aa r X i v : . [ m a t h . C V ] N ov ON THE THEORY OF STEIN MANIFOLDS
DUSTIN TRAN
Abstract.
This paper examines the broad structure on Stein manifolds andhow it generalizes the notion of a domain of holomorphy in C n . Along withthis generalization, we see that Stein manifolds share key properties from do-mains of holomorphy, and we prove one of these major consequences. In par-ticular, we investigate an equivalence, similar to domains of holomorphy andpseudoconvexity, on the class of manifolds. Then, we examine the canonicalsymplectic structure of Stein manifolds inherited from this equivalence, andhow its symplectic topology develops. Introduction
The class of Stein manifolds was first introduced by Karl Stein in 1951 [14], orig-inally under the name of holomorphically complete manifolds. Based on a set ofthree axioms, Stein sought to generalize the concept of a domain of holomorphyin C n to complex manifolds. After initial developments, further contributions weremade by H. Cartan [3], who proved vast generalizations of the first and secondCousin problems, now known as Cartan’s Theorem A and B. Other notable resultswere discovered by K. Oka [9], known previously for his solutions to the Cousinproblems and his work on domains of holomorphy, and E. Bishop [2], who provedan embedding theorem for Stein manifolds. For the interested reader, R. Remmert’scomprehensive survey article [10] provides a thorough historical review.This paper concerns Stein manifolds based on its primary motivation, and buildsup some familiar properties from domains of holomorphy. In particular, we showthat being Stein is equivalent to a certain notion of pseudoconvexity. Then by usingthe argument in the proof, we prove as a corollary that one of the axioms definingStein manifolds is implicit from the other two. Lastly, we see how the admittanceof a plurisubharmonic function leads us into questioning the natural symplecticstructure that arises, laying down modern results on symplectic topology.2. Preliminary Concepts
Domains of Holomorphy, Holomorph-Convex, and Pseudoconvex.
We first review some preliminary notions prior to defining Stein manifolds, definingthree equivalent definitions for a particular class of open sets in C n . Given an openset Ω ⊂ C n , denote A (Ω) as the set of holomorphic functions on Ω, C k (Ω) as the Date : May 14, 2012. class of C k functions on Ω, and C k ( p,q ) as the set of all forms of type ( p, q ) withcoefficients in C k . Definition 2.1.
An open set Ω ⊂ C n is a domain of holomorphy if there do notexist non-empty open sets U ⊂ Ω and U ⊂ C n , such that: • U is connected and U Ω ; • For every f ∈ A (Ω) , there is a function g ∈ A ( U ) such that f = g on U . Informally, we can interpret a domain of holomorphy as an open set, where some f ∈ A (Ω) cannot be holomorphically extended to a bigger set.For a compact subset K ⊂ Ω, the holomorphically convex hull of K is defined as(2.1) b K := n z ∈ Ω : | f ( z ) | ≤ sup w ∈ K | f ( w ) | for all f ∈ A (Ω) o . Definition 2.2.
An open set Ω ⊂ C n is holomorph-convex if for every compactsubset K ⊂ Ω , its holomorphically convex hull b K ⊂ Ω is compact. Recall that a function φ : Ω → R is plurisubharmonic if it is upper semicontinuousand τ f ( z + τ w ) is subharmonic for all z, w ∈ C n . An equivalent definition onthe class of C functions is that φ ∈ C (Ω) is plurisubharmonic if(2.2) n X j,k =1 ∂ φ ( z ) ∂z j ∂z k w j w k ≥ , z ∈ Ω , w ∈ C n . That is, the above hermitian form must be positive semi-definite. Moreover, wesay that φ is strictly plurisubharmonic if the hermitian form in (2.2) is >
0, i.e.,positive definite.
Definition 2.3.
An open set Ω ⊂ C n is pseudoconvex if there exists a plurisubhar-monic function φ ∈ C (Ω) such that • For all c ∈ R , Ω c := { z ∈ Ω : φ ( z ) < c } is relatively compact in Ω . Moreover, we say that Ω is strictly pseudoconvex if the function φ is strictly plurisub-harmonic. We now introduce the following theorem which relates these three classestogether. Theorem 2.4.
