arXiv:1811.02150v2 [math.DG] 20 Jan 2020 nue yteicuini netv Lma31.O h te ad a he hand, isomorphism; other an is the (1.1) On that yields 3.1). [AG62] (Lemma Grauert injective and is Andreotti inclusion the by induced (1.1) morphism the if metric Hermitian flat ucetysal yuigteBt-hr ls,w ilso that show will we class, Bott-Chern the using By small. sufficiently M suoovxra hypersurface real pseudoconvex aiodi ti aiodo ieso tlattre e eak4.2. Remark see three; least U at dimension of manifold Stein a is manifold (Pro form 1.2. contact Corollary pseudo-Einstein a of existence the to condition U ope manifold complex tlattre oepeiey ewl rv h following complex the 1.1. of prove Theorem manifold will we complex precisely, fixed More sta a this three. in show least will hypersurface at we real paper, a is this as In form mations contact structure. CR pseudo-Einstein a a of invariant, of deformations an existence such the of variation whether the arises consider Gaus we renormalized When the of [Mar16]. term formula boundary the and [CY13,Hir14] ture sncsayfrdfiigsm lblC nains h total the invariants: CR global some defining for necessary is hr safltHriinmti ntecnnclbnl of bundle canonical the on metric Hermitian flat a is there of of eew iea uln fapofo hoe ..Tk uua neigh tubular a Take 1.1. Theorem of proof a of outline an give ambient we an Here when known already been have may stability this that Note an necessary a and theorem this from follows wiggles for stability The suoEnti otc om hc a rtitoue yLee by introduced first was which form, contact pseudo-Einstein A 2010 e od n phrases. and words Key TBLT FTEEITNEO PSEUDO-EINSTEIN A OF EXISTENCE THE OF STABILITY = M M ∂ Ω ahmtc ujc Classification. Subject Mathematics evdudrdfrain sara yesraei xdco fixed three. a least in at cont hypersurface dimension pseudo-Einstein real complex a a of of as deformations existence under m the served CR that pseudoconvex prove strictly we closed paper, of invariants global some Abstract. in in disaped-isencnatfr.Te hr xssan a exists there Then form. contact pseudo-Einstein a admits X X h xsec faped-isencnatfr on form contact pseudo-Einstein a of existence The . uhta t aoia udehsafltHriinmetric. Hermitian flat a has bundle canonical its that such Let Let X suoEnti otc ompasacuilrl ndefini in role crucial a plays form contact pseudo-Einstein A fcmlxdmnina es he.Asm htisboundar its that Assume three. least at dimension complex of Ω Ω , earltvl opc titypedcne oani a in domain pseudoconvex strictly compact relatively a be suoEnti otc om e ojcue Bott-Chern conjecture; Lee form; contact pseudo-Einstein X , H M 1 OTC FORM CONTACT ( and , M U, 1. UATAKEUCHI YUYA ′ O Introduction in ) U → U rmr 20,Scnay3E0 32V15. 32E10, Secondary 32V05, Primary ea nTerm11 hnaystrictly any Then 1.1. Theorem in as be H 1 disaped-isencnatform. contact pseudo-Einstein a admits 1 ( U ∩ Ω , O ) nfls nthis In anifolds. c omi pre- is form act pe manifold mplex U ∩ oiin2.1). position rsre under preserved s-Bonnet-Chern Q fw take we if Ω M iiyfrdefor- for bility piecurva- -prime K ew s the use we re h question the mle that implies eighborhood U sufficient d ng dimension disa admits class. borhood eutof result complex [Lee88], U y 2 YUYA TAKEUCHI assumption that the complex dimension of X is at least three. A proof of this fact will be given in Section 4. Before the end of the introduction, we remark a relation between our result and the Lee conjecture. Lee [Lee88, Proposition D] has proved that the first Chern 1,0 1,0 2 class c1(T M) of T M is equal to zero in H (M, R) if M admits a pseudo- Einstein contact form, and conjectured that the converse also holds if M is closed; this is called the Lee conjecture. There are some affirmative results on this conjec- ture [Lee88, Dra94, CC07, CSW12, CCT14], but it is still open. (Remark that we need an extra assumption on the pseudo-Hermitian torsion in [CCT14, Theorem 1.1], which has been pointed out in the erratum [CCT16].) The stability of the existence of a pseudo-Einstein contact form follows from the Lee conjecture since the first Chern class of a CR structure is invariant under deformations of a CR structure. In other words, Corollary 1.2 can be considered as one of affirmative results on the Lee conjecture.
Acknowledgements This paper is a part of the author’s Ph.D. thesis. He is grateful to his supervisor Kengo Hirachi for various valuable suggestions. He also would like to thank Taiji Marugame for helpful comments on Remark 4.2. This work was partially supported by JSPS Research Fellowship for Young Scientists, JSPS KAKENHI Grant Num- bers JP16J04653 and JP19J00063, and the Program for Leading Graduate Schools, MEXT, Japan.
2. Preliminaries We first recall some facts on strictly pseudoconvex domains; see [GPR94, Chapter V] and references therein for details. In what follows, the word “domain” means a relatively compact connected open set. Let Ω be a domain with smooth boundary M in an (n + 1)-dimensional complex manifold X. A defining function of Ω is 1 1 a smooth function on X such that Ω = ρ− (( , 0)), M = ρ− (0), and dρ = 0 −∞ 6 on M. A domain Ω is said to be strictly pseudoconvex if we can take a defining function of Ω that is strictly plurisubharmonic near M. It is known that any strictly pseudoconvex domain Ω is holomorphically convex, and consequently, there exist a Stein space Z and a proper surjective holomorphic map ϕ: Ω Z having some → good properties, called the Remmert reduction of Ω. In our setting, ϕ is described as follows. A compact analytic subset E of positive dimension at every point in Ω is called a maximal compact analytic subset of Ω if it is maximal among such subsets with respect to inclusion relations; this E is determined uniquely by Ω. The map ϕ contracts each connected component of E to a point, and induces a biholomorphism Ω E Z ϕ(E). In particular, Z has at most finite normal isolated singularities. \ → \ We next give a brief introduction to CR manifolds. Let M be a (2n + 1)- dimensional manifold without boundary. A CR structure is an n-dimensional com- plex subbundle T 1,0M of the complexified tangent bundle TM C satisfying the ⊗ following conditions:
T 1,0M T 0,1M =0, [Γ(T 1,0M), Γ(T 1,0M)] Γ(T 1,0M), ∩ ⊂ where T 0,1M is the complex conjugate of T 1,0M in TM C. A typical example ⊗ of a CR manifold is a real hypersurface M in a complex manifold X; it has the STABILITY OF THE EXISTENCE OF A PSEUDO-EINSTEIN CONTACT FORM 3 natural CR structure 1 0 1 0 T , M = T , X (TM C). ∩ ⊗ A CR structure T 1,0M is said to be strictly pseudoconvex if there exists a nowhere- vanishing real one-form θ on M such that θ annihilates T 1,0M and √ 1dθ(Z, Z) > 0, 0 = Z T 1,0M; − − 6 ∈ we call such a one-form a contact form. Note that the boundary of a strictly pseudoconvex domain is a strictly pseudoconvex CR manifold with respect to its natural CR structure. It is known that there is a canonical one-to-one correspondence between contact forms on M and Hermitian metrics on the canonical bundle of M. A contact form is said to be pseudo-Einstein if the corresponding Hermitian metric is flat; see [HMM17, Section 2.3] for details. Remark that this definition coincides with that given by Lee [Lee88] if n 2. In this paper, however, we do not use this definition ≥ but the following necessary and sufficient condition to the existence of a pseudo- Einstein contact form in terms of a Hermitian metric on the canonical bundle of the ambient complex manifold.
Proposition 2.1 ([HMM17, Proposition 2.6]). Let M be a strictly pseudoconvex real hypersurface in a complex manifold X. Then M admits a pseudo-Einstein contact form if and only if the canonical bundle KX of X has a Hermitian metric that is flat on the pseudoconvex side near M.
