THE INDEX THEOREM FOR QUASI-TORI

THE INDEX THEOREM FOR QUASI-TORI DISSERTATION zur Erlangung des DOKTORGRADES (DR. RER. NAT.) der FAKULT¨ AT F¨ UR MATHEMATIK, PHYSIK UND INFORMATIK der UNIVERSIT¨ AT BAYREUTH vorgelegt von TSZ ON MARIO CHAN aus Hong Kong 1. Gutachter: Prof. Dr. Fabrizio Catanese 2. Gutachter: Prof. Dr. Philippe Eyssidieux 3. Gutachter: Prof. Dr. Ngaiming Mok BAYREUTH Tag der Einreichung: 27. November, 2012 Tag der Kolloquiums: 15. Februar, 2013 Erkl¨ arung Ich versichere eidesstattlich, dass ich diese Arbeit selbst¨ andig verfasst habe, und ich keine anderen als die von mir angegebenen Quellen und Hilfsmittel benutzt habe. Ich best¨ atige, dass Hilfe von gewerblichen Promotionsberatern bzw. -vermittlern oder ¨ ahnlichen Dienstleistern weder bisher in Anspruch genommen wurde noch k¨ unftig in Anspruch genommen wird. Ich best¨ atige, dass ich keine fr¨ uhere Promotionsversuche gemacht habe. Unterschrift des Autors i Acknowledgements It is my pleasure to express here my gratitude to my supervisor Prof. Fabrizio Catanese for suggesting me this research problem and for his continual guidance, as well as sharing his point of view about Mathematics and a lot of his personal experience in life. My gratitude also goes to Prof. Ingrid Bauer for encouraging me to explore different fields of Mathematics. Moreover, her care to me during my sickness made me feel like home while I was staying in a country distant from mine. Many thanks to all current and former colleagues in the Lehrstuhl Mathematik VIII of Universit¨at Bayreuth, in particular to Michael L¨onne, Fabio Perroni, Masaaki Murakami, Stephen Coughlan, Matteo Penegini, Wenfei Liu and Yifan Chen, for their help on my thesis, inspiring discussions on Mathematical ideas, sharing about the cultures and lifestyles of their own countries, and, most importantly, their encouragements which helped me to get through the most depressing period of my Ph.D. study. Thanks also to our secretary Leni Rostock who helped to sort out all the troubles during my stay in Bayreuth, from getting the residence permit to finding a medical doctor. Thanks to her, we have never missed the birthday of anybody in Lehrstuhl VIII. Wish that she would enjoy her life after retirement. Special thanks to my M.Phil. supervisor Prof. Ngaiming Mok, who taught me the basics about the Bochner–Kodaira formulas; and to Michael L¨onne, Florian Schrack, Sascha Weigl and Christian Gleißner who helped me to translate the abstract and summary into German. I would also like to thank DAAD for their support under the Forschungsstipendien f¨ ur Doktoranden. Lastly, I would like to declare that I owe my friends outside the Mathematics community in both Hong Kong and Germany a lot. Without their comforts and encouragements, this thesis could never be finished. My debts to them can never be fully redeemed. I am also badly indebted to my parents, who have given me freedom to do whatever I wish. ii Abstract The Index theorem for holomorphic line bundles on complex tori asserts that some cohomology groups of a line bundle vanish according to the signature of the associated hermitian form. In this article, this theorem is generalized to quasi-tori, i.e. connected complex abelian Lie groups which are not necessarily compact. In view of the Remmert–Morimoto decomposition of quasi-tori as well as the K¨unneth formula, it suffices to consider only Cousin-quasi-tori, i.e. quasi-tori which have no non-constant holomorphic functions. The Index theorem is generalized to holomorphic line bundles, both linearizable and non-linearizable, on Cousin-quasi-tori using L2-methods coupled with the Kazama–Dolbeault isomorphism and Bochner– Kodaira formulas. iii Zusammenfassung Ein Quasi-Torus ist eine zusammenh¨ angende komplexe abelsche Lie-Gruppe X=Cn/Γ, wobei Γ eine diskrete Untergruppe von Cnist. Xheißt Cousin-Quasi-Torus, wenn alle holomorphen Funktionen auf Xkonstant sind. Ist Xkompakt, so ist Xein komplexer Torus. Nach einem Satz von Remmert und Morimoto (vgl. [Mo2] oder [CC1, Prop. 1.1]) gibt es f¨ ur jeden Quasi-Torus Xeine Zerlegung X∼ =Ca×(C∗)b×X′, wobei X′ein Cousin-Quasi-Torus ist. Das Ziel des vorliegenden Artikels ist, das Verschwinden von Kohomologiegruppen von Geradenb¨ undeln auf Xzu untersuchen. Die K¨ unnethformel (vgl. [Kau]) besagt, dass sich die Kohomologiegruppen von X in direkte Summen von topologischen Tensorprodukten von Kohomologiegruppen von Ca×(C∗)bund des Cousin-Quasi-Torus X′zerlegen lassen. Man wird dadurch auf den Fall gef¨ uhrt, dass Xein Cousin-Quasi-Torus ist, da Ca×(C∗)bSteinsch ist und somit alle h¨ oheren Kohomologiegruppen (mit Grad ≥1) von koh¨ arenten Garben verschwinden. Es wird also im vorliegenden Artikel angenommen, dass X ein Cousin-Quasi-Torus ist. Sei Fder maximale komplexe Unterraum von RΓ und m:= dimCF. Wie im kompakten Fall kann jedem holomorphen Geradenb¨ undel Leine hermitesche Form Hauf Cnzugeordnet werden, deren Imagin¨ arteil Im Hmit der ersten Chernklasse c1(L) von Lassoziiert ist und ganzzahlige Werte in Γ ×Γ annimmt. Im Unterschied zum kompakten Fall ist Hnicht eindeutig. Lediglich die Einschr¨ ankung von Im Hauf RΓ×RΓ, und somit H|F×F, ist eindeutig bestimmt. Dies macht zumindest plausibel, dass nur H|F×Fanstelle von Hf¨ ur die Eigenschaften von Lverantwortlich ist. Die vorliegende Dissertation widmet sich dem Beweis des folgenden Satzes: Index-Satz f¨ ur Cousin-Quasi-Tori. Sei X=Cn/Γein Cousin-QuasiTorus, Fder maximale komplexe Unterraum von RΓ,Lein holomorphes Geradenb¨ undel auf Xund Heine mit Lassoziierte hermitesche Form auf Cn×Cn. Sei m:= dimCF. Die Einschr¨ ankung H|F×Fhabe s− Fnegative und s+ Fpositive Eigenwerte. Dann gilt Hq(X, L) = 0 f¨ ur q < s− Foder q > m −s+ F. Dieser Satz wird zur¨ uckgef¨ uhrt auf den Index-Satz f¨ ur komplexe Tori, wie er von Mumford [Mum], Kempf [Kem], Umemura [U], Matsushima [Ma] und Murakami [Mur] f¨ ur kompakte Xbewiesen wurde. Da Xstark (m+ 1)-vollst¨ andig ist (vgl. [Kaz1]; siehe auch §2.2), enth¨ alt der Satz auch einen Spezialfall des Resultats von Andreotti und Grauert, das besagt, dass Hq(X, F) = 0 ist f¨ ur alle q≥m+ 1 und f¨ ur jede koh¨ arente analytische Garbe Fauf X(vgl. [AGr]). Das Verschwinden von Hq(X, L) kann unter Verwendung der Dolbeault-Isomorphismen auf gewisse ∂-Gleichungen f¨ ur L-wertige (0, q)-Formen zurckgef¨ uhrt werden. Diese k¨ onnen mit L2-Methoden gel¨ ost werden. Man zeigt zun¨ achst die Existenz einer formalen L¨ osung einer ∂-Gleichung in einem Hilbertraum, indem man die ben¨ otigte L2-Absch¨ atzung nachweist, und beweist dann die Glattheit der L¨ osung. Letzteres iv ZUSAMMENFASSUNG v kann mit Hilfe der Regularit¨ atstheorie von ∂-Operatoren erledigt werden, also ist der entscheidende Schritt der Nachweis der ben¨ otigten L2-Absch¨ atzungen. Diese kann man durch Anwendung der Bochner–Kodaira-Ungleichungen bekommen. Jeder Cousin Quasi-Torus Xhat eine Faserb¨ undelstruktur ¨ uber einem komplexen Torus