Given an open set Ω ⊂ C n , the following conditions are equivalent: • Ω is a domain of holomorphy. • Ω is holomorph-convex. • Ω is pseudoconvex. Readers interested in the proof may consult [11, Theorem 7.3.2], which uses stan-dard metric techniques. Although we do not use Theorem 2.4 to prove the resultsin this paper, the theorem provides a powerful intuition in recognizing the implicitstructure from Stein manifolds.
N THE THEORY OF STEIN MANIFOLDS 3
Remark 2.5.
This equivalence is actually part of a larger set of equivalences,including Levi convexity and what is now known as the local Levi property. Whenthe equivalence of these five conditions was first proven, the main difficulty liedin proving that the local Levi property implies domain of holomorphy. This becameknown as the Levi problem, named after E.E. Levi in 1911, and it remained unsolveduntil 1953, first proven by K. Oka [8] . Stein manifolds.
We now provide the original definition of Stein manifolds,and later investigate what equivalent definitions come about. Recall that a complexmanifold X n is a real 2 n -manifold (so n = dim C X ), equipped with holomorphictransition maps. Definition 2.6.
A Stein manifold is a complex manifold X n which satisfies thefollowing axioms: • Convexity: X is holomorph-convex; • Separation: If z = z ∈ X , then there exists f ∈ A ( X ) such that f ( z ) = f ( z ); • Local Coordinates: For all z ∈ X , there exists n functions f , ..., f n ∈ A ( X ) which form a coordinate system at z . By the definition, it is easy to see that every closed complex submanifold of a Steinmanifold is Stein, and that the Cartesian product X × X of two Stein manifoldsis also Stein. Example 2.7.
Let X be a noncompact Riemann surface (complex manifold ofdimension one). Then by Behnke-Stein (1948) , X must be a Stein manifold, havingproved the result by using a version of Runge’s approximation theorem. Example 2.8.
Every domain of holomorphy in C n is a Stein manifold X n . Indeed, because the structure on domains of holomorphy is an underlying motiva-tion for Stein manifolds, many theorems concerning domains of holomorphy applyequally well to Stein manifolds. First, we recall the following theorem, which is justthe generalization of the same argument for open sets in C . Theorem 2.9.
Let Ω be an open set in C n . For every compact set K ⊂ Ω andevery neighborhood U of K , there exist constants C α for all multi-orders α suchthat (2.3) sup K | ∂ α u | ≤ C α || u || L ( U ) , u ∈ A (Ω) . We now note the following properties, which naturally extend to Stein manifolds byanalogous arguments from open sets in C n . For brevity, we omit the exact proofs,though the analogous arguments are more or less straightforward. Theorem 2.10.
Let Ω be an open set in C n , or Ω n a complex manifold. Also,suppose φ is a strictly plurisubharmonic function in C ∞ (Ω) such that Ω c := { z ∈ Ω : φ ( z ) ≤ c } is relatively compact in Ω for all c ∈ R . Then every function whichis holomorphic in a neighborhood of Ω can be approximated uniformly over Ω byfunctions in A (Ω) . TRAN
Theorem 2.11.
Let Ω is pseudoconvex open set in C n , or Ω n a complex manifoldsuch that there exists a strictly plurisubharmonic function φ ∈ C ∞ ( X ) such that Ω c := { z ∈ X : φ ( z ) < c } ⊂⊂ X for every c ∈ R . Then the equation ∂u = f has asolution u ∈ C ∞ ( p,q ) ( X ) for every f ∈ C ∞ ( p,q +1) ( X ) such that ∂f = 0 . Readers interested in their proofs may look at [6, Theorems 4.3.1 and 5.2.8] and [6,Corollaries 4.2.6 and 5.2.6] respectively.3.
