An implicit function theorem for sprays and applications to Oka theory
aa r X i v : . [ m a t h . C V ] A p r AN IMPLICIT FUNCTION THEOREM FOR SPRAYSAND APPLICATIONS TO OKA THEORY
YUTA KUSAKABE
Abstract.
We solve fundamental problems in Oka theory by establishing animplicit function theorem for sprays. As the first application of our implicitfunction theorem, we obtain an elementary proof of the fact that approximationyields interpolation. This proof and L´arusson’s elementary proof of the conversegive an elementary proof of the equivalence between approximation and inter-polation. The second application concerns the Oka property of a blowup. Weprove that the blowup of an algebraically Oka manifold along a smooth algebraiccenter is Oka. In the appendix, equivariantly Oka manifolds are characterizedby the equivariant version of Gromov’s condition Ell , and the equivariant lo-calization principle is also given. Introduction
In the present paper, we establish an implicit function theorem for (holomorphic)sprays to solve fundamental problems in modern Oka theory initiated by Gromov[4] in 1989 (cf. [2, 3]). Here is the definition of sprays.
Definition 1.1.
Let X be a complex space and Y be a complex manifold.(1) A (local) spray over a holomorphic map f : X → Y is a holomorphic map s : U → Y from an open neighborhood of the zero section of a holomorphicvector bundle E → X such that s (0 x ) = f ( x ) for all x ∈ X . Particularly inthe case of U = E , s is also called a global spray .(2) A spray s : U → Y is dominating if s | U x : U x → Y is a submersion at 0 x foreach x ∈ X .The following is our implicit function theorem for sprays. Here, the space ofholomorphic maps O ( U, Y ) and the space of holomorphic sections Γ(Ω , U ) areequipped with the compact-open topologies.
Theorem 1.2.
Let X be a Stein space, Ω ⋐ X be a relatively compact open subset, Y be a complex manifold and s f : U → Y be a dominating spray over a holomorphicmap f : X → Y . Then there exist an open neighborhood V ⊂ O ( U, Y ) of s f and acontinuous map ϕ : V →
Γ(Ω , U ) such that ϕ ( s f ) = 0 and s ◦ ϕ ( s ) = f | Ω for all s ∈ V . Mathematics Subject Classification.
Primary 32E10, 32Q56; Secondary 32M05, 32S45.
Key words and phrases.
Stein space, Oka manifold, dominating spray, ellipticity, blowup.
1N IMPLICIT FUNCTION THEOREM FOR SPRAYS AND APPLICATIONS 2
The proof of Theorem 1.2 is given in Section 2. The relative version of thistheorem (Remark 2.2) was already used implicitly in our previous paper to provethe implication from CAP to convex ellipticity [7, Proposition 5.3]. In this paper,we give two more applications to Oka theory. To state the first application, letus recall the following Oka properties. Here, Op K denotes a non-specified openneighborhood of K . Definition 1.3.
Let Y be a complex manifold.(1) Y enjoys the Basic Oka Property with Approximation (BOPA) if for any Steinspace X , any compact O ( X )-convex subset K ⊂ X and any continuous map f : X → Y such that f | Op K is holomorphic, there exists a homotopy f t : X → Y ( t ∈ [0 , f t | Op K is holomorphic and approximates f uniformlyon K for each t ∈ [0 , f : X → Y is holomorphic.(2) Y enjoys the Basic Oka Property with Interpolation (BOPI) if for any Steinspace X , any closed complex subvariety X ′ ⊂ X and any continuous map f : X → Y such that f | X ′ is holomorphic, there exists a homotopy f t : X → Y ( t ∈ [0 , f t | X ′ = f | X ′ for each t ∈ [0 , f : X → Y isholomorphic.It is one of the results in L´arusson’s paper [10] that BOPI implies BOPA. Theconverse is also true by Forstneriˇc’s Oka principle (see [3, § Oka manifold . Com-pared with L´arusson’s proof, however, the proof of Forstneriˇc’s Oka principle isnot easy. In Section 3, we give an elementary proof of the following fact by usingTheorem 1.2.
Corollary 1.4.