Tmit steinschen Fasern (siehe §2.1 und (eq 2.3)). Mit Hilfe der Lerayschen Spektralsequenz folgt Hq(X, L)∼ =Hq(T, p∗OX(L)) f¨ ur alle q≥0, wobei p:X→Tdie Projektion aus (eq 2.3) ist. Die Idee ist jetzt zu zeigen, dass der Dolbeault Komplex der Garben (A0,• T⊗OTp∗OX(L), ∂), eine azyklische Aufl¨ osung von p∗OX(L) auf Tist und das Verschwinden der Kohomologie durch L¨ osen der ∂-Gleichungen zu zeigen. Kazama [Kaz2] und Kazama–Umeno [KU2] geben eine leicht ver¨ anderte Formulierung, sie betrachten die Aufl¨ osung von OX(L) durch einen Unterkomplex (H0,•(L), ∂)von (A0,• X(L), ∂)(siehe §2.3 f¨ ur die Definition von H0,q(L)). Der Teilkomplex ist ebenfalls eine azyklische Aufl¨ osung von OX(L) auf Xund liefert damit den Kazama–Dolbeault Isomorphismus (vgl. [KU2], siehe auch Theorem 2.3.1). Letzterer Ansatz wird hier aufgegriffen. Das Ziel der Darstellung ist dann die L¨ osung der ∂-Gleichung ∂ξ =ψf¨ ur ein gegebenes ψ∈Γ(X, H0,q(L)) mit ∂ψ = 0. Jedes Geradenb¨ undel Lauf Xkann durch ein System von Automorphiefaktoren definiert werden, die in eine zur Appell–Humbert-Normalform analoge Normalform ¨ ubergef¨ uhrt werden k¨ onnen, die gegeben ist durch (vgl. [CC1,§2.2] und [V,§2]) ϱ(γ)eπH(z,γ)+ π 2H(γ,γ)+fγ(z)∀γ∈Γ, wobei ϱein Halbcharakter auf Γ und {fγ(z)}γ∈Γein additiver Kozykel ist (vgl. [CC1,§2.2] und [V,§2], siehe auch (eq 2.8)). Wenn {fγ(z)}γ∈Γein Korand ist, so wird Lals linearisierbar bezeichnet; andernfalls als nicht linearisierbar. Indem man den Trick verwendet, den Murakami in [Mur] f¨ ur den kompakten Fall benutzt hat (siehe §3.3), n¨ amlich die Metrik gso abzu¨ andern, dass der vom linearen Teil (dem zahmen Teil) von Lin den Basisrichtungen kommende Kr¨ ummungsterm von unten beschr¨ ankt ist, wenn qim gegebenen Bereich liegt, kann man die ben¨ otigten L2-Absch¨ atzungen erhalten, wenn Llinearisierbar ist (siehe §4). Dies beweist den Index-Satz f¨ ur linearisierbare L(siehe Theorem 4.1.1). Beim Nachweis der ben¨ otigten L2-Absch¨ atzungen f¨ ur nicht linearisierbare Lauf Xgibt eine zus¨ atzliche technische Schwierigkeit, die von dem vom nichtlinearen Teil (dem wilden Teil) von Lkommenden Kr¨ ummungsterm herr¨ uhrt. F¨ ur diesen wird Takayama’s schwaches ∂∂-Lemma ([Taka2, Lemma 3.14]; siehe auch §5.1) angewandt, um den Term auf relativ kompakten Teilmengen von Xzu beschr¨ anken. Dadurch erh¨ alt man die ben¨ otigten L2-Absch¨ atzungen nicht auf X, sondern lediglich auf der aussch¨ opfenden Familie {Kc}c∈R>0von pseudokonvexen relativ kompakten Teilmengen. Man erh¨ alt dann eine Folge {ξν}ν≥1von lokalen L¨ osungen, so dass ∂ξν=ψ|Kνist f¨ ur ein gegebenes ψ∈Γ(X, H0,q(L)) ∩ker ∂und f¨ ur alle ganzen Zahlen ν≥1. Indem man ein Argument im Beweis von Theorem B f¨ ur Steinsche R¨ aume in [GR, Ch. IV, §5] nachvollzieht, speziell indem man eine Approximation vom Runge-Typ verwendet, kann man die lokalen L¨ osungen ξνso korrigieren, dass sie auf jedem Kckonvergieren, was dann eine globale L¨ osung f¨ ur alle qim gegebenen Bereich liefert (siehe §5.4). Der Beweis des Index-Satzes ist damit vollst¨ andig. Contents Erkl¨ arung i Acknowledgements ii Abstract iii Zusammenfassung iv Chapter 1. Introduction and the main theorem 1 1.1. The main theorem 2 1.2. Methodology 3 Chapter 2. Preliminaries 4 2.1. A (C∗)n−m-principal bundle structure on X4 2.2. An exhaustive family of pseudoconvex subsets 5 2.3. Kazama sheaves and Kazama–Dolbeault isomorphism 5 2.4. Holomorphic line bundles on X6 2.5. A hermitian metric on L7 2.6. An L2-norm, the L2-spaces L2 0,(q′,q′′) c,χ and differential operators 9 Chapter 3. L2estimates 11 3.1. Existence of a solution of ∂ξ =ψ11 3.2. Bochner–Kodaira formulas 16 3.3. Murakami’s trick 20 Chapter 4. The linearizable case 27 4.1. Proof of Theorem 1.1.1 for linearizable L27 Chapter 5. The non-linearizable case 28 5.1. Bounds on the wild curvature terms 28 5.2. Existence of weak solutions on Kc29 5.3. A Runge-type approximation 30 5.4. Proof of Theorem 1.1.1 for general L32 List of Symbols 35 Bibliography 36 vi CHAPTER 1 Introduction and the main theorem Aquasi-torus is a complex abelian Lie group X=Cn/Γ, where Γ is a discrete subgroup of Cn.Xis said to be a Cousin-quasi-torus if all holomorphic functions on Xare constant functions.1Xis the familiar complex torus when it is compact, i.e. when rk Γ = 2n. The study of quasi-tori dates back to the early 20th century when Cousin studied the triply periodic functions of two complex variables ([Cou]). There he showed the existence of 2-dimensional quasi-tori without non-constant holomorphic functions. He also gave, among other things, a complete description of holomorphic line bundles on quasi-tori of dimension 2 and their sections using a method of asymptotic counting of zeros of the sections. In the 60’s, Kopfermann ([Kop]) studied systematically toroidal groups of arbitrary dimensions with a view to generalize the theory of abelian functions on complex tori. He also gave an example of a noncompact toroidal group with no non-constant meromorphic functions. Morimoto ([Mo1] and [Mo2]) studied Cousin-quasi-torus as the maximal toroidal subgroup of a complex (not necessarily abelian) Lie group, aiming to classify non-compact complex Lie groups. He classified all 3-dimensional abelian complex Lie groups. In the early 70’s, Andreotti and Gherardelli gave seminars on quasi-abelian varieties, i.e. Cousin-quasi-tori which possess structures of quasi-projective algebraic varieties ([AGh]). They showed that, among other things, a Cousin-quasi-torus is a quasiabelian variety if and only if the Generalized Riemann Relations are satisfied on it. Later on, among other contributors, Kazama ([Kaz1] and [Kaz2]), Pothering ([P]), Hefez ([Hef]), Vogt ([V]), Huckleberry and Margulis ([HM]), Abe ([Ab1] and [Ab2]), Capocasa and Catanese ([CC1] and [CC2]), and Takayama ([Taka2]) made some direct contributions to the theory of quasi-tori and Cousin-quasi-tori. A brief exposition of the historical development of the Generalized Riemann Relations can be found in [CC1, p. 29], and the Introduction of [AK] describes a brief chronology of the study of toroidal groups in general. The current research stems from the study of Capocasa and Catanese (ref. [CC1] and [CC2]). In [CC1], they gave an affirmative answer to a long standing problem of whether the existence of a non-degenerate meromorphic function on a quasi-torus is equivalent to the Generalized Riemann Relations. In [CC2], they moved on to prove the Lefschetz type theorems on quasi-tori in the best form, based on a statement of Abe with an erroneous proof in [Ab3, Thm. 6.4] (see [CC2, Corollary 1.2]).2Abe’s statement is then substituted by a result proven by Takayama ([Taka1, Thm. 1.3 and 1A Cousin-quasi-torus is also called a toroidal group or (H, C)-group in literature, where the latter means that all holomorphic functions are constant (ref. [AK, Def. 1.1.1]). 