Stein Manifolds and Pseudoconvexity
We now prove a non-trivial extension of a property on domains of holomorphy.In particular, we shall establish a equivalence of definitions, similar to domainsof holomorphy and pseudoconvexity, in the sense of manifolds. Not surprisingly,this equivalence requires smoothness and nondegeneracy on the plurisubharmonicfunction, as opposed to only continuity for C n . Lemma 3.1. If X n is a Stein manifold, then X admits a strictly plurisubharmonicfunction φ ∈ C ∞ ( X ) such that • For all c ∈ R , X c := { z ∈ X : φ ( z ) < c } is relatively compact in X .Proof. Suppose X n is a Stein manifold, and let K ⊂ X be compact and U an openneighborhood of b K , the holomorphically convex hull of K . By the convexity axiomon X , we can choose a sequence of compact subsets of Ω,(3.1) K = b K ⊂ K = c K ⊂ K = c K ⊂ ..., such that c K j = K j and ∪ K j = X for all j ∈ N . Let U j be open sets such that K j ⊂ U j ⊂ K j +1 and U ⊂ U. Since b K j = K j , then for all j , we can choose functions f jk ∈ A ( X ) for k = 1 , ..., k j ,such that(3.2) | f jk | < ⇒ k j X k =1 | f jk ( z ) | < − j , z ∈ K j (3.3) max k | f jk ( z ) | > ⇒ k j X k =1 | f jk ( z ) | > j, z ∈ K j +2 \ U j . We note the implications are true, by taking sufficiently large powers of f jk andrearranging. Now by the local coordinate axiom, we can also choose n functionsforming a system of local coordinates at any z ∈ K j . Consider the function(3.4) φ : X → R ,z (cid:16) X j ∈ N k j X k =1 | f jk ( z ) | (cid:17) − . N THE THEORY OF STEIN MANIFOLDS 5
By (3.2), the inner series is bounded by 2 − j , so the full series converges, and by(3.3), we see that φ ( z ) > j − z ∈ X \ U j . Note that the series(3.5) X j,k f jk ( z ) f jk ( ζ )converges uniformly on compact subsets of X × X , so the sum is holomorphic on z and its complex conjugate is holomorphic on ζ . Then φ is a smooth function. Byabove, it is also easy to see that φ is plurisubharmonic.It remains to show that φ is strictly plurisubharmonic. Suppose for some z ∈ X ,(3.6) n X i =1 w i ∂f jk ( z ) ∂z i = 0 for all j, k .Then because there always exists n functions f jk forming a local coordinate systemat z , w = 0. Thus, (3.6) is positive-definite, so we are done. (cid:3) Remark 3.2.
Note that the proof does not require the separation axiom, only usingthe holomorph-convex and local coordinates axioms. This will become importantwhen establishing an equivalence between the two conditions of the lemma, as it willprove that the separation axiom is not necessary in defining Stein manifolds.
We now prove the implication from the other side, in order to obtain the followingtheorem. Note that the proof essentially mimicks the argument seen in [6].
Theorem 3.3.
A complex manifold X n is a Stein manifold if and only if it admitsa strictly plurisubharmonic function φ ∈ C ∞ ( X ) such that • For all c ∈ R , X c := { z ∈ X : φ ( z ) < c } is relatively compact in X .Proof. By Lemma 3.1, we have established the forward implication. Now supposewe have such a strictly plurisubharmonic function φ ∈ C ∞ ( X ). We first assert thefollowing claim. Claim.
For all w ∈ X , there exists a neighborhood U and function f ∈ A ( U ) ,such that f ( w ) = 0 and (3.7) Re f ( z ) < φ ( z ) − φ ( w ) , z = w ∈ U. Proof.