For any complex manifold,
BOPA implies
BOPI . The same argument gives an elementary proof of the implication from POPAto POPI (Corollary 3.3). Recently, Kutzschebauch, L´arusson and Schwarz [8]introduced the equivariant version of Oka properties. Our elementary proof isapplicable also to the equivariant setting (see Remark 3.2).The second application concerns the Oka property of a blowup. In Oka theory, itis important to understand when the blowup of an Oka manifold along a smoothcenter is Oka. It is known that there exists a discrete set D ⊂ C such thatthe blowup Bl D C of C along D is not Oka [6, Example A.3]. On the otherhand, L´arusson and Truong proved that the blowup of a manifold of Class A (e.g. C n ) along a smooth algebraic center is Oka [11, Main Theorem] (see also [5]). Amanifold of Class A is known to be algebraically Oka. To state the definition ofalgebraically Oka manifolds, recall that a complex manifold Y is Oka if and only if Y satisfies Gromov’s condition Ell , i.e. for any holomorphic map f : X → Y froma Stein manifold there exists a dominating global spray over f [6, 7]. Analogously,an algebraically Oka manifold is defined to be an algebraic manifold Y satisfyingthe algebraic version of Ell , i.e. for any regular map f : X → Y from an affine N IMPLICIT FUNCTION THEOREM FOR SPRAYS AND APPLICATIONS 3 manifold there exists an algebraic dominating global spray over f (cf. [12]). InSection 4, we generalize the result of L´arusson and Truong as follows. Corollary 1.5.
Let Y be an algebraically Oka manifold and A ⊂ Y be a closedalgebraic submanifold. Then the blowup Bl A Y of Y along A is Oka. In the appendix, a few results on equivariant Oka theory are given. More pre-cisely, we give the equivariant versions of the characterization of Oka manifolds byEll and the localization principle for Oka manifolds (Corollary A.4 and TheoremA.5). As an application of the latter, we prove that every smooth toric varietywith its torus action is equivariantly Oka (Example A.6).2. Proof of Theorem 1.2
In order to prove Theorem 1.2, let us recall Rouch´e’s theorem (see [1, § µ a ( f ) of a holomorphic map). Theorem 2.1 (cf. [1, p. 110]) . Let U ⋐ C N be a bounded domain, f, g : U → C N be continuous maps which are holomorphic on U and satisfy | f − g | < | f | on ∂U .Then the numbers of zeroes of f and g in U (counted with multiplicities) are equal: X f ( a )=0 µ a ( f | U ) = X g ( b )=0 µ b ( g | U ) . Proof of Theorem 1.2.
By the definition of a spray, U is an open neighborhood ofthe zero section of a holomorphic vector bundle p : E → X . Let E ′ denote theholomorphic vector subbundle of E with fibers E ′ x = ker(d( s f | U x ) x : E x = T x U x → T f ( x ) Y ) , x ∈ X. Since X is Stein, there exists another holomorphic vector subbundle E ′′ of E suchthat E = E ′ ⊕ E ′′ (cf. [3, Corollary 2.6.6]). Then the map ( p, s f ) | U ∩ E ′′ : U ∩ E ′′ → X × Y restricts to a biholomorphic map from an open neighborhood U ′′ ⊂ U ∩ E ′′ of the zero section onto an open neighborhood of the graph of f . Since Ω ⋐ X is relatively compact, we may assume that U ′′ | Ω ⊂ U is also relatively compact.Note that if a spray s : U → Y is sufficiently close to s f , the restriction s | U ′′ | Ω isuniformly close to s f | U ′′ | Ω . Then by using Rouch´e’s theorem (Theorem 2.1) on eachfiber of p | U ′′ | Ω : U ′′ | Ω → Ω, we can conclude that there exists a unique holomorphicsection ϕ ( s ) : Ω → U ′′ which is close to the zero section and satisfies s ◦ ϕ ( s ) = f | Ω .By construction, ϕ ( s ) depends on s continuously and satisfies ϕ ( s f ) = 0. (cid:3) Remark 2.2.
The same proof also gives the relative version (for relative sprays[7, Definition 2.1]) and the parametric version of Theorem 1.2 (use the argumentsin [3, p. 254]).
N IMPLICIT FUNCTION THEOREM FOR SPRAYS AND APPLICATIONS 4 Elementary proof of that approximation implies interpolation
Let us recall the following fact which ensures the existence of dominating localsprays. It is an easy application of Siu’s theorem [13] and Cartan’s Theorem A.Here, a compact subset of a complex space is called a
Stein compact if it admits abasis of open Stein neighborhoods.