2Th´eor`eme 6.4 in [Ab3] asserts that, on a non-compact toroidal group X, there exists a constant c > 0 such that, for any holomorphic line bundle Lwith an associated hermitian form Hon Cnsuch that H|F×F> cIm(where Imis the m×m-identity matrix and Fis the maximal complex subspace of RΓ; see §2), H0(X, L) is non-trivial, and in fact infinite-dimensional. 1 2 1. INTRODUCTION AND THE MAIN THEOREM Thm. 6.1]).3These results clarify some basic properties of meromorphic functions and global sections of holomorphic line bundles on quasi-tori. This article goes a step further into the investigation of the higher cohomology groups of holomorphic line bundles on quasi-tori. The aim is to generalize the Index theorem on tori to quasi-tori. 1.1. The main theorem Denote the C-span and R-span of Γ by CΓ and RΓ respectively. Let π:Cn→X be the natural projection. Then K:= π(RΓ) = RΓ/Γ is the maximal compact subgroup of X, and F:= RΓ∩√−1RΓ is the maximal complex subspace in RΓ. By a theorem of Remmert and Morimoto (ref. [Mo2], see also [CC1, Prop. 1.1]), if Xis a quasi-torus, there is a decomposition X∼ =Ca×(C∗)b×X′, where X′is Cousin. The aim of this article is to investigate the vanishing of cohomology groups of holomorphic line bundles on X. The K¨unneth formula (ref. [Kau]) asserts that the cohomology groups on Xdecompose into direct sum of topological tensor products of cohomology groups on Ca×(C∗)band the Cousin-quasi-torus X′. In view of this, since Ca×(C∗)bis Stein and thus all higher cohomology groups (with degree ≥1) of coherent sheaves vanish, one is reduced to the case where Xis Cousin. In what follows, Xis assumed to be a non-compact Cousin-quasi-torus unless otherwise stated. In this case, CΓ = Cn, and rk Γ = dimRRΓ = n+mfor some integer m such that 0 < m < n. Note that mis the complex dimension of F. Given a holomorphic line bundle Lon X, it is analogous to the compact case that there is a hermitian form Hon Cn×Cnassociated to L, whose imaginary part Im Htakes integral values on Γ ×Γ and corresponds to the first Chern class c1(L) of L(ref. [CC1]). Im His uniquely determined only on RΓ×RΓ, so His uniquely determined only on F×F. The following theorem is a generalization of the Index theorem on complex tori (ref. [Mum, p. 150], [Mur] or [BL,§3.4])4to Cousin-quasi-tori, which is the main result of this article. Theorem 1.1.1.Let X=Cn/Γbe a Cousin-quasi-torus, Fthe maximal complex subspace of RΓ,La holomorphic line bundle on X, and Ha hermitian form on Cn×Cnassociated to L. Let m:= dimCF. Suppose H|F×Fhas respectively s− F negative and s+ Fpositive eigenvalues. Then one has Hq(X, L) = 0 for q < s− For q > m −s+ F. Let Ωp Xbe the sheaf of germs of holomorphic p-forms on X, and set Ωp X(L) := Ωp X⊗OXOX(L). Since the cotangent bundle of Xis trivial, one has Ωp X(L)∼ = ⊕(n p)OX(L), and thus Hq(X, Ωp X(L)) ∼ =⊕(n p)Hq(X, L). Therefore, one has the following 3Theorem 1.3 and 6.1 in [Taka1] together asserts that, for any positive line bundle Lon a noncompact toroidal group X, there exists an explicitly given integer µ0>0 such that H0(X, L⊗µ) is non-trivial for all µ≥µ0. Corollary 1.2 in [CC2] holds true by applying Takayama’s result and Proposition 1.1 in [CC2]. Takayama also gives a different proof of a weaker form of Lefschetz type theorems in [Taka2]. 4The Index theorem on complex tori was first proven by Mumford [Mum] and Kempf [Kem] in the algebraic case, and later by Umemura [U], Matsushima [Ma] and Murakami [Mur] in the analytic case. 2.6. AN L2-NORM, THE L2-SPACES L 2 0,(q′,q′′ ) c,χ AND DIFFERENTIAL OPERATORS 9 With the chosen ηtand ηw, a hermitian metric on Lis defined by (eq 2.10) η(z) := ηt(z)ηw(z) = e−πH(z,z)−2 Re ℏδ(z). The curvature form of Lwith respect to ηis then given by ΘT+ ΘW, which represents the class 2πc1(L) in H2(X, R) (while ΘTrepresents 2πc1(L) in 2πH2(X, Z)). 2.6. An L2-norm, the L2-spaces L2 0,(q′,q′′) c,χ and differential operators Let gbe a hermitian metric on X. Fix an apt coordinate system. For the purpose of this article, gis chosen to be a translational invariant metric such that the decomposition T1,0=T1,0 u⊕T1,0 vis orthogonal. Denote by ω:= −Im gthe associated (1,1)-form as usual. Fix any holomorphic line bundle L. Consider any 0 < c ≤ ∞ and 0 ≤q≤n. Denote the pointwise 2-norm on A0,q(Kc;L) induced from the hermitian metrics g and ηby |·|g,η. Let also eχ:R≥0→Rbe a smooth function and set χ:= eχ◦φ. For the purpose of this article, eχis always assumed to be a non-negative convex increasing function. In this case, χis plurisubharmonic. Set |ζ|2 g,η,χ := |ζ|2 g,η e−χ. Let µbe the measure induced from the volume form ω∧n n!. Define ∥ζ∥Kc,χ := √∫Kc|ζ|2 g,η,χ dµ for any ζ∈A0,q(Kc;L). Then ∥·∥Kc,χ defines an L2-norm with weight e−χ(or simply χ) on A0,q 0(Kc;L), the space of sections in A0,q(Kc;L) with compact support. To simplify notation, dµ in the integral is made implicit in what follows. The inner product corresponding to ∥·∥Kc,χ is denoted by ⟨·,·⟩Kc,χ. The norm is written as ∥·∥Kc,g,η,χ to emphasize its dependence on gand ηwhen necessary. Denote by L2 0,q c,χ := L2 0,q χ(Kc;L) the Hilbert space of (µ-)measurable L-valued (0, q)-forms ζon Kcsuch that ∥ζ∥Kc,χ <∞. It is well known that A0,q 0(Kc;L)⊂ L2 0,q c,χ is a dense subspace under the norm ∥·∥Kc,χ. For any 0 ≤p′, q′≤n−mand 0 ≤p′′, q′′ ≤m, define A(p′,p′′),(q′,q′′ ):= A(T∗p′,q′ u∧T∗p′′,q′′ v), i.e. a sheaf of germs of smooth sections of T∗p′,q′ u∧T∗p′′,q′′ v(defined in §2.1). For other values of p′, p′′, q′and q′′, set A(p′,p′′ ),(q′,q′′):= 0. Note that, for 0 ≤p, q ≤n, there is a decomposition (eq 2.11) Ap,q =⊕ p′+p′′=p q′+q′′=q A(p′,p′′),(q′,q′′ ). This decomposition depends on the choice of the decomposition (eq 2.4). Since the fibre and base directions are orthogonal to each other with respect to g, the decomposition is also orthogonal with respect to g. As only those sheaves with p′+p′′ = 0 are considered in what follows, set A0,(q′,q′′):= A(0,0),(q′,q′′) 10 2. PRELIMINARIES for notational convenience. Notice that H0,q′′ (L) is a subsheaf of A0,(0,q′′)(L) for 0≤q′′ ≤m. For any c > 0, denote also the space of sections in A0,(q′,q′′)(Kc;L) with compact support by A0,(q′,q′′) 0(Kc;L). Define L2 0,(q′,q′′) c,χ := L2 0,(q′,q′′) χ(Kc;L) := A0,(q′,q′′) 0(Kc;L), i.e. the closure of A0,(q′,q′′) 0(Kc;L) in (L2 0,q′+q′′ c,χ ,∥·∥Kc,χ). Note that the decomposition (eq 2.12) L2 0,q c,χ =⊕ q′+q′′=q L2 0,(q′,q′′) c,χ induced from (eq 2.11) is also an orthogonal decomposition. The operator ∂is decomposed into ∂[u]+∂[v]according to the decomposition (eq 2.4), where ∂[u]and ∂[v]are operators such that ∂[u]:A0,(q′,q′′)(Kc;L)→A0,(q′+1,q′′)(Kc;L) and ∂[v]:A0,(q′,q′′)(Kc;L)→A0,(q′,q′′+1)(Kc;L). Denote the formal adjoints of ∂[u]and ∂[v]above respectively by ϑ[u]:A0,(q′+1,q′′)(Kc;L)→A0,(q′,q′′)(Kc;L) and ϑ[v]:A0,(q′,q′′+1)(Kc;L)→A0,(q′,q′′)(Kc;L) (see, for example, [D1, Ch. VI, 1.5] for the definition). Some basic facts about differential operators on Hilbert spaces are recalled here. Extend the action of these operators to L2 0,(q′,q′′) c,χ in the sense of distributions (or currents). Then, they define closed (i.e. having closed graph) and densely defined linear operators on L2 0,(q′,q′′) c,χ (see, for example, [H¨or2, Ch. 1] and [D2, Prop. 4.9]) with domain given by (eq 2.13) Dom(q′,q′′) Kc,χ T(or Dom T) := {ζ∈L2 0,(q′,q′′) c,χ :∥Tζ∥Kc,χ <∞}, where Tdenotes any of the above operators. Note that Tis densely defined since A0,(q′,q′′) 0(Kc;L)⊂Dom(q′,q′′) Kc,χ T. An operator will be written as (T, Dom T) when the domain is emphasized. Given ∂[u]:L2 0,(q′,q′′) c,χ →L2 0,(q′+1,q′′) c,χ and ∂[v]:L2 0,(q′,q′′) c,χ →L2 0,(q′,q′′+1) c,χ with domains given as in (eq 2.13), their Hilbert space adjoints (also called Von Neumann’s adjoints, see for example [D1, Ch. VIII, §1] for a discussion on them) are denoted respectively by ∂∗ [u]:L2 0,(q′+1,q′′) c,χ →L2 0,(q′,q′′) c,χ and ∂∗ [v]:L2 0,(q′,q′′+1) c,χ →L2 0,(q′,q′′) c,χ , which are closed and densely defined operators on L2 0,(q′+1,q′′) c,χ and L2 0,(q′,q′′+1) c,χ respectively. Denote also their domains of definition respectively by Dom(q′+1,q′′) Kc,χ ∂∗ [u]and Dom(q′,q′′+1) Kc,χ ∂∗ [v]. In general, one has Dom(q′+1,q′′) Kc,χ ∂∗ [u]⊂Dom(q′+1,q′′) Kc,χ ϑ[u]and ∂∗ [u]ζ=ϑ[u]ζfor all ζ∈Dom(q′+1,q′′) Kc,χ ∂∗ [u](see, for example, [D1, Ch. VIII, §3]). The same holds true for ∂∗ [v]and ϑ[v]. CHAPTER 3 L2estimates 3.1. Existence of a solution of ∂ξ =ψ The aim of this section is to show that, for 0 ≤q≤m, given ψ∈H0,q(Kc;L)∩ L2 0,(0,q) c,χ such that ∂ψ = 0 on Kc, there exists a weak solution ξ∈L2 0,(0,q−1) c,χ of the ∂-equation ∂ξ =ψprovided that an L2estimate is satisfied. When c=∞, there exists a strong solution which lies in H0,q−1(X;L). First recall the following classical theorems for L2estimates (see, for example, [H¨or3, Lemmas 4.1.1 and 4.1.2] or [D1, Ch. VIII, Thm. 1.2]). Let (H1,⟨·,·⟩1), (H2,⟨·,·⟩2) and (H3,⟨·,·⟩3) be some Hilbert spaces, and let (S, Dom S) and (T, Dom T) be two closed (i.e. closed graph) and densely defined linear operators with domains Dom S⊂H2and Dom T⊂H1respectively such that H1 T //H2 S //H3 and S◦T= 0, i.e. T(Dom T)⊂ker S:= {ζ∈Dom S:Sζ = 0}. Let S∗and T∗denote the Hilbert space adjoints of Sand Trespectively, which are also closed, densely defined and satisfies T∗◦S∗= 0 (see, for example, [D1, Ch. VIII, Thm. 1.1]). Theorem 3.1.1 (see [H¨or3, Lemmas 4.1.1 and 4.1.2]).If there exists a constant C > 0such that (eq 3.1) ∥Sζ∥2 3+∥T∗ζ∥2 1≥C∥ζ∥2 2for all ζ∈Dom S∩Dom T∗, then (1) for every ψ∈ker S, there exists ξ∈im T∗∩Dom Tsuch that Tξ =ψand ∥ξ∥2 1≤1 C∥ψ∥2 2. In other words, ker S= im T(and thus im Tis closed as ker Sis so); (2) for every Ψ∈(ker T)⊥= im T∗, there exists Ξ∈im T∩Dom T∗such that T∗Ξ = Ψ and ∥Ξ∥2 2≤1 C∥Ψ∥2 1. In other words, im T∗= im T∗. Remark 3.1.2.By exchanging the roles of Sand T∗, one also gets ker T∗= im S∗ and im S= im Sif the L2estimate (eq 3.1) is satisfied. When Xis compact, consider the complex L2 0,q−1(X;L)∂ //L2 0,q(X;L)∂ //L2 0,q+1(X;L). Murakami [Mur] shows that the L2estimates (eq 3.1) hold for q < s−or q > n−s+ by choosing the hermitian metric gsuitably. The L2estimate on L2 0,q(X;L) implies that the harmonic L-valued (0, q)-forms must vanish. Elements in Hq(X, L) are represented by harmonic forms when Xis compact, so this proves the vanishing of Hq(X, L) in the compact case. In the current situation, although elements in Hq(X, L) are not represented by harmonic forms in general, the L2estimate (eq 3.1) is still useful in solving ∂- equations which leads to the vanishing of Hq(X, L) for suitable q’s according to Theorem 3.1.1 (1). 11 12 3. L2ESTIMATES Due to the existence of non-linearizable line bundles, it turns out it is necessary to solve ∂-equation on Kcfor any 0 < c < ∞(see §5.1). Therefore, the aim now is to solve the ∂-equation ∂ξ =ψ|Kcfor a given ψ∈H0,q(X;L) with ∂ψ = 0. In view of the fibre bundle structure (eq 2.3), instead of considering the complex L2 0,q−1 c,χ ∂ //L2 0,q c,χ ∂ //L2 0,q+1 c,χ , it is natural (see the discussion in §1.2) to consider the subcomplex (eq 3.2) L2 0,(0,q−1) c,χ Tq−1 //L2 0,q c,χ <2> T∗ q−1 oo Sq //L2 0,q+1 c,χ <3> S∗ q oo, where Tq−1and Sqact as ∂on L2 0,(0,q−1) c,χ and L2 0,q c,χ <2>respectively, and T∗ q−1and S∗ q are their Hilbert space adjoints.1The Hilbert spaces in the complex are defined as A0,q <2>(Kc;L) := A0,(1,q−1) ⊕A0,(0,q)(Kc;L), A0,q+1 <3>(Kc;L) := A0,(2,q−1) ⊕A0,(1,q)⊕A0,(0,q+1)(Kc;L) ; L2 0,q c,χ <2>:= A0,q 0<2>(Kc;L) = L2 0,(1,q−1) c,χ ⊕L2 0,(0,q) c,χ , L2 0,q+1 c,χ <3>:= A0,q+1 0<3>(Kc;L) = L2 0,(2,q−1) c,χ ⊕L2 0,(1,q) c,χ ⊕L2 0,(0,q+1) c,χ . Recall from (eq 2.11) and (eq 2.12) that all the direct sums on the right hand sides above are orthogonal decompositions. Denote the norms on L2 0,(0,q−1) c,χ ,L2 0,q c,χ <2>and L2 0,q+1 c,χ <3>respectively by ∥·∥1,∥·∥2and ∥·∥3, and their inner products by ⟨·,·⟩ with the corresponding subscripts. Write the Hilbert space adjoint of ∂:L2 0,q−1 c,χ →L2 0,q c,χ as ∂∗. Let pr: L2 0,q−1 c,χ → L2 0,(0,q−1) c,χ be the orthogonal projection. For later use, (T∗ q−1,Dom T∗ q−1) is described more explicitly. Proposition 3.1.3.With the notation described above, one has Dom T∗ q−1= DomKc,χ ∂∗∩L2 0,q c,χ <2> = Dom(1,q−1) Kc,χ ∂∗ [u]⊕Dom(0,q) Kc,χ ∂∗ [v]. Moreover, for any ζ=ζ′+ζ′′ ∈Dom T∗ q−1where ζ′∈Dom(1,q−1) Kc,χ ∂∗ [u]and ζ′′ ∈ Dom(0,q) Kc,χ ∂∗ [v], one has T∗ q−1ζ= pr ∂∗ζ=∂∗ [u]ζ′+∂∗ [v]ζ′′. Proof. Define operators (W1,Dom W1) and (W2,Dom W2) from L2 0,q c,χ <2>into L2 0,(0,q−1) c,χ such that Dom W1:= DomKc,χ ∂∗∩L2 0,q c,χ <2>, Dom W2:= Dom(1,q−1) Kc,χ ∂∗ [u]⊕Dom(0,q) Kc,χ ∂∗ [v], and W1ζ:= pr ∂∗ζfor ζ∈Dom W1, W2ζ:= ∂∗ [u]ζ′+∂∗ [v]ζ′′ for ζ=ζ′+ζ′′ ∈Dom W2. 1The symbol Tq−1(resp. Sq) is used instead of ∂so that the domains and codomains of the two operators can be distinguished. More precisely, if ι:L2 0,(0,q−1) c,χ ,→L2 0,q−1 c,χ and pr: L2 0,q c,χ →L2 0,q c,χ <2> are respectively the inclusion and projection, then Tq−1= pr ◦∂◦ι. Therefore, T∗ q−1and ∂∗are different operators. 