We examine w in local coordinates and obtain the inequality by lookingat its Taylor expansion. Let ( z , .., z n ) represent the local coordinates at w in aneighborhood U , such that the corresponding coordinate for w is the origin. Thenby Taylor series expansion, we can represent φ as(3.8) φ ( z ) = φ (0) + Re f ( z ) + n X j,k =1 ∂ φ (0) ∂z j ∂z j z j z k + O ( | z | ) , where f is a polynomial of degree ≤ f (0) = 0, represented by f ( z ) = az + bz .By definition of φ , the hermitian form in (3.8) is positive definite, so for non-zero z ∈ U ,(3.9) φ ( z ) > φ (0) + Re f ( z ) = ⇒ Re f ( z ) < φ ( z ) − φ ( w ) . TRAN (cid:3)
Now let w ∈ X . By the above claim, there exists a neighborhood U and function g ∈ A ( U ) , so that w U and w is covered by a set of local coordinates. Also, let U , U be other neighborhoods of w , such that U is relatively compact in U , and U is relatively compact in U. Let ψ be a smooth function compactly supported on X by U and ψ ( U ) = 1 . We aim to use ψ as a smooth cutoff function. Note that supp( ∂ψ ) ∈ U \ U , sothere exists c > ψ ( w ) and ǫ >
0, such that(3.10) Re g ( z ) < − ǫ, z ∈ supp( ∂ψ ) and φ ( z ) < c. From the proof of Theorem 2.11 [6, p.126], there exists a function φ c ∈ C ∞ ( X c ) , bounded from below in X c , such that the solution ∂u = f for every f ∈ L , ( X c , φ c )with ∂f = 0 has a solution u ∈ L ( X c , φ c ) where(3.11) || u || φ c ≤ || f || φ c . Also by Theorem 2.11, note that u must be smooth if f is. Now let g ∈ A ( U ) andconsider with a sufficiently large parameter t ,(3.12) g t = ψge tg − u t , where we aim to construct u t such that g t ∈ A ( X c ) and u t is small. Note that g t isholomorphic if(3.13) ∂ ( ψge tg − u t ) = 0 = ⇒ ∂u t = ge tg ∂ψ. Let f t := ∂u t . Then by (3.10), || f t || ψ c = O ( e − ǫt ) for fixed g , so we can apply (3.11).Thus, f t has a solution u t such that(3.14) || g t || ψ c = O ( e − ǫt ) , t → ∞ . Note that by (3.13), u t is holomorphic in the complement of the support of ∂ψ in X c . By Theorem 2.9, we obtain for w = w ∈ X ,(3.15) g t ( w ) = u t ( w ) → g t ( w ) = g ( w ) + u t ( w ) → g ( w ) , t → ∞ . Without loss of generality, let g ( w ) = 1, so g t ( w ) = g t ( w ) for t sufficently large.By Theorem 2.10, we can locally approximate g t , on X φ ( w ) , by a function h ∈ A ( X )such that h ( w ) = h ( w ). Thus, we have satisfied the separation axiom.Moreover, (3.14) implies that for every c < φ ( w ) ∈ R , (3.16) Z X c | g t | dV → , t → ∞ . By Theorem 2.9, g t → X c . If c ′ < c , then | g t | < on X c for sufficiently large t while g t ( w ) →
1. By Theorem 2.10, wecan approximate g t again by functions in A ( X ), so w cannot belong in b Ω c ′ , theholomorphically convex hull of Ω c ′ , for any c ′ < φ ( w ) . Thus, b Ω c ′ = Ω c ′ for all c ′ .Thus, we have satisfied the holomorph-convexity axiom. N THE THEORY OF STEIN MANIFOLDS 7
We now prove local coordinates axiom. Without loss of generality, we can suppose g ∈ A ( U ) vanishes at w . Then because ∂g t = ∂g − ∂u t at w and ∂u t → w by Theorem 2.9, we have that ∂g t → ∂g at w when t → ∞ . Now let g , ..., g n define a coordinate system at w formed by functions whichvanish there. Then the Jacobian of the corresponding functions g t , ..., g nt ∈ A ( X c )with respect to g , ..., g n converges to 1 as t → ∞ . Now by Theorem 2.10, we canapproximate g t , ..., g nt by h , ..., h n ∈ A ( X ) such that the Jacobian of h , ..., h n withrespect to g , ..., g n is also non-zero at w . This proves the last axiom. (cid:3) Thus, a generalized version of Theorem 2.4 still holds on the class of manifolds.This becomes important in deciphering how other problems belonging to severalcomplex variables in C n may translate more loosely to manifolds and sheaves, be-coming instrumental in Cartan’s later work in particular. We also see that with theadmittance of a particularly nice function, we obtain a canonical symplectic struc-ture, built from the manifold’s complex structure and plurisubharmonic function.Such consequences shall be delved into deeper within the following section. Butfirst, we note a remarkable property on Stein manifolds. Corollary 3.4.
