0 ). Kodaira’s embedding theorem and Theorem 2 can be unified into the following.

M is called a weakly pseudoconvex manifold (= weakly 1-complete manifold) if it admits a C ∞ plurisubharmonic exhaustion function. It is known that every complex Lie group is weakly pseudoconvex (cf. [42]). Fixing any C ∞ plurisubharmonic exhaustion function φ : M → R , we put M c = { x ; φ ( x ) < c } for any c ∈ R . By the use of sufficiently rapidly increasing convex functions on [ − ∞ , c ) , the following can be deduced without difficulty, based on a standard L 2 method in [50]. (For the main result of [50], see Proposition 2 in §2.)

Therefore, if M is connected and noncompact in the situation of Proposition 1, one has

This observation enables us to construct singular fiber metrics (see §2) on L to conclude the following.

If K M < 0 , by applying Theorem 4 to produce singular fiber metrics of K M * with positive curvature current and with enough singularities along Σ , one has the following.

Theorem 5 is essentially a small addendum of Theorem 4. A somewhat bigger one is the following.

So, in view of Theorems 1–3, we are left with the following questions since 1998.

Q1. Does Theorem 4 remain true under the weaker assumption that L ∣ M ⧹ U > 0 for some U ⋐ M ?

Q2. What about Theorem 6?

As for Theorem 5, one may ask the following.

Q3. Does Theorem 5 remain true if K M < 0 is replaced by K M ∣ M ⧹ U < 0 for some U ⋐ M ?

Recently, it turned out that the answer to Q3 is affirmative. Namely, the following holds true.

It turned out that the answers to Q1 and Q2 are affirmative (cf. [60]). The next section will be devoted to showing an outline of the proof of Theorem 7.

2 Levi problem on weakly pseudoconvex manifolds

The proof of Theorem 7 is a combination of Takayama’s proof of Theorem 5 and the following.

Recall that H n , q ( M , E ) is canonically isomorphic to H 0 , q ( M , K M ⊗ E ) for any holomorphic vector bundle E → M . We note that (3) is an extension of Runge’s approximation theorem of Oka-Weil type (see [23]). (2) can be regarded as its extension to higher cohomology groups (cf. [1]). Because of (1), (2) is a consequence of the extension of (3) to the cohomology of higher degrees, but it can also be understood as a unique continuation of the cohomology classes (cf. [31, Part V, Commentary]).

In order to apply an argument in the proof of Theorem 5, one needs to produce a singular fiber metric of K M * in such a way that the L 2 method is available to find enough holomorphic functions to conclude that M is holomorphically convex.

Here, by a singular fiber metric of a holomorphic line bundle L , we shall mean a system of measurable functions of the form { h α e − ψ α } α associated with a trivializing open covering { U α } of M for L in such a way that h α ∈ C ∞ ( U α ) , h α > 0 , ψ α ∈ PSH ( U α ) ∩ L 1 ( U α ) , and h α e − ψ α = h β e − ψ β ∣ η β α ∣ 2 are satisfied on U α ∩ U β for a system of transition functions { η α β } of L .

Proposition 2 is applied to show the following.

For any singular fiber metric h of L , we denote by L ( 2 ) , loc p , q ( M , L , h ) the set of measurable L -valued ( p , q ) forms u on M such that ∣ u ∣ h , x 2 ∈ ℒ loc , x 1 for all x ∈ M and set

By applying Proposition 3, the proof of Proposition 2 can be generalized without difficulty to show the following.

One can deduce the following from Proposition 4.

In short, on weakly 1-complete domains, line bundles that are positive near the boundary have sufficiently many holomorphic sections over compact sets if sufficiently high tensor power is taken, so that it is possible to construct singular fiber metrics by using them in order to solve an interpolation problem. On locally pseudoconvex bounded domains in complex manifolds, the L 2 method works to solve similar interpolation problems. Some of the specific outcomes will be reviewed in the following.

3 Pseudoconvexity and the Bergman kernel

Every holomorphic map

is extendable to a holomorphic map D 2 → Ω (Hartogs). If Ω ⊂ C n and ∂ Ω is a C 2 -smooth real hypersurface with a defining function ρ , ∂ ∂ ¯ ρ ∣ Ker ∂ ρ is called the Levi form of ∂ Ω . Every domain of holomorphy Ω ⊂ C n with C 2 -smooth boundary has a defining function ρ whose Levi form is everywhere semipositive on ∂ Ω (Levi).