Lemma 3.1 (cf. [3, Lemma 5.10.4]) . Let K be a Stein compact in a complex spaceand Y be a complex manifold. Then for any holomorphic map f : Op K → Y thereexist an open neighborhood W ⊂ C N of and a dominating spray Op K × W → Y over f .Proof of Corollary 1.4. Assume that a complex manifold Y enjoys BOPA. Let X , X ′ and f be as in the definition of BOPI (Definition 1.3). Since X is Stein, thereexists an exhausting sequence ∅ = K ⊂ K ⊂ K ⊂ · · · ⊂ [ j ∈ N K j = X, K j ⊂ K ◦ j +1 of compact O ( X )-convex subsets. Set t j = 1 − − j for each j ∈ N ∪ { } , andtake a complete distance function d on Y and a positive number ε >
0. For each j ∈ N , we shall inductively construct homotopy f t : X → Y ( t ∈ [ t j − , t j ]) suchthat f t j | Op K j is holomorphic and the following hold for each t ∈ [ t j − , t j ]; • f t | X ′ = f | X ′ , • f t | Op K j − is holomorphic andsup x ∈ K j − d ( f t ( x ) , f t j − ( x )) < ε j . Then f = lim t → f t : X → Y exists and the homotopy f t ( t ∈ [0 , f t : Op K j → Y ( t ∈ [ t j − , t j ]) with the above properties since we can extendit as f t j − +( t − t j − ) χ ( x ) ( x ) by using a continuous function χ : X → [0 ,
1] such that χ | Op K j ≡ χ is contained in a small open neighborhood of K j .Assume that we already have f t : X → Y ( t ∈ [ t j − , t j ]) for some j ∈ N ∪ { } where t − = 0 and K − = ∅ . Set L j = K j ∪ ( X ′ ∩ K j +1 ) and note that it is O ( X )-convex. By using the semiglobal holomorphic extension theorem (cf. [3,Theorem 3.4.1]), after deforming f t j if necessary, we may assume that f t j | Op L j is holomorphic. Then by Lemma 3.1 there exists a dominating local spray s t j :Op L j × Op D N → Y over f t j | Op L j . By using a retraction of C N onto D N anda continuous function χ ′ : X → [0 ,
1] such that χ ′ | Op L j ≡ χ ′ is con-tained in a small open neighborhood of L j , we may assume that s t j is extendedto X × C N → Y continuously so that s t j ( · ,
0) = f t j . By BOPA of Y , there ex-ists a homotopy s t : Op K j +1 × C N → Y ( t ∈ [ t j , t j +1 ]) such that s t | Op L j × Op D N is holomorphic and approximates s t j on Op L j × Op D N for each t ∈ [ t j , t j +1 ], N IMPLICIT FUNCTION THEOREM FOR SPRAYS AND APPLICATIONS 5 and s t j +1 is holomorphic. Then Theorem 1.2 implies the existence of a homo-topy of holomorphic maps ϕ t : Op L j → C N ( t ∈ [ t j , t j +1 ]) such that ϕ t j ≡ s t ◦ (id , ϕ t ) = f t j | Op L j for each t ∈ [ t j , t j +1 ]. We can extend this homotopyto ϕ t : Op K j +1 → C N continuously so that ϕ t j ≡
0. Then by the parametricCartan–Oka–Weil theorem (cf. [3, Theorem 2.8.4]), there exists a homotopy ofholomorphic maps e ϕ t : Op K j +1 → C N ( t ∈ [ t j , t j +1 ]) such that e ϕ t j ≡ t ∈ [ t j , t j +1 ]; • e ϕ t | X ′ ∩ Op K j +1 = ϕ t | X ′ ∩ Op K j +1 , and • e ϕ t | Op K j approximates ϕ t uniformly on K j .Then the homotopy f t = s t ◦ (id , e ϕ t ) : Op K j +1 → Y ( t ∈ [ t j , t j +1 ]) has the desiredproperties. (cid:3) Remark 3.2. (1) By taking a nontrivial compact O ( X )-convex subset K ⊂ X and a small number ε >
0, we can see that the above argument also gives anelementary proof of that BOPA implies BOPAI (see [3, § G -BOPA implies G -BOPI for a finite group G (cf. [8, Corollary 4.2]).(3) L´arusson’s proof of that interpolation implies approximation [10] works even if Y is a singular complex space (see [9, Proposition 1.5]). However, our proof of theconverse is not applicable since we do not have a good definition of dominatingsprays for singular targets.It is known that the parametric Oka properties POPA and POPI are also equiv-alent (see [3, § Corollary 3.3.