The separation axiom is a consequence of the other two axioms,and so Stein manifolds are classified by holomorph-convexity and how it evaluatespoints locally.Proof.
Let X be a complex manifold satisfying the holomorph-convex and localcoordinates axiom. As explained in Remark 3.2, we can still apply Lemma 3.1since its proof does not use the separation axiom. Then by Theorem 3.3, X mustbe Stein. (cid:3) A Glimpse of its Symplectic Structure
Liouville domains and manifolds.
By the admittance of a strictly plurisub-harmonic function, Stein manifolds fall into a broader class of manifolds, one whichhas seen much development in the past decade. Here, we denote M n to be a man-ifold of real dimension n . Recall that if a symplectic manifold M n is exact, thenits symplectic form ω M has a corresponding primitive θ M . That is, there exists a1-form θ M satisfying ω M = dθ M . The Liouville vector field Z M is the vector fieldsuch that ω M ( Z M , · ) = θ M . Definition 4.1.
A Liouville domain is a compact exact symplectic manifold withboundary M n , such that Z M points strictly outwards along ∂M . Given a manifold M n , h : M → R is an exhausting function if it is bounded belowand proper (continuous such that the inverse image of compact subsets is compact).An exact symplectic structure is complete if the flow of Z M exists for all time. Definition 4.2.
A Liouville manifold is a complete exact symplectic manifold M n which admits an exhausting function h : M → R with the following property: TRAN
There exists a sequence { c k } k ∈ N for c k ∈ R , approaching + ∞ as k → ∞ , such that dh ( Z M ) > along h − ( c k ) . We say that a Liouville manifold M n has finite type if dh ( Z M ) > C ⊂ M . Note that for any Liouville manifold M n , its correspondingsublevel sets yield an exhaustion of M by Liouville domains. Property 4.3.
A Stein manifold X n is a Liouville manifold f M n . Moreover, ifits plurisubharmonic function can be taken to have a compact critical set, then theStein manifold is of finite type.Proof. Now let X n be a Stein manifold. By Theorem 3.3, X admits a smoothstrictly plurisubharmonic function φ : X → R , such that each open set X c isrelatively compact. Note that this implies φ is our desired exhausting function.Furthermore, X has a 1-form(4.1) θ X := − d c φ, where d c is defined by d c ( φ )( X ) := dφ ( JX ), J being the complex structure whichevaluates cotangent vectors. X then carries the canonical symplectic form withprimitive θ X , so(4.2) ω X := − dd c φ. Fixing c k to be a regular value of φ , the sublevel set(4.3) M n := φ − ( −∞ , c k ] , equipped with ω X , is a Liouville domain. Thus, every Stein manifold X n is aLiouville manifold f M n . The last remark regarding the critical set of φ follows byconstruction. (cid:3) Symplectic cohomology.
We now introduce a particularly useful invariantfor distinguishing symplectic manifolds in one of the Liouville classes, first consid-ering Liouville domains. We denote SH ∗ as short-hand for symplectic cohomology.For a more comprehensive introduction to symplectic cohomology, one may consultSeidel’s survey article [12]. Fix K to be the desired field of coefficients. Definition 4.4. If M is a Liouville domain of dimension n , then the symplecticcohomology of M with K -coefficients, SH ∗ ( M ) , is a Z / -graded K -vector space witha natural Z / -graded map (4.4) H ∗ + n ( M ; K ) → SH ∗ ( M ) . Given two Liouville domains U , M where dim U = dim M , and an embedding ǫ : U → M such that ǫ ∗ θ M − cθ U is exact for some constant c >
0, then Viterbo’sconstruction [15] assigns to each embedding a pull-back restriction map(4.5) SH ∗ ( ǫ ) : SH ∗ ( M ) → SH ∗ ( U ) . This is homotopy invariant within the space of all such embeddings, and functorialwith respect to composition of embeddings. More generally, by a parametrizationargument of Viterbo’s construction, this is invariant under isotopies of embeddings,within the same class.