3.1. EXISTENCE OF A SOLUTION OF ∂ξ =ψ13 These are closed and densely defined linear operators on L2 0,q c,χ <2>. Since ∥Tq−1ζ∥2 2= ∂ζ2 2=∂[u]ζ2 2+∂[v]ζ2 2for all ζ∈L2 0,(0,q−1) c,χ , it follows that Dom Tq−1= Dom ∂∩L2 0,(0,q−1) c,χ = Dom(0,q−1) Kc,χ ∂[u]∩Dom(0,q−1) Kc,χ ∂[v]. First is to show that (T∗ q−1,Dom T∗ q−1) = (W1,Dom W1). Note that, for any f∈L2 0,(0,q−1) c,χ and any ζ∈Dom W1, one has ⟨f, W1ζ⟩1=⟨f, pr ∂∗ζ⟩1=⟨f, ∂∗ζ⟩Kc,χ . For any ˜ ζ∈L2 0,q c,χ =L2 0,q c,χ <2>⊕(L2 0,q c,χ <2>)⊥, write ˜ ζ=ζ+ζ⊥where ζ∈L2 0,q c,χ <2> and ζ⊥∈(L2 0,q c,χ <2>)⊥=⊕q q′=2 L2 0,(q′,q−q′) c,χ . Note that ∂∗ζ⊥∈⊕q−1 q′=1 L2 0,(q′,q−1−q′) c,χ = (L2 0,(0,q−1) c,χ )⊥, thus ⟨f, ∂∗ζ⊥⟩Kc,χ = 0 for any f∈L2 0,(0,q−1) c,χ . Therefore, for any f∈L2 0,(0,q−1) c,χ , one has f∈Dom W∗ 1 :⇐⇒ ∃ C > 0: ∀ζ∈Dom W1,|⟨f, W1ζ⟩1|=⟨f, ∂∗ζ⟩Kc,χ≤C∥ζ∥2 ⇐⇒ ∃ C > 0: ∀˜ ζ∈DomKc,χ ∂∗, ⟨f, ∂∗˜ ζ⟩Kc,χ=⟨f, ∂∗ζ⟩Kc,χ≤C˜ ζKc,χ ⇐⇒ f∈Dom ∂∩L2 0,(0,q−1) c,χ = Dom Tq−1as (∂∗)∗=∂ (ref. [D1, Ch. VIII, §1] for the definition of the domain of Hilbert space adjoints), and thus Dom W∗ 1= Dom Tq−1. It follows that ⟨f, W1ζ⟩1=⟨f, ∂∗ζ⟩Kc,χ =⟨∂f, ζ⟩2= ⟨Tq−1f, ζ⟩2for any f∈Dom Tq−1and ζ∈Dom W1. As a result, (Tq−1,Dom Tq−1) = (W∗ 1,Dom W∗ 1), and hence (T∗ q−1,Dom T∗ q−1) = (W1,Dom W1) (ref. [D1, Ch. VIII, Thm. 1.1]). The proof of (T∗ q−1,Dom T∗ q−1) = (W2,Dom W2) is similar. Notice that ∥ζ∥2 2= ∥ζ′∥2 2+∥ζ′′∥2 2and thus ∥ζ′∥2+∥ζ′′∥2≤√2∥ζ∥2for all ζ=ζ′+ζ′′ ∈L2 0,q c,χ <2>. Then, for any f∈L2 0,(0,q−1) c,χ , one has f∈Dom W∗ 2 :⇐⇒ ∃ C > 0: ∀ζ=ζ′+ζ′′ ∈Dom W2, |⟨f, W2ζ⟩1|=⟨f, ∂∗ [u]ζ′+∂∗ [v]ζ′′⟩1≤C∥ζ∥2 ⇐⇒ ∃ C > 0: ∀ζ′∈Dom(1,q−1) Kc,χ ∂∗ [u]and ∀ζ′′ ∈Dom(0,q) Kc,χ ∂∗ [v], ⟨f, ∂∗ [u]ζ′⟩1≤C∥ζ′∥2and ⟨f, ∂∗ [v]ζ′′⟩1≤C∥ζ′′∥2 ⇐⇒ f∈Dom(0,q−1) Kc,χ ∂[u]∩Dom(0,q−1) Kc,χ ∂[v]= Dom Tq−1, and thus Dom W∗ 2= Dom Tq−1. Note that ⟨f, W2ζ⟩1=⟨∂[u]f, ζ′⟩2+⟨∂[v]f, ζ′′⟩2= ⟨∂[u]f+∂[v]f, ζ′+ζ′′⟩2=⟨Tq−1f, ζ⟩2for f∈Dom Tq−1and ζ∈Dom W2, since 14 3. L2ESTIMATES L2 0,(1,q−1) c,χ ⊥L2 0,(0,q) c,χ . Therefore, one has (Tq−1,Dom Tq−1) = (W∗ 2,Dom W∗ 2), and thus (T∗ q−1,Dom T∗ q−1) = (W2,Dom W2) (ref. [D1, Ch. VIII, Thm. 1.1]). □ Suppose now given 0 < c ≤ ∞ and ψ∈H0,q(Kc;L)∩L2 0,(0,q) c,χ ⊂L2 0,q c,χ <2>such that Sqψ=∂ψ = 0. Theorem 3.1.1 (1) asserts that, if the L2estimate (eq 3.1) is satisfied, then there exists ξ∈im T∗ q−1⊂L2 0,(0,q−1) c,χ such that (eq 3.3) Tq−1ξ=∂ξ =ψin L2 0,(0,q) c,χ . One can have a further reduction. When c=∞, since (X, g) is complete in the sense of Riemannian geometry, A0,q 0<2>(X;L) is dense in DomXT∗ q−1∩DomXSqunder the above graph norm (see, for example, [D1, Ch. VIII, Thm. 3.2]). Therefore, it suffices to establish the required L2estimates (eq 3.1) for ζ∈A0,q 0<2>(X;L). Suppose c < ∞. Note that A0,q <2>(Kc;L)⊂Dom Sq. Since ∂Kcis smooth and χis smooth on a neighborhood of Kc, using [H¨or1, Prop. 2.1.1] together with an argument of partition of unity, it yields the following Proposition 3.1.4.A0,q <2>(Kc;L)∩Dom T∗ q−1is dense in Dom T∗ q−1∩Dom Sq under the graph norm √T∗ q−1ζ2 1+∥Sqζ∥2 3+∥ζ∥2 2. Proof. Note that the statement follows from [H¨or1, Prop. 2.1.1] when T∗0,q X and Lare both trivial by using a partition of unity. The aim now is to handle the case when Lis non-trivial. Take a locally finite open cover {Uα}α∈Aof Xsuch that every Uαis a coordinate chart of Xand Lis trivialized on each Uαwith transition functions σαβ ∈O∗ X(Uα∩ Uβ) for all α, β ∈A. Then, for any ζ∈L2 0,q χ(X;L) with ζαrepresenting ζover Uα under the trivialization, one has ζα=σαβζβon Uα∩Uβ. Fix any ζ∈Dom T∗ q−1∩Dom Sq. It suffices to show that ζcan be approximated by a sequence {ζ(ν)}ν∈N⊂A0,q <2>(Kc;L)∩Dom T∗ q−1under the given graph norm. Extend ζby zero to a section on X. Using a partition of unity which decomposes ζinto a sum of finitely many compactly supported sections, one can assume that ζis compactly supported in a coordinate chart U:= U0∈ {Uα}α∈A. Then the hermitian metric ηon Lcan be viewed as a function eη:= η0on U=U0(under the given trivialization), and any L-valued form f∈L2 0,q g,η,χ(U;L) can be viewed as a OX-valued form e f:= f0∈L2 0,q g,eη,χ(U). Let W:= U∩Kc. Note that one has e fW,g,eη,χ =∥f∥W,g,η,χ,∂e fW,g,eη,χ =∂fW,g,η,χ and ∂∗e fW,g,eη,χ =∂∗fW,g,η,χ for all f∈L2 0,q g,η,χ(W;L). Then ζ∈Dom T∗ q−1∩Dom Sqimplies e ζ∈DomW,g,eη,χ ∂∗∩ DomW,g,eη,χ ∂∩L2 0,q g,eη,χ <2>(W). Since gand χare fixed in what follows, subscripts of them are omitted from the notations below. By [H¨or1, Prop. 2.1.1] (or applying [H¨or1, Prop. 1.2.4] directly), there exists a sequence {e ζ(ν)}ν∈N⊂A0,q(W)∩DomW,eη∂∗such that ∂∗(e ζ(ν)−e ζ) 2 W,e η+∂(e ζ(ν)−e ζ) 2 W,eη+e ζ(ν)−e ζ 2 W,eη→0 as ν→ ∞ and supp e ζ(ν)⋐Ufor all ν∈N. As e ζ(ν)’s are obtained from convolutions between smoothing kernels and e ζwhich do not change the type of forms, it follows that e ζ(ν)∈A0,q <2>(W). The sections ζ(ν)∈A0,q <2>(W;L) defined by ζ(ν) α:= 1 σ0αe ζ(ν) on Uα∩U=∅are compactly supported in U(hence ζ(ν)∈A0,q <2>(Kc;L)) and 3.1. EXISTENCE OF A SOLUTION OF ∂ξ =ψ15 satisfy g ζ(ν)=e ζ(ν). Therefore, one obtains a sequence {ζ(ν)}ν∈N⊂DomKc,η ∂∗∩ A0,q <2>(Kc;L) = Dom T∗ q−1∩A0,q <2>(Kc;L) (see Proposition 3.1.3) such that T∗ q−1(ζ(ν)−ζ)2 1+Sq(ζ(ν)−ζ)2 3+ζ(ν)−ζ2 2 ≤∂∗(ζ(ν)−ζ) 2 W,η +∂(ζ(ν)−ζ)2 W,η +ζ(ν)−ζ2 W,η as T∗ q−1= pr ∂∗ by Prop. 3.1.3 =∂∗(e ζ(ν)−e ζ) 2 W,eη+∂(e ζ(ν)−e ζ) 2 W,eη+e ζ(ν)−e ζ 2 W,eη →0 as ν→ ∞ as required. □ As a result, it suffices to establish the required L2estimates (eq 3.1) for ζ∈ A0,q <2>(Kc;L)∩Dom T∗ q−1. The above discussion is summarized in the following Proposition 3.1.5.Suppose 0< c ≤ ∞. If there exists a constant C > 0such that (eq 3.4) ∥Sqζ∥2 3+T∗ q−1ζ2 1≥C∥ζ∥2 2 for all ζ∈{A0,q <2>(Kc;L)∩Dom T∗ q−1when c < ∞, A0,q 0<2>(X;L)when c=∞, then, for every ψ∈H0,q(Kc;L)∩L2 0,(0,q) χ(Kc;L)such that ∂ψ = 0, there exists ξ∈L2 0,(0,q−1) χ(Kc;L)such that ∂ξ =ψin L2 0,(0,q) χ(Kc;L). Remark 3.1.6.Let L2 0,q−1(Kc;L; loc) denote the space of locally L2L-valued (0, q −1)-forms on Kc, which contains L2 0,(0,q−1) χ(Kc;L) as a subspace. It follows from the classical regularity theory for ∂-operator or elliptic operators (ref. [H¨or3, Thm. 4.2.5 and Cor. 4.2.6] or [H¨or2, Thm. 4.1.5 and Cor. 4.1.2]) that the existence of ξ∈L2 0,q−1(Kc;L; loc) satisfying the equation (eq 3.3) in L2 0,q(Kc;L; loc) implies that there exists ξ∈A0,q−1(Kc;L) (but not necessarily in A0,(0,q−1)(Kc;L)) satisfying the same equation in A0,q(Kc;L). In case c=∞, Theorem 2.3.1 implies that there even exists a solution ξ∈H0,q−1(X;L) such that ∂ξ =ψon X. Remark 3.1.7.Write H0,q L2(Kc;L) := H0,q(Kc;L)∩L2 0,(0,q) c,χ . Following the idea discussed in §1.2, it would be more natural to consider the L2estimate on H0,q c,χ := H0,q L2(Kc;L) rather than L2 0,q c,χ <2>, where the closure is taken in L2 0,(0,q) c,χ . However, the author faces the difficulty in obtaining the required estimate from the Bochner–Kodaira inequalities when H0,q c,χ instead of L2 0,q c,χ <2>is considered. Write ∂∗ Hcas the Hilbert space adjoint of ∂=∂[v]:H0,q c,χ →H0,q+1 c,χ . It can be shown that ∂∗ Hc= prc◦∂∗ [v]on Dom(0,q) Kc,χ ∂∗ Hc, where prc:L2 0,(0,q) c,χ →H0,q c,χ is the orthogonal projection. Set ð∗ ⊥c:= ∂∗ [v]−∂∗ Hc, then ∂∗ Hcζand ð∗ ⊥cζare orthogonal to each other for all ζ∈Dom(0,q) Kc,χ ∂∗ Hcand ∂∗ [v]ζ 2 Kc,χ =∂∗ Hcζ 2 Kc,χ +ð∗ ⊥cζ2 Kc,χ . 16 3. L2ESTIMATES From the Bochner–Kodaira inequalities, one obtains ∂ζ2 Kc,χ +∂∗ [v]ζ 2 Kc,χ ≥∫Kc Curv(ζ, ζ) for all ζ∈H0,q L2(Kc;L)∩Dom(0,q) Kc,χ ∂∩Dom(0,q) Kc,χ ∂∗ [v], where ∫KcCurv(ζ, ζ) is the curvature term arising from the curvature of L. By choosing suitably the metrics gand η, the curvature term can be bounded below by C∥ζ∥2 Kc,χ for some constant C > 0. Therefore, in order to obtain the desired estimate ∂ζ2 Kc,χ +∂∗ Hcζ 2 Kc,χ ≥ C′′ ∥ζ∥2 Kc,χ for some constant C′′ >0, one has to show that ∥ð∗ ⊥cζ∥2 Kc,χ ≤C′∥ζ∥2 Kc,χ for some constant C′>0 such that C > C′. However, the constant C′depends on g in general and one may not be able to make C′smaller than Cby altering g. That’s why the L2estimate on L2 0,q c,χ <2>instead of H0,q c,χ is considered in this article. 