This proposition implies that any strictly pseudoconvex real hypersurface in a complex manifold X admits a pseudo-Einstein contact form if KX has a flat Hermitian metric. Thus we can derive Corollary 1.2 from Theorem 1.1. In the remainder of this paper, we will prove Theorem 1.1.
3. Bott-Chern class and the existence of a flat Hermitian metric 1,1 R Let X be a complex manifold. The real Bott-Chern cohomology HBC (X, ) of bi-degree (1, 1) is defined by
1,1 H (X, R)= d-closed real (1, 1)-forms on X / √ 1∂∂ψ ψ C∞(X, R) . BC { } − | ∈ Let f : Y X be a holomorphic map between complex manifolds. Then it defines → 1,1 1,1 the natural morphism f ∗ : H (X, R) H (Y, R) induced by the pullback of BC → BC (1, 1)-forms. BC For a holomorphic line bundle L over X, the first Bott-Chern class c1 (L) 1,1 R ∈ HBC (X, ) is defined as follows. Take a Hermitian metric h of L. Then the curvature (√ 1/2π)Θh = (√ 1/2π)∂∂ log h is a d-closed real (1, 1)-form on X, − −1,1 − R and defines an element of HBC (X, ). This cohomology class is independent of the BC BC choice of h, denoted by c1 (L). From the definition, c1 (L) = 0 if and only if BC BC L admits a flat Hermitian metric. Note that c1 is natural; that is, f ∗c1 (L) = BC c1 (f ∗L) for any holomorphic map f : Y X. 1 1 → The cohomology H , (X, R) has also a sheaf-theoretic interpretation. Let p,q BC A be the sheaf of smooth (p, q)-forms and be that of pluriharmonic functions. Then P there exists the following exact sequence of sheaves [Big69, Teorema (2.1)]:
0,0 √ 1∂∂ 1,1 d 2,1 1,2 0 R − R ( )R. →P→A −−−−−→A −→ A ⊕ A 4 YUYA TAKEUCHI
Here the subscript R means the subsheaf consisting of real forms. This exact 1 1,1 R sequence implies that H (X, ) is isomorphic to HBC (X, ). Note that a holo- P 1 1 morphic map f : Y X induces a natural morphism f ∗ : H (X, ) H (Y, ), → 1,1 1,1 P → P which is compatible with f ∗ : H (X, R) H (Y, R) defined above. BC → BC This formulation gives a sufficient condition to the existence of a flat Hermitian metric. Lemma 3.1. Let X and Y be complex manifolds and f : Y X be a holomor- → phic map. Assume that f induces injective morphisms H1(X, ) ֒ H1(Y, ) and O → O ,H2(X, R) ֒ H2(Y, R), and a surjective morphism H1(X, R) ։ H1(Y, R). Then → for any holomorphic line bundle L over X, it admits a flat Hermitian metric if so does f ∗L.