N THE THEORY OF STEIN MANIFOLDS 9
Note that any Liouville domain can be canonically extended to a finite type Liouvillemanifold by attaching an infinite cone to the boundary. Conversely, any finite typeLiouville manifold can be truncated to a Liouville domain which is a sufficientlylarge sublevel set of the exhausting function. So with the natural correspondencebetween Liouville domains and Liouville manifolds of finite type, we can naturallydefine symplectic cohomology on the latter. We aim to generalize this feature forLiouville manifolds of finite type to arbitrary Liouville manifolds.
Definition 4.5. If M is a Liouville manifold, then (4.6) SH ∗ ( M ) = lim ← SH ∗ ( U ) , where the limit is over all pairs ( U, k ) such that the Liouville vector field Z M sat-isfying θ M + dk points strictly outwards along ∂U , where U ⊂ M is a compactcodimension zero submanifold and k is a function on M . Exotic symplectic structures.
Note that any smooth complex affine alge-braic variety is a Stein manifold. By choosing a suitable K¨ahler form, we see thatsmooth complex affine algebraic varieties naturally correspond to Liouville man-ifolds of finite type. The most recent result on exotic symplectic structures, byAbouzaid-Seidel [1] in 2010, shows that there exists infinitely many distinct exoticstructures on smooth complex affine varieties of real dimension n ≥ Theorem 4.6 (Mc-Lean) . There exists an exotic Stein manifold for all complexdimension n > . Moreover, for complex dimension n > , there exists infinitelymany exotic Stein manifolds of finite type, each of which are pairwise distinct. The main tool used in distinguishing exotic structures from the standard one hasbeen using symplectic (co)homology as an invariant on particular classes of man-ifolds. Abouzaid-Seidel’s argument for their paper uses the fact that symplecticcohomology is invariant under symplectomorphisms between Liouville manifolds offinite type, yet it remains open whether or not this is true for Liouville manifoldsin general. The proof of the invariance for finite type relies heavily on the cylin-drical end property exclusive to finite type Liouville manifolds, and so no possibleextension of the current proof can be taken.Other symplectic invariants remain useful as well. Harris [5] proves the existence ofexotic structures by examining how some symplectic properties change under smalldeformations of its symplectic form. He also shows that symplectic cohomologyfails to distinguish them from the standard form, while this invariant succeeds.Ultimately, while symplectic cohomology continues to be a powerful tool, moderndevelopments in other tools and concepts are in the works, in order to discover newproperties from a new perspective.
Last remarks.
Stein manifolds provide a rich algebraic, complex, and sym-plectic structure, all of which stem from the manifold’s primary motivation as adomain of holomorphy. For open sets in C , we see that there always exists a holo-morphic function on the open set, such that it cannnot be holomorphically extendedonto a bigger set. This is not true for C n in general, and so domains of holomorphybecome our natural consideration for this nice structure of open sets in C .In turn, Stein manifolds become the natural consideration for this structure on theclass of manifolds. From this generalization, we obtain many of the same prop-erties, thus connecting many of the general concepts regarding sheaves, varieties,and manifolds, back down to the more concrete, identifiable spaces with alreadyheavily-developed theorems and properties. As a result, Stein manifolds remain anincredibly useful space to analyze, bridging the gap between two seemingly separatecategories. References [1] M. Abouzaid and P. Seidel. Altering symplectic manifolds by homologous recombination.Preprint arXiv:1007.3281, 2010.[2] E. Bishop. Mappings of partially analytic spaces.
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Paris: S´eminaire E.N.S. ,1951-52.[4] K. Cieliebak and Y. Eliashberg. Symplectic geometry of stein manifolds. In preparation.[5] R. Harris. Distinguishing between exotic symplectic structures. Preprint arXiv:1102.0517,2012.[6] L. H¨ormander. An introduction to complex analysis in several variables.
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Comment. Math.Helv. , 80(4):859-881, 2005.[14] K. Stein. Analytische funktionen mehrerer komplexer ver¨anderlichen zu vorgegebenen peri-odizit¨atsmoduln und das zweite Cousinsche Problem.
Math. Ann. 123 , 1951 (p.201-222).[15] C. Viterbo. Functors and computations in Floer homology with applications, Part I.