3.2. Bochner–Kodaira formulas Let ∇:A(T∗•,•⊗L)→A(T∗C⊗T∗•,•⊗L), where T∗C:= T∗1,0⊕T∗0,1, be the connection on T∗•,•⊗Linduced from the Chern connections on the holomorphic hermitian vector bundles (T1,0, g) and (L, ηe−χ). Therefore, ∇is compatible with the pointwise norm |·|g,η,χ. Under a chosen apt coordinate system, set ∂k:= ∂ ∂zkand ∂k:= ∂ ∂zkfor 1 ≤k≤n. These define global vector fields on X. Set ∇k:= ∇∂kand ∇k:= ∇∂kfor 1 ≤k≤n. Set also ∇vj:= ∇n−m+j=∇∂ ∂vjand ∇vj:= ∇n−m+j=∇∂ ∂vj(and define ∂vjand ∂vj similarly) for 1 ≤j≤mfor notational convenience. Since the hermitian metric gis translational invariant on X, the Christoffel symbols given from gvanish and thus one has locally (eq 3.5) ∇k=∂k+∂klog (ηe−χ), ∇k=∂k for 1 ≤k≤n. For later use, note that the commutator of ∇kand ∇ℓis given by Θkℓ := [∇k,∇ℓ] = −∂k∂ℓlog (ηe−χ), and the curvature form of Lendowed with the metric ηe−χis given by (eq 3.6) Θ := −√−1∂∂ log (ηe−χ)=√−1 n ∑ k,ℓ=1 Θkℓ dzk∧dzℓ. Write the curvature tensor associated to Θ as R:= n ∑ k,ℓ=1 Θkℓ dzk⊗dzℓ. Since the base and fibre directions are orthogonal to each other with respect to g, the identification between Ap,q and Ap,q =Aq,p := A(Tq,p) induced from g respects the decomposition (eq 2.4) (Ap,q here means the complex conjugate of Ap,q). For later use, set A(p′,p′′),(q′,q′′ ):= A(Tp′,q′ u∧Tp′′,q′′ v)and A(p′,p′′),0:= A(p′,p′′ ),(0,0) for 0≤p′, q′≤n−mand 0 ≤p′′, q′′ ≤m. For any ζ∈Ap,0⊗A0,q, let ζ∨denote the image of ζin A0,p ⊗Aq,0via the isomorphism induced from g. Then, for example, if ζ∈A0,(q′,q′′ ), one has ζ∨∈A(q′,q′′ ),0. 3.2. BOCHNER–KODAIRA FORMULAS 17 As a bilinear form on A1,0⊗A1,0,Rcan be decomposed according to the decomposition (eq 2.4) into the sum of Ruu := R|A(1,0),0⊗A(1,0),0,Ruv := R|A(1,0),0⊗A(0,1),0, Rvu := R|A(0,1),0⊗A(1,0),0,Rvv := R|A(0,1),0⊗A(0,1),0. Since Ris a hermitian form, it follows that Ruu =Ruu,Rvv =Rvv and Ruv =Rvu. Let Trg:A0,q ⊗Aq,0→A0,0be the trace operator which is defined in such a way that ζ⊗ξ7→ ξ∨⌟ζ, where ζ∈A0,q,ξ∈Aq,0and ξ∨⌟ζdenotes the complete contraction between ζand ξ∨. Denote by Trg,η the similar contraction for L-valued forms. Fix any 0 <c<∞. Denote the Hilbert space adjoint of ∂:L2 0,q−1 c,χ →L2 0,q c,χ by ∂∗:L2 0,q c,χ →L2 0,q−1 c,χ . Identify A1,1and A1,0⊗A0,1via the isomorphism dzk∧dzℓ7→ dzk⊗dzℓfor any 1 ≤k, ℓ ≤n. Let R∨(ζ⊗ζ) (resp. (∂∂φ)∨(ζ⊗ζ)) denotes the natural contraction between R∨(resp. (∂∂φ)∨) and ζ⊗ζ. Let ∇=∇(1,0) +∇(0,1) be the decomposition of ∇into (1,0)- and (0,1)-types. The ∇-Bochner–Kodaira formula (cf. [Siu, (2.1.4) and (1.3.3)]) is then given by (eq 3.7) ∂ζ2 Kc,χ +∂∗ζ 2 Kc,χ =∫∂Kc e−χ |dφ|gTrg,η (∂∂φ)∨(ζ⊗ζ) +∇(0,1)ζ2 Kc,χ +∫Kc e−χTrg,η R∨(ζ⊗ζ) for all ζ∈A0,q(Kc;L)∩DomKc,χ ∂∗. Remark 3.2.1.Note that the measure for the boundary integral is induced from ((dφ)∨ |dφ|g ⌟ω∧n n!)∂Kc . In order to compare notations in [Siu, (2.1.4)] and those in (eq 3.7), write [x]Siu to mean the symbol xused in [Siu]. Then [∇]Siu =∇(0,1) ,[∇]Siu =∇(1,0) ,[ρ]Siu =φ−c |dφ|g ,[Rijkl]Siu = 0 , and [−Ωαβst]Siu = components of R= Θkℓ . Note that [Rijkl]Siu = 0 as the Chern connection on (T1,0, g) is flat. Also be aware of the typos of the signs preceding the curvature integrals involving [Ωs αβ t]Siu and [Rs t]Siu in [Siu, (2.1.4)]. The correct signs can be found in [Siu, (1.3.3)]. To see that the boundary term in (eq 3.7) coincides with the one in [Siu, (2.1.4)], note that at every z∈∂Kc, ∂∂ (φ−c |dφ|g)(z) = ∂∂φ |dφ|g (z)−∂φ ∧∂|dφ|g |dφ|2 g (z)−∂|dφ|g∧∂φ |dφ|2 g (z). After taking ∨and contracting with ζ⊗ζwhere ζ∈A0,q(Kc;L)∩DomKc,χ ∂∗, the last two terms on the right hand side vanish because, for ζ∈A0,q(Kc;L), (∂φ)∨⌟ζ= 0 on ∂Kcif and only if ζ∈DomKc,χ ∂∗(ref. [H¨or1, pg. 101] or [Siu, (2.1.1)]). The boundary terms therefore coincides. When the subcomplex (eq 3.2) is considered, the ∇-Bochner–Kodaira formula (eq 3.7) is restricted to ζ∈A0,q <2>(Kc;L)∩DomKc,χ ∂∗=A0,q <2>(Kc;L)∩DomKc,χ T∗ q−1 (see Proposition 3.1.3). The (0,1)-connection splits into ∇(0,1) =∇(0,1) u+∇(0,1) v 18 3. L2ESTIMATES according to the decomposition (eq 2.4). Write ∇u:= ∇(0,1) uand ∇v:= ∇(0,1) vfor notational convenience. Let also prF:A0,q ⊗A0,s →A0,(0,q)⊗A0,(0,s)be the canonical projection (where A0,s (resp. A0,(0,s)) is the complex conjugate of A0,s (resp. A0,(0,s))). Set (eq 3.8) Bd(ζ, ζ) := ∫∂Kc e−χ |dφ|gTrg,η (∂∂φ)∨(ζ⊗ζ) for notational convenience. Then (eq 3.7) gives the following Lemma 3.2.2.For any ζ=ζ′+ζ′′ ∈A0,q <2>(Kc;L)∩Dom T∗ q−1, where ζ′∈ A0,(1,q−1)(Kc;L)∩Dom ∂∗ [u]and ζ′′ ∈A0,(0,q)(Kc;L), one has (eq 3.9) ∥Sqζ∥2 3+T∗ q−1ζ2 1= Bd(ζ, ζ) + ∂[u]ζ′′2 3+∂[v]ζ′2 3 +∥∇uζ′∥2 Kc,χ +∥∇vζ′′∥2 Kc,χ +∫Kc e−χTrg,η prF(R∨(ζ⊗ζ)). Proof. On DomKc,χ ∂∗, one has ∂∗=ϑ[u]+ϑ[v]. Then, for all ζ=ζ′+ζ′′ ∈ Dom T∗ q−1= DomKc,χ ∂∗∩L2 0,q c,χ <2>(see Proposition 3.1.3), one has ∂∗ζ=ϑ[u]ζ′+ϑ[u]ζ′′ +ϑ[v]ζ′+ϑ[v]ζ′′ =T∗ q−1ζ+ϑ[v]ζ′, as T∗ q−1ζ=∂∗ [u]ζ′+∂∗ [v]ζ′′ (see Proposition 3.1.3) and ϑ[u]ζ′′ = 0. Note also that ∇(0,1)ζ=∇uζ′+∇uζ′′ +∇vζ′+∇vζ′′, and ∂ζ =Sqζ. Since the decomposition (eq 2.4) is orthogonal with respect to g, it follows that ∂∗ζ 2 Kc,χ =T∗ q−1ζ2 1+∥ϑ[v]ζ′∥2 Kc,χ and ∇(0,1)ζ2 Kc,χ =∥∇uζ′∥2 Kc,χ +∥∇uζ′′∥2 Kc,χ +∥∇vζ′∥2 Kc,χ +∥∇vζ′′∥2 Kc,χ . Note that ∥∇uζ′′∥2 Kc,χ =∂[u]ζ′′2 3. Following the argument in [H¨or1, pg. 101] with ∂[v]in place of ∂, it follows that, for any ζ∈A0,(q′,q′′)(Kc;L), ζ∈Dom(q′,q′′ ) Kc,χ ∂∗ [v]if and only if (∂[v]φ)∨⌟ζ= 0 on ∂Kc. Since ∂[v]φ= 0, it follows that A0,(q′,q′′ )(Kc;L)⊂Dom(q′,q′′ ) Kc,χ ∂∗ [v]. In particular, ζ′∈ Dom(1,q−1) Kc,χ ∂∗ [v]for all ζ′∈A0,(1,q−1)(Kc;L). Then, since the decomposition (eq 2.4) is orthogonal with respect to g, by taking the analogy between the decompositions Ar=⊕p+q=rAp,q and Ap,q =⊕p=p′+p′′ q=q′+q′′ A(p′,p′′),(q′,q′′ )and putting ∂[v]in place of ∂, one can follow the derivation of (eq 3.7) as in [Siu,§1 and §2] to obtain ∂[v]ζ′2 Kc,χ +∥ϑ[v]ζ′∥2 Kc,χ =∫∂Kc e−χ |dφ|gTrg,η (∂[v]∂[v]φ)∨(ζ′⊗ζ′) +∥∇vζ′∥2 Kc,χ +∫Kc e−χTrg,η R∨ vv(ζ′⊗ζ′) for any ζ′∈A0,(1,q−1)(Kc;L). The boundary term vanishes as ∂[v]∂[v]φ= 0. Therefore, combining the above results with (eq 3.7) yields ∥Sqζ∥2 3+T∗ q−1ζ2 1= Bd(ζ, ζ) + ∂[u]ζ′′2 3+∂[v]ζ′2 3+∥∇uζ′∥2 Kc,χ +∥∇vζ′′∥2 Kc,χ +∫Kc e−χTrg,η R∨(ζ⊗ζ)−∫Kc e−χTrg,η R∨ vv(ζ′⊗ζ′). 