Proof. Assume that f ∗L has a flat Hermitian metric. As we noted above, this is BC BC equivalent to f ∗c1 (L)= c1 (f ∗L) = 0. Hence it is enough to prove the injectivity 1 1 of f ∗ : H (X, ) H (Y, ). Consider the following exact sequence of sheaves: P → P √ 1 Re 0 R − 0. → −−−→O −−→P → This induces the following commutative diagram with exact rows: H1(X, R) H1(X, ) H1(X, ) H2(X, R) O P
H1(Y, R) H1(Y, ) H1(Y, ) H2(Y, R). O P The injectivity of H1(X, ) H1(Y, ) follows from an easy diagram chasing. P → P 4. Proof of Theorem 1.1 Let X, Ω, and M be as in Theorem 1.1. We first reduce the problem on X to that on a Stein space. Take a defining function ρ of Ω that is strictly plurisubharmonic near the boundary. Without loss of generality, we may assume that ρ: X R is → proper. Then, for sufficiently small δ > 0, there exists a diffeomorphism 1 χ: ( 2δ, 2δ) M ρ− (( 2δ, 2δ)) − × → − such that χ(0,p)= p and ρ(χ(t,p)) = t. Replacing δ to a smaller one if necessary, 1 we may assume that ρ is strictly plurisubharmonic on ρ− (( 2δ, 2δ)). In partic- 1 − ular, Ω′ = ρ− (( ,δ)) is a strictly pseudoconvex domain in X containing Ω. −∞ Consider the Remmert reduction ϕ: Ω′ Z. From the strict plurisubharmonicity → of ρ, it follows that the maximal compact analytic subset of Ω′ cannot intersect 1 1 with ρ− (( δ, δ)); in particular, ϕ is a biholomorphism on ρ− (( δ, δ)). Without − − loss of generality, we may assume that ρ descends to a smooth function Z R; → use the same letter ρ for abbreviation. It is sufficient to show the existence of a 1 1 neighborhood U ρ− (( δ, δ)) of M = ρ− (0) such that KU has a flat Hermitian ⊂ − metric. To this end, we need to construct a “good” exhaustion function on Z. Lemma 4.1. Fix 0 <α<δ. There exists a smooth non-negative strictly plurisub- harmonic exhaustion function φ on Z satisfying the following conditions: 1 φ− (0) coincides with the singular set A of Z; • φ is of the form • ρ(p) φ(p)= + K δ(δ ρ(p)) − STABILITY OF THE EXISTENCE OF A PSEUDO-EINSTEIN CONTACT FORM 5
1 on ρ− (( α, δ)) for a constant K > 0 ; − 1 φ
H1(Ω( K, 0), ) H1(Ω( ǫ, 0), ). − O − O From [AG62, Th´eor`eme 15], it follows that each row is an isomorphism; here we use the assumption that the complex dimension of X is at least three. Hence it is sufficient to show the left column is an isomorphism. Since Ω( ,ǫ) and −∞ Ω( , 0) are Stein spaces, we obtain the following commutative diagram whose −∞ rows are isomorphisms: 1 2 H (Ω( K,ǫ), ) ≃ H (Ω( ,ǫ), ) − O A −∞ O
1 2 H (Ω( K, 0), ) ≃ H (Ω( , 0), ). − O A −∞ O On the other hand, the right column of the above diagram is also an isomorphism by the excision property of the local cohomology. This completes the proof. What is left is to show Lemma 4.1, the existence of a “good” exhaustion function φ on Z. Proof of Lemma 4.1. As noted in Section 2, the singular set A of Z is finite, given by A = p1,...,pk Z. We first construct a smooth non-negative strictly { } ⊂ 1 plurisubharmonic exhaustion function ψ on Z with ψ− (0) = A. There exists a proper holomorphic regular embedding f : Z CN for sufficiently large N [Nar60, → Theorem 6]; in what follows, we identify Z with the image of f. Then ψ0 = 2 k 2 N z + log z pj is a strictly plurisubharmonic exhaustion function on C | | j=1 | − | P 6 YUYA TAKEUCHI
1 2 k 2 with ψ0− ( )= A. Hence ψ = exp ψ0 = exp z j=1 z pj is a smooth non- −∞ | | | −N | 1 negative strictly plurisubharmonic exhaustion function Q on C with ψ− (0) = A. Choose β R with α<β<δ, and take a smooth function λ: R [0, 1] on R ∈ → such that λ 1 on ( , β) and λ 0 on ( α, ). Then the function ≡ −∞ − ≡ − ∞ φ1(p)= λ(ρ(p))ψ(p)