For any complex manifold,
POPA implies
POPI . The Oka properties are generalized to the relative setting and it is known thatapproximation and interpolation are equivalent also for a holomorphic submer-sion (see [7, § Blowups of algebraically Oka manifolds
We first recall the Convex Approximation Property (CAP) and its algebraicversion aCAP.
Definition 4.1.
A complex manifold (resp. an algebraic manifold) Y enjoys CAP (resp. aCAP ) if any holomorphic map from an open neighborhood of a compactconvex set K ⊂ C n ( n ∈ N ) to Y can be uniformly approximated on K byholomorphic maps (resp. regular maps) C n → Y . N IMPLICIT FUNCTION THEOREM FOR SPRAYS AND APPLICATIONS 6
The following facts relate the above properties to the Oka properties.
Theorem 4.2 (cf. [3, § . (1) A complex manifold enjoys CAP if and only if it is Oka.(2) An algebraic manifold enjoys aCAP if it is algebraically Oka.
Let us prove the stability of aCAP under blowups which implies Corollary 1.5(compare with the instability of CAP [6, Example A.3]).
Corollary 4.3.
Let Y be an algebraic manifold and A ⊂ Y be a closed algebraicsubmanifold. If Y enjoys aCAP , then the blowup Bl A Y also enjoys aCAP . Inparticular, the blowup Bl A Y is Oka.Proof. Let π : Bl A Y → Y denote the blowup. Take a compact convex set K ⊂ C n and a holomorphic map f : Op K → Bl A Y . By Lemma 3.1, there existsa dominating local spray s ′ : Op K × Op D N → Y over π ◦ f . Since Y enjoysaCAP, there exists a regular map s : C n × C N → Y which approximates s ′ onOp K × Op D N . Then Theorem 1.2 implies the existence of a holomorphic section ϕ : Op K → Op K × C N of the trivial bundle which is close to the zero section andsatisfies s ◦ ϕ = π ◦ f . Let Sing( s ) denote the singular locus of s and Σ ⊂ C n + N bethe intersection of Sing( s ) and the non-Cartier locus of the scheme theoretic inverseimage s − ( A ). Then Σ ⊂ C n + N is a closed algebraic subvariety of codimension atleast two because C n + N is smooth. Since s | Op K × C N is dominating and ϕ is closeto the zero section, we may assume that ϕ ( K ) ⊂ C n + N \ Sing( s ) ⊂ C n + N \ Σ.Note that the pullback s ∗ Bl A Y → C n + N \ Σ of Bl A Y → Y along the restriction s : C n + N \ Σ → Y coincides with the blowup Bl s − ( A ) \ Σ ( C n + N \ Σ) → C n + N \ Σ over C n + N \ Sing( s ). By the universality of the pullback, there exists a holomorphicmap e ϕ : Op K → Bl s − ( A ) \ Σ ( C n + N \ Σ) such that the following diagram commutes:Op K Bl s − ( A ) \ Σ ( C n + N \ Σ) Bl A Y C n + N \ Σ Y fϕ e ϕ π ∗ ss ∗ π πs By definition, if s − ( A ) \ Σ is non-Cartier at z ∈ s − ( A ) \ Σ then s − ( A ) \ Σ issmooth at z . Thus the result of L´arusson and Truong [11, Theorem 1] implies thatthe blowup Bl s − ( A ) \ Σ ( C n + N \ Σ) is algebraically Oka. By Theorem 4.2, there existsa regular map ˜ f : C n → Bl s − ( A ) \ Σ ( C n + N \ Σ) which approximates e ϕ uniformly on K . Then the regular map π ∗ s ◦ ˜ f : C n → Bl A Y approximates f on K . (cid:3) Remark 4.4.
In his seminal paper [4], Gromov asked whether the algebraic Okaproperty is a birational invariant [4, Remark 3.5.E ′′′ ] and this problem is still
N IMPLICIT FUNCTION THEOREM FOR SPRAYS AND APPLICATIONS 7 open. Since Corollary 4.3 and the converse of (2) in Theorem 4.2 would imply thestability of the algebraic Oka property under blowups, it is natural to ask whetherthe converse of (2) in Theorem 4.2 holds. It will be studied in future work.