In 1933, Bergman observed the following in some special cases:

where, B Ω ( z , w ) denotes the Bergman kernel function of Ω (cf. [5]).

In 1965, Hörmander [40] proved that, given a domain Ω ⊂ C n ,

if the range of the ∂ ¯ -operator L ( 2 ) 0 , 0 ( Ω ) → L ( 2 ) 0 , 1 ( Ω ) is closed and ∂ Ω is strongly pseudoconvex at z 0 . Here, L ( 2 ) p , q ( Ω ) denotes the space of L 2 ( p , q ) -forms on Ω .

In 1974, Fefferman proved for strongly pseudoconvex domains Ω with C ∞ -smooth boundary that

holds for some C ∞ functions φ and ψ on Ω ¯ . The following is an application.

Fefferman’s theorem. Every biholomorphic map between two strongly pseudoconvex bounded domains Ω 1 and Ω 2 with C ∞ -smooth boundary extends as a diffeomorphism from Ω 1 ¯ to Ω 2 ¯ .

By an L 2 extension theorem in [62],

Ω ⋐ ψ c v x C n and ∂ Ω ∈ L i p ⇒ δ Ω ( z ) − 2 ≲ B Ω ( z , z ) .^[6]

Recently, Chen [13] proved that δ Ω ( z ) − 2 ≲ B Ω ( z , z ) also holds if ∂ Ω is locally the graph of a continuous function.

A connection between the weighted Bergman kernels and pluripotential theory is shown in the following diagram:

where RKHS ≔ reproducing kernel Hilbert space, CONS ≔ complete orthonormal system, and

Demailly’s approximation theorem in [17] asserts that

holds for any φ ∈ PSH ( Ω ) if Ω is pseudoconvex.

Looking for even better approximations, Demailly [18] asked whether or not ∣ f ∣ 2 e − φ ∈ L 1 ( Ω ) for φ ∈ PSH ( Ω ) and f ∈ O ( Ω ) implies ∣ f ∣ 2 e − p φ ∈ L loc 1 ( Ω ′ ) for Ω ′ ⋐ Ω and for p > 1 sufficiently close to 1. Recently, a sharp and effective affirmative answer was given by Guan [32].

If Ω admits a divisor A ⊂ Ω such that Ω ⧹ A is Stein, generalization of (4) holds for the families of weighted Bergman kernels, say B h m , for the spaces H ( 2 ) n , 0 ( Ω , L , h m ) ( m = 1 , 2 , … ) for the line bundles L → M with singular fiber metrics h with positive curvature current. An important property of B h is that

holds if Ω ⋐ Ω (= Ω is compact), where ℐ h denotes the multiplier ideal sheaf of h defined by

Therefore, it is tempting to ask for the behavior of log B h m ( z , z ) as m → ∞ and as z → ∂ Ω . In particular, the asymptotics of functions

might be interesting for any fixed fiber metric h 0 of a fixed line bundle L 0 → M .

4 Some geometry beyond holomorphic convexity

In Oka’s theory, important existence theorems are tied together by an approximation theorem of Runge type, so that the existence of a PSH exhaustion function is crucial to let the limiting arguments run. Hence,

Bergman, Oka → Grauert, Hörmander → Fefferman, et al.

≈

restricted class of PSH exhaustions ⇒ sharper analytic results.

On the other hand, the principal idea of [29] can be understood as follows:

In short, as far as the existence theorems are concerned, one should be able to replace the assumption of strict pseudoconvexity of Ω , in many cases, by the combination of the weak pseudoconvexity of Ω and the positivity of bundles along the boundary of Ω . Hence, if one wants to study the bundle-valued Bergman kernels on complex manifolds, one has to extend the application of the L 2 method to the situation where the domain does not admit PSH exhaustion functions in canonical ways. Such an extension of the objects seems to be indispensable because the following questions remain open for Kähler manifolds M .

Continuation of analytic objects on such Ω and M ˜ will be accompanied with interesting questions. For instance, one may ask O ( X ) ↠ ? O ( X ⧹ K ) if X = Ω or M ˜ as above when X is connected, dim X ≥ 2 and K ⋐ X . The answer is no in general, but something not totally stupid can be said in some cases. For instance, if Ω is a smooth and locally pseudoconvex bounded domain in a Kähler manifold of dimension ≥ 2 whose complement is not locally pseudoconvex, then ∂ Ω must be connected (cf. [52]).