3.3. MURAKAMI’S TRICK 25 Proof. For q= 0, it follows from (eq 3.18) that π∫Kc e−χTrg,η prF((e H(M))∨(ζ⊗ζ))=πM ∥ζ∥2 2≥π 4M∥ζ∥2 2, so this case is done. Assume q= 0. Since H∨ uv is a bounded linear operator on L2 0,(1,0) c,χ ⊗L2 0,(0,1) c,χ (where L2 0,(0,1) c,χ here means the complex conjugate of L2 0,(0,1) c,χ ), it follows that there is a bounded linear operator N:L2 0,(0,q) c,χ →L2 0,(1,q−1) c,χ such that ∫Kc e−χTrg,η H∨ uv(ζ′⊗ζ′′) = ⟨ζ′,Nζ′′⟩2 for all ζ′∈L2 0,(1,q−1) c,χ and ζ′′ ∈L2 0,(0,q) c,χ . In fact, after a linear change of coordinates such that gbecomes the Euclidean metric while keeping the decomposition (eq 2.4) orthogonal, one has Trg,η H∨ uv(ζ′⊗ζ′′) = η∑′ Jq−1 n−m ∑ i=1 m ∑ j=1 ζ′ iJq−1(Hvu)ji ζ′′ jJq−1, where ∑′ Jq−1denotes summation over all ordered multiindices Jq−1such that 1 ≤ j1<··· < jq−1≤m, and (Hvu)ji’s are the components of Hvu =Huv. Therefore, under such coordinates, (Nζ′′)iJq−1= m ∑ j=1 (Hvu)ji ζ′′ jJq−1. Moreover, |Nζ′′|2 g,η =η∑′ Jq−1 n−m ∑ i=1  m ∑ j=1 (Hvu)ji ζ′′ jJq−1 2 ≤η∑′ Jq−1 n−m ∑ i=1 (m ∑ j=1 (Hvu)ji2)(m ∑ j=1 ζ′′ jJq−1 2)by Cauchy– Schwarz ineq., =|Hvu|2 g·q|ζ′′|2 g,η =|Huv|2 g·q|ζ′′|2 g,η as Huv =Hvu . Since both Huv and gare translational invariant forms, |Huv|2 gis a constant. Set ν:= √q|Huv|g. Then, one has (∗ν)∥Nζ′′∥2≤ν∥ζ′′∥2 for all ζ′′ ∈L2 0,(0,q) c,χ . Note that νdepends only on q,Huv and g. It is independent of HEin particular. Since the decomposition (eq 2.4) is orthogonal with respect to g,gcan be decomposed into gE+gFsuch that gEis a hermitian metric on T1,0 uand gFis that on T1,0 v. Choose a real number λ > 0 such that (∗λ)λ≥max {M 2,2ν2 M,4ν}. Since νis independent of HE, by varying the real part of the matrix of HEunder the chosen apt coordinates according to Proposition 2.4.2, HEcan be chosen such that HE≥λgE, 26 3. L2ESTIMATES and therefore, ∫Kc e−χTrg,η H∨ E(ζ′⊗ζ′)≥λ∥ζ′∥2 2 for all ζ′∈A0,(1,q−1)(Kc;L). It follows from (eq 3.18) that, for any ζ=ζ′+ζ′′ ∈A0,q <2>(Kc;L), ∫Kc e−χTrg,η prF(e H(M))∨(ζ⊗ζ) ≥λ∥ζ′∥2 2+ 2 Re ⟨ζ′,Nζ′′⟩2+M∥ζ′′∥2 2 =λζ′+1 λNζ′′ 2 2−1 λ∥Nζ′′∥2 2+M∥ζ′′∥2 2by completing square , ≥λζ′+1 λNζ′′ 2 2−ν2 λ∥ζ′′∥2 2+M∥ζ′′∥2 2by (∗ν), ≥M 2(ζ′+1 λNζ′′ 2 2 +∥ζ′′∥2 2)by (∗λ), thus ν2 λ≤M 2, =M 2ζ+1 λNζ′′ 2 2 as L2 0,(1,q−1) c,χ ⊥L2 0,(0,q) c,χ . Furthermore, since ζ+1 λNζ′′2≥ ∥ζ∥2−1 λ∥Nζ′′∥2 ≥ ∥ζ∥2−ν λ∥ζ′′∥2by (∗ν), ≥(1−ν λ)∥ζ∥2as ∥ζ′′∥2≤ ∥ζ∥2, ≥3 4∥ζ∥2≥0 by (∗λ), one has M 2ζ+1 λNζ′′ 2 2≥M 2·(3 4)2 ∥ζ∥2 2≥M 4∥ζ∥2 2. This completes the proof. □ CHAPTER 4 The linearizable case 4.1. Proof of Theorem 1.1.1 for linearizable L The proof of Theorem 1.1.1 for linearizable Lis given here so that one can see clearly how the proof works without having to handle additional technicality required for the case of non-linearizable line bundles. Theorem 4.1.1.Suppose Lis linearizable and q < s− For q > m −s+ F. Then, for any ψ∈H0,q(X;L)such that ∂ψ = 0, there exists ξ∈H0,q−1(X;L)such that ∂ξ =ψon X. (In case q= 0 < s− F, this means ψ= 0.) In other words, by virtue of Theorem 2.3.1, Hq(X, L) = 0 for any qin the given range. Proof. Fix any ψ∈H0,q(X;L)∩ker ∂. An L2-norm ∥·∥X,χ is chosen as follows. Since Lis linearizable, one can take ℏ= 0 (see §2.5 for the definition of ℏ). Then, choose δ= 0 and thus ℏδ=ℏ−δ= 0. Choose the translational invariant hermitian metric gof the form as described in the proof of Lemma 3.3.2 for q > m −s+ For Lemma 3.3.4 for q < s− F, with M= 1. For the hermitian form Hassociated to L, choose HE:= H|E×Eas described in the proof of Lemma 3.3.6. A hermitian metric ηon Lis then defined as in §2.5. Choose a convex increasing smooth function eχ(thus χ:= eχ◦φis plurisubharmonic, i.e. √−1∂∂χ ≥0) such that ∥ψ∥X,χ <∞. An L2-norm ∥·∥X,χ is then fixed and ψ∈L2 0,(0,q) χ(X;L). Note that every ζ∈A0,q 0<2>(X;L) is contained in A0,q 0<2>(Kc;L) for some sufficiently large but finite c > 0. Consequently, the conclusion of Corollary 3.3.3 when q > m−s+ For Corollary 3.3.5 when q < s− F, as well as that of Lemma 3.3.6, holds for all ζ=ζ′+ζ′′ ∈A0,q 0<2>(X;L), where ζ′∈A0,(1,q−1) 0(X;L) and ζ′′ ∈A0,(0,q) 0(X;L). Since ℏδ= 0, W(ζ, ζ) (see (eq 3.14)) and W′ F(ζ′′, ζ′′) (see (eq 3.19)) both vanish for all ζ=ζ′+ζ′′ ∈A0,q 0<2>(X;L). Since χis plurisubharmonic on Xand ∂[v]χ= 0 = ∂[v]χ, one can choose at every point z∈Xthe coordinates such that both gand √−1∂[u]∂[u]χare simultaneously diagonalized while keeping the decomposition (eq 2.4) orthogonal, and see that Trg,η prF((∂∂χ)∨(ζ⊗ζ))= Trg,η (∂[u]∂[u]χ)∨(ζ′⊗ζ′)≥0. Therefore, wt(ζ, ζ)≥0 (see (eq 3.14)). As a result, combining Lemma 3.3.6 as well as the above facts about W,W′ F and wt with Corollary 3.3.3 or Corollary 3.3.5, one obtains ∥Sqζ∥2 3+T∗ q−1ζ2 1≥π 4∥ζ∥2 2 for all ζ∈A0,q 0<2>(X;L). This is the required L2estimate. Proposition 3.1.5 and Remark 3.1.6 then assert that there exists ξ∈H0,q−1(X;L) such that ∂ξ =ψon X.□ 27 CHAPTER 5 The non-linearizable case For a non-linearizable line bundle L, the wild curvature terms W(see (eq 3.14)) and W′ F(see (eq 3.19)) are not identically zero. In order to get the estimates for these terms, Takayama’s Weak ∂∂-Lemma (ref. [Taka2, Lemma 3.14]) is invoked. One is then forced to restrict attention to each of the Kc’s and obtain the required L2estimates there. What then remains is to show that the existence of a solution of the ∂-equation ∂ξ =ψon every Kcimplies the existence of a global solution. The argument for this latter part is essentially the same as the one in [GR, Ch. IV, §1, Thm. 7]. An apt coordinate system is fixed throughout this section. 5.1. Bounds on the wild curvature terms Takayama proves in [Taka2] the following Weak ∂∂-Lemma. Weak ∂∂-Lemma 5.1.1 (cf. [Taka2, Lemma 3.14]).Let ωbe a positive real (1,1)-form on X, and let θbe a smooth real 1-form on Xsuch that θ=β+βfor some smooth (0,1)-form β, and dθ is of type (1,1). Then for every positive number εand every relatively compact open subset Wof X, there exists a smooth function δon Xsuch that −εω < dθ −2√−1∂∂ Re δ < εω on W . Moreover, if β∈H0,1(X), then δcan be chosen such that δ∈H(X). In the current situation, the role of βin Lemma 5.1.1 is taken by √−1∂ℏ (therefore dθ = 2√−1∂∂ Re ℏ), and that of Wby Kc. Remark 5.1.2.In Takayama’s formulation, the assertion of the Weak ∂∂-Lemma is that there exists a smooth real valued function fεW := 2(Im f0+ Im ΨM0) on X such that −εω < dθ −√−1∂∂fεW < εω on W, in which f0is a smooth function on Xsuch that β=ϕ+∂f0for some real analytic (0,1)-form ϕin H0,1(X), and ΨM0is some real analytic function in H(X). Therefore, the smooth function δhere is given by δ:= −√−1(f0+ ΨM0) in Takayama’s notation. If β∈H0,1(X), then one has f0∈H(X) as ∂[u]f0= 0, so δ∈H(X) also. Remark 5.1.3.As a side remark, following the construction of δin [Taka2, Lemma 3.14], ∂ℏδ=∂ℏ−∂δ is real analytic on X, so ℏδis real analytic on Cn. It follows that the hermitian metric ηon Lis real analytic. Suitable estimates for the wild curvature terms Wand W′ Fare obtained by choosing a proper δ∈H(X) according to the Weak ∂∂-Lemma. Lemma 5.1.4.Suppose a hermitian metric gon Xand a choice of HEare fixed. Then, on every Kcwhere 0< c < ∞, given any real number εw>0and for any 28 5.2. EXISTENCE OF WEAK SOLUTIONS ON Kc29 q≥0, one can choose δc∈H(X)which yields a hermitian metric ηcon Lsuch that, for any given weight χ, |W(ζ, ζ)| ≤ εwq∥ζ∥2 Kc,ηc,χ (eq 5.1) |W′ F(ζ′′, ζ′′)| ≤ εwm∥ζ′′∥2 Kc,ηc,χ ≤εwm∥ζ∥2 Kc,ηc,χ (eq 5.2) for all ζ=ζ′+ζ′′ ∈A0,q <2>(Kc;L)where ζ′∈A0,(1,q−1)(Kc;L)and ζ′′ ∈A0,(0,q)(Kc;L). Proof. First the estimate for Wis considered. Recall that ωis the (1,1)-form associated to g. The Weak ∂∂-Lemma asserts that, for any εw>0, there exists δc∈H(X) such that (eq 5.3) −2εwω < 2√−1∂∂ Re ℏδc<2εwωon Kc. Such δcyields a hermitian metric ηcon Lgiven the fixed choice of HE. Then, it