Appendix A. Elliptic characterization and localization ofequivariantly Oka manifolds
As we mentioned in the introduction, Oka manifolds (resp. algebraically Okamanifolds) are characterized by Ell (resp. the algebraic version of Ell ). Thesecharacterizations imply the following localization principles. Theorem A.1 (cf. [6, Theorem 1.4] and [3, Proposition 6.4.2]) . Let Y be acomplex manifold (resp. an algebraic manifold). Assume that each point of Y hasan open Oka (resp. algebraically Oka) neighborhood with respect to the analytic(resp. algebraic) Zariski topology . Then Y is Oka (resp. algebraically Oka). In this appendix, we give the equivariant versions of these facts. To this end,we need the following definitions.
Definition A.2.
Let G be a reductive complex Lie group and Y be a G -manifold(i.e. a complex manifold endowed with a holomorphic action of G ).(1) Y is a G -Oka manifold if for any reductive closed subgroup H of G the fixed-point manifold Y H is Oka (cf. [8, § Y satisfies Condition G - Ell if for any G -equivariant holomorphic map f : X → Y from a Stein G -manifold there exists a G -equivariant dominatingglobal spray s : E → Y over f where E → X is a holomorphic G -vectorbundle. Theorem A.3.
For any reductive complex Lie group G , a G -manifold Y is G -Okaif it satisfies Condition G - Ell .Proof. Let H be a reductive closed subgroup of G , X be a Stein manifold and f : X → Y H be a holomorphic map. Consider the holomorphic action of G on theStein manifold ( G/H ) × X defined by g · ([ g ′ ] , x ) = ([ gg ′ ] , x ) and the G -equivariantholomorphic map ˜ f : ( G/H ) × X → Y , ([ g ] , x ) g · f ( x ). By G -Ell of Y , thereexists a G -equivariant dominating global spray s : E → Y over ˜ f . Consider theholomorphic H -vector bundle E = E | { [1] }× X over X ∼ = { [1] } × X . Then the fixed-point manifold E H is a holomorphic vector bundle over X (cf. the proof of [8,Proposition 3.2]) and the restriction E H → Y H of s is a dominating global sprayover f : X → Y H . Therefore Y H satisfies Ell and thus it is Oka. (cid:3) In the case when G is finite, Kutzschebauch, L´arusson and Schwarz [8, Corol-lary 4.2] proved that the G -Oka property is equivalent to G -BOPAJI which easily A subset of Y is open with respect to the analytic (resp. algebraic) Zariski topology if itscomplement is a closed complex subvariety (resp. a closed algebraic subvariety). N IMPLICIT FUNCTION THEOREM FOR SPRAYS AND APPLICATIONS 8 implies Condition G -Ell (see also the proof of [8, Corollary 4.3]). Thus we obtainthe following characterization of G -Oka manifolds. Corollary A.4.
For any finite group G , a G -manifold is G -Oka if and only if itsatisfies Condition G - Ell . The following is the equivariant version of the localization principle.
Theorem A.5.
Let G be a reductive complex Lie group and Y be a G -manifold.Assume that each point of Y has a G -invariant open G -Oka neighborhood withrespect to the analytic Zariski topology. Then Y is G -Oka.Proof. Assume that Y is covered by G -invariant open G -Oka subsets U λ ( λ ∈ Λ)with respect to the analytic Zariski topology. Then for a reductive closed subgroup H of G the fixed-point manifold Y H is covered by Zariski open Oka subsets U Hλ ( λ ∈ Λ). Thus Y H is Oka by the usual localization principle (Theorem A.1). (cid:3) The equivariant localization principle gives the following example of an equiv-ariantly Oka manifold.
Example A.6.
Let Y be a smooth toric variety with the torus action ( C ∗ ) n y Y . Then Y is covered by ( C ∗ ) n -invariant Zariski open affine subsets of the form C k × ( C ∗ ) n − k . Their fixed-point manifolds with respect to the action of a reductiveclosed subgroup of ( C ∗ ) n are also of the form C j × ( C ∗ ) l , and hence they are Oka.Thus the smooth toric variety Y is ( C ∗ ) n -Oka. Acknowledgement
I wish to thank my supervisor Katsutoshi Yamanoi for helpful comments. I alsothank Franc Forstneriˇc and Finnur L´arusson for valuable discussions at Ljubljanain 2018. An elementary proof of that approximation implies interpolation wasasked by L´arusson at that time. This work was supported by JSPS KAKENHIGrant Number JP18J20418.
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Department of Mathematics, Graduate School of Science, Osaka University,Toyonaka, Osaka 560-0043, Japan
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