1 1 is strictly plurisubharmonic on ρ− (( , β)) and identically zero on ρ− (( α, δ)). −∞ − − Next, take a non-negative smooth function g1 on R with
1 1 supp g1 ((2δ)− , (β + δ)− ), g1(t)dt =1, ⊂ ZR and set t s g2(t)= g1(r)drds. Z0 Z0
This g2 is a non-negative and non-decreasing convex smooth function on R, vanishes 1 identically on ( , (2δ)− ], and −∞ 1 1 g2(t)= t δ− + g2 δ− > 0 − 1 on a neighborhood of [(β + δ)− , ). The function ∞ 1 φ2(p)= g2 δ ρ(p) − 1 1 vanishes identically on ρ− (( , δ]), is plurisubharmonic on ρ− (( δ, δ)), and −∞ − − ρ(p) 1 φ2(p)= + g2 δ− > 0. δ(δ ρ(p)) − 1 on a neighborhood of ρ− ([ β,δ)). Hence, for any ǫ> 0, the sum φ = ǫφ1 + φ2 is a − non-negative smooth exhaustion function on Z such that it is strictly plurisubhar- 1 1 monic on ρ− (( , β) ( α, δ)), and satisfies φ− (0) = A. Since φ2 is strictly −∞ − ∪ − 1 plurisubharmonic on the compact set ρ− ([ β, α]), the function φ is also strictly − − plurisubharmonic there for sufficiently small ǫ. Replacing ǫ by a smaller one, we 1 1 also have φ
References
[AG62] A. Andreotti and H. Grauert, Th´eor`eme de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193–259. [Big69] B. Bigolin, Gruppi di Aeppli, Ann. Scuola Norm. Sup. Pisa (3) 23 (1969), 259–287. [CC07] J. Cao and S.-C. Chang, Pseudo-Einstein and Q-flat metrics with eigenvalue estimates on CR-hypersurfaces, Indiana Univ. Math. J. 56 (2007), no. 6, 2839–2857. [CCT14] D.-C. Chang, S.-C. Chang, and J. Tie, Calabi-Yau theorem and Hodge-Laplacian heat equation in a closed strictly pseudoconvex CR manifold, J. Differential Geom. 97 (2014), no. 3, 395–425. [CCT16] D.-C. Chang, S.-C. Chang, and J. Tie, Erratum to “Calabi-Yau theorem and Hodge- Laplacian heat equation in a closed strictly pseudoconvex CR (2n + 1)-manifold”, J. Differential Geom. 102 (2016), no. 2, 351. [CSW12] J.-T. Chen, T. Saotome, and C.-T. Wu, The CR almost Schur lemma and Lee conjec- ture, Kyoto J. Math. 52 (2012), no. 1, 89–98. [CY13] J. S. Case and P. Yang, A Paneitz-type operator for CR pluriharmonic functions, Bull. Inst. Math. Acad. Sin. (N.S.) 8 (2013), no. 3, 285–322. [Dra94] S. Dragomir, On a conjecture of J. M. Lee, Hokkaido Math. J. 23 (1994), no. 1, 35–49. [GPR94] H. Grauert, Th. Peternell, and R. Remmert (eds.), Several complex variables. VII, Encyclopaedia of Mathematical Sciences, vol. 74, Springer-Verlag, Berlin, 1994. Sheaf- theoretical methods in complex analysis, A reprint of ıt Current problems in mathemat- ics. Fundamental directions. Vol. 74 (Russian), Vseross. Inst. Nauchn. i Tekhn. Inform. (VINITI), Moscow. [Hir14] K. Hirachi, Q-prime curvature on CR manifolds, Differential Geom. Appl. 33 (2014), no. suppl., 213–245. [HMM17] K. Hirachi, T. Marugame, and Y. Matsumoto, Variation of total Q-prime curvature on CR manifolds, Adv. Math. 306 (2017), 1333–1376. [Lee88] J. M. Lee, Pseudo-Einstein structures on CR manifolds, Amer. J. Math. 110 (1988), no. 1, 157–178. [Mar16] T. Marugame, Renormalized Chern-Gauss-Bonnet formula for complete K¨ahler- Einstein metrics, Amer. J. Math. 138 (2016), no. 4, 1067–1094. [Nar60] R. Narasimhan, Imbedding of holomorphically complete complex spaces, Amer. J. Math. 82 (1960), 917–934. [Tak18] Y. Takeuchi, A constraint on Chern classes of strictly pseudoconvex CR manifolds, 2018. arXiv:1808.02209. [Yau81] S. S. T. Yau, Kohn-Rossi cohomology and its application to the complex Plateau prob- lem. I, Ann. of Math. (2) 113 (1981), no. 1, 67–110.
Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan E-mail address: [email protected]