We also note that Ω ⋐ s t r . ψ c v x M is an intrinsic property of Ω (cf. [47]), whereas Ω ⋐ loc . ψ c v x M is not, as one can see from the following examples.

1. CP 2 ⊃ Ω = CP 2 ⧹ { p } ≅ O ( 1 ) CP 1 ⊂ loc . ψ c v x CP 2 blown-up at p .

2. CP n × CP 1 ⊃ ( C n ⧹ { 0 } ) × { ζ ∈ C ; 1 < ∣ ζ ∣ < exp ( 2 π 2 ∕ log 2 ) } (not locally pseudoconvex if n ≥ 2 )

(See [51] for n = 1 and [20] for n ≥ 2 . See also [49].)

Therefore, one has to impose conditions more than the mere local convexity on ∂ Ω to extend Grauert’s theorem for E ∣ ∂ Ω > 0 along the idea (Gr-4).

As for conjecture 2, partial answers are given in [24] and [10].

5 From Riemann to Demailly

In [19], Demailly remarked as follows.

It is remarkable that Bernhard Riemann already anticipated in [66] the use of L 2 estimates and the idea of minimizing energy, even though his terminology was very different from the one currently in use.

As is well known, Riemann’s idea was realized, or rather justified, by Hilbert and Weyl and then further extended by Hodge and Kodaira. In particular, Kodaira characterized projective algebraic varieties as compact complex manifolds that admit positive line bundles, by establishing a cohomology vanishing theorem.

Demailly’s thesis [16] is one of the generalizations of Kodaira’s vanishing theorem. Demailly proved a vanishing theorem with L 2 estimates on complete Kähler manifolds under the semipositivity conditions on the curvature of the bundles. It was first observed by Grauert [26] that complete Kähler metrics live naturally on Stein manifolds as well as on quasi-projective manifolds. The reason why Demailly’s L 2 vanishing theorem is effective in algebraic geometry is that L 2 holomorphic functions extend analytically across proper analytic subsets of the domains in C n as in the case of Riemann’s removable singularity theorem in one variable.

The method of Demailly is a natural extension of Skoda’s variant of Andreotti-Vesentini-Hörmander’s refinement of Oka-Kodaira’s solution of the generalized Cousin problem. According to what I heard, Skoda, who was the adviser of Demailly, explored his method of solving a division problem with L 2 estimates after reading Oka’s paper recommended by Lelong. Demailly’s works have clearly shown that the method of L 2 estimates was a big breakthrough in SCV and complex geometry.

6 Bundle-convexity theorems

Here is a pseudo-chronologically ordered collection of works of the author related to Demailly’s:

Note that “ L 2 extension” is closely related to “ log B Ω t ( z , z ) ∈ PSH w.r.t. ( t , z ) ,” which was first discovered in [41,44] in special cases and established in [6] for Stein families { Ω t } . L 2 extension problems have been solved in this context from various viewpoints [4,7,8,19,32–34,63]. Another connection between the L 2 extension and the Bergman kernel is given by Demailly’s aforementioned approximation theorem.

Let E → M be a holomorphic vector bundle equipped with a fiber metric h , and let Ω ⊂ M be a relatively compact open set. In this situation, the ( E , h ) -convexity of Ω can be expressed more concisely, i.e.,

By an abuse of language, we shall confuse E -convexity with ( E , h ) -convexity for the bounded domains. Nontrivial bundle-convexity theorems were first obtained by Pinney [64] and Asserda [3].

Theorem 9 realizes the idea (Gr-4) under ∂ Ω ∈ C 2 , but only partially because E > 0 seems obviously superfluous.

Since Theorem 10 becomes false if one replaces E > 0 by E ∣ ∂ Ω > 0 (see Example 1 in §5), it is still necessary to impose some condition on ∂ Ω to extend it in the direction of (Gr-4).

Theorems 9 and 10 have been extended in [58] as follows.

Consequently, one can extend Theorem 7:

See also [65], which concludes holomorphic convexity in a similar situation but under a stronger assumption. On the other hand, negative but interesting examples exist also in this direction (cf. [14]).

A result on the kernel asymptotics for the case ∂ Ω ∈ C 2 is the following.

Sketchy accounts of the proofs of Theorems 11 and 12 are given below.