follows from (eq 3.14) that, for any weight χ, −εw∫Kc e−χTrg,ηcprF(g∨(ζ⊗ζ))≤W(ζ, ζ)≤εw∫Kc e−χTrg,ηcprF(g∨(ζ⊗ζ)) for any ζ=ζ′+ζ′′ ∈A0,q <2>(Kc;L) (εwinstead of 2εwin the bounds because of the factor 1 2in ω=−Im g=√−1 2∑k,ℓ gkℓdzk∧dzℓ). Note that ∫Kc e−χTrg,ηcprF(g∨(ζ⊗ζ))=∥ζ′∥2 Kc,ηc,χ +q∥ζ′′∥2 Kc,ηc,χ ≤q∥ζ∥2 Kc,ηc,χ when q≥1. When q= 0, the integral on the left hand side is zero, so the above inequality is still valid. As a result, one obtains −εwq∥ζ∥2 Kc,ηc,χ ≤W(ζ, ζ)≤εwq∥ζ∥2 Kc,ηc,χ and hence (eq 5.1). For the estimate for W′ F, note that (eq 5.3) implies −2εwprFω < 2√−1∂[v]∂[v]Re ℏδc<2εwprFωon Kc. Then, one has −εwm < 2 Trg∂[v]∂[v]Re ℏδc< εwmwith the same εwand δcas above. Therefore, it follows from (eq 3.19) that −εwm∥ζ′′∥2 Kc,ηc,χ ≤W′ F(ζ′′, ζ′′)≤εwm∥ζ′′∥2 Kc,ηc,χ for any ζ′′ ∈A0,(0,q)(Kc;L), and hence (eq 5.2). □ 5.2. Existence of weak solutions on Kc With the bounds given in §5.1 for the wild curvature terms, it is easy to follow the proof of Theorem 4.1.1 and get the following Proposition 5.2.1.Suppose Lis a holomorphic line bundle on X(which can possibly be non-linearizable), and suppose q < s− For q > m−s+ F. Then, there exists a suitable hermitian metric gon Xsuch that the following holds: for any 0< c < ∞, a hermitian metric ηcon Lcan be chosen such that, given any plurisubharmonic weight χ, the L2estimate ∥Sqζ∥2 Kc,ηc,χ +T∗ q−1ζ2 Kc,ηc,χ ≥π 4∥ζ∥2 Kc,ηc,χ for all ζ∈A0,q <2>(Kc;L)∩DomKc,ηc,χ T∗ q−1is satisfied. 30 5. THE NON-LINEARIZABLE CASE Proof. Choose the translational invariant hermitian metric gas described in the proof of Lemma 3.3.2 for q > m −s+ For Lemma 3.3.4 for q < s− F, with M= 2. For the hermitian form Hassociated to L, choose HEas described in the proof of Lemma 3.3.6. These choices are independent of c. Consider Kcfor some fixed 0 < c < ∞. Take any εw>0 such that (∗)εw(q+m)≤π 4 and choose δc∈H(X) according to Lemma 5.1.4 such that, for any given weight χ, the inequalities (eq 5.1) and (eq 5.2) hold under the induced L2-norm ∥·∥Kc,ηc,χ. By the choices of the metrics, the conclusion of Corollary 3.3.3 when q > m −s+ F or Corollary 3.3.5 when q < s− F, as well as that of Lemma 3.3.6, holds for all ζ= ζ′+ζ′′ ∈A0,q <2>(Kc;L)∩DomKc,ηc,χ T∗ q−1, where ζ′∈A0,(1,q−1)(Kc;L)∩Dom(1,q−1) Kc,ηc,χ ∂∗ [u] and ζ′′ ∈A0,(0,q)(Kc;L). Since χis plurisubharmonic, wt(ζ, ζ)≥0 for all ζ∈A0,q <2>(Kc;L) as in the proof of Theorem 4.1.1. As a result, from Corollary 3.3.3 or 3.3.5 as well as Lemma 3.3.6, one obtains ∥Sqζ∥2 Kc,ηc,χ +T∗ q−1ζ2 Kc,ηc,χ ≥{π 2∥ζ∥2 Kc,ηc,χ +W(ζ, ζ) for q > m −s+ F π 2∥ζ∥2 Kc,ηc,χ +W′ F(ζ′′, ζ′′) + W(ζ, ζ) for q < s− F ≥π 2∥ζ∥2 Kc,ηc,χ −εw(m+q)∥ζ∥Kc,ηc,χ by (eq 5.1) and (eq 5.2), and εwq < εw(m+q) ≥π 4∥ζ∥2 Kc,ηc,χ by (∗). This gives the required L2estimate. □ Since, for any ψ∈H0,q(X;L), one has ψ|Kc∈L2 0,(0,q)(Kc;L) (unweighted) for any 0 < c < ∞, it follows the following corollary of Propositions 3.1.5 and 5.2.1. Corollary 5.2.2.Consider the exhaustive sequence {Kν}ν∈N>0of relatively compact open subsets of X. Suppose q < s− For q > m −s+ F. Then one can choose a suitable hermitian metric gon Xand a sequence of hermitian metrics {ην}ν∈N>0 on Las in Proposition 5.2.1 such that, for any ψ∈H0,q(X;L)∩ker ∂, there exists a sequence of solutions {ξ′ ν}ν∈N>0such that ξ′ ν∈L2 0,(0,q−1) ην(Kν;L)(unweighted) and ∂ξ′ ν=ψ|Kνin L2 0,(0,q) ην(Kν;L). Remark 5.2.3.Since χhas to be smooth on a neighborhood of Kc(as required by [H¨or1, Prop. 2.1.1] so that A0,q <2>(Kc;L)∩Dom T∗ q−1is dense in Dom T∗ q−1∩Dom Sq under the suitable graph norm), if ψ∈H0,q(Kc;L), there may not exist such χsuch that ∥ψ∥Kc,χ <∞. To avoid technical difficulty, the author does not attempt to solve the ∂-equation for any ψ∈H0,q(Kc;L) such that ∂ψ = 0 by means of L2 estimates directly. 5.3. A Runge-type approximation This section is devoted to proving a Runge-type approximation which is required to construct a global solution to the equation ∂ξ =ψfrom the solutions on Kν’s given in Corollary 5.2.2. 5.3. A RUNGE-TYPE APPROXIMATION 31 In what follows, qis assumed to be 0 < q < s− For q > m−s+ F, and the hermitian metric gas well as the family of hermitian metrics {ηc}c>0as asserted by Proposition 5.2.1 is fixed. Then, according to the choices of the ηc’s in the proof of Proposition 5.2.1, for any c′, c > 0, one has ηc=ηc′e2 Re(δc′−δc)=: ηc′eδc′c. Note that eδc′c>0 on X. It is understood that the hermitian metric ηcon Lis chosen when the L2-norm on Kcis considered, so write L2 0,(0,q) ηc,χ (Kc;L) as L2 0,(0,q) χ(Kc;L), ⟨·,·⟩Kc,ηc,χ as ⟨·,·⟩Kc,χ and so on to simplify notation. When the weight χis absent from the notation, e.g. L2 0,(0,q)(Kc;L) or ⟨·,·⟩Kc, it is understood that the corresponding object is unweighted, i.e. χ= 0. For any finite c′> c > 0 and for any Ψ ∈L2 0,(0,q−1)(Kc;L), if Ψ is extended by zero to a section in L2 0,(0,q−1)(Kc′;L), then it follows that (eq 5.4) ⟨ζ, Ψ⟩Kc=⟨ζ, Ψeδc′c⟩Kc′ for any ζ∈L2 0,(0,q−1)(Kc′;L). Define (kerKc′Tq−1)Kcto be the image of kerKc′Tq−1under the restriction map L2 0,(0,q−1)(Kc′;L)→L2 0,(0,q−1)(Kc;L). Note that Tq−1commutes with the restriction map (as c > 0), so one has(kerKc′Tq−1)Kc⊂kerKcTq−1. The following proof of the required Runge-type approximation is an analogue of the one for strongly pseudoconvex manifolds given in [H¨or3, Lemma 4.3.1]. Proposition 5.3.1.Suppose 0< q < s− For q > m −s+ F, and gand ηc’s are chosen according to Proposition 5.2.1. Then, for any finite c′> c > 0, the closure of (kerKc′Tq−1)Kcin L2 0,(0,q−1)(Kc;L)is kerKcTq−1. In other words, (kerKc′Tq−1)Kc is dense in kerKcTq−1. Proof. By virtue of the Hahn-Banach theorem, it suffices to show that for every Ψ∈L2 0,(0,q−1)(Kc;L), if the induced bounded linear functional L2 0,(0,q−1)(Kc;L)∋ζ7→ ⟨ζ, Ψ⟩Kc vanishes on (kerKc′Tq−1)Kc, then it also vanishes on kerKcTq−1.1 Suppose that Ψ ∈L2 0,(0,q−1)(Kc;L) satisfies the above assumption. Extend Ψ by zero to Kc′as a section in L2 0,(0,q−1)(Kc′;L). Now it suffices to show that there exists Ξ∈L2 0,q <2>(Kc′;L) such that Ξ ≡0 on Kc′\Kcand (†)⟨ζ, Ψeδc′c⟩Kc′=⟨Tq−1ζ, Ξ⟩Kc′ for any ζ∈DomKc′Tq−1, which then implies that (‡)⟨ζ, Ψ⟩Kc=⟨Tq−1ζ, Ξe−δc′c⟩Kc for any ζ∈DomKc′Tq−1due to (eq 5.4). The equality (‡) holds true for ζ∈ A0,(0,q−1) 0(Kc′;L) in particular, and A0,(0,q−1)(Kc;L) is dense in DomKcTq−1under the graph norm √∥ζ∥2 Kc+∥Tq−1ζ∥2 Kcby [H¨or1, Prop. 2.1.1], so (‡) also holds 1If there exists ζ∈kerKcTq−1which does not lie in the closure of (kerKc′Tq−1)Kc in L2 0,(0,q−1)(Kc;L), then the Hahn-Banach theorem asserts that there is a bounded linear functional Λ such that (kerKc′Tq−1)Kc⊂ker Λ and Λζ= 1. 32 5. THE NON-LINEARIZABLE CASE true for ζ∈DomKcTq−1. It follows that ⟨ζ, Ψ⟩Kc=⟨Tq−1ζ, Ξe−δc′c⟩Kc= 0 for all ζ∈kerKcTq−1⊂DomKcTq−1as required. It remains to show the existence of such Ξ. Take a sequence of smooth convex increasing functions eχν:R→Rsuch that eχν(x) = 0 for all x≤c, and eχν(x)↗+∞as ν→ ∞ for every x>c. Note that eχν≥0 for any ν≥0 by such choice. Set χν:= eχν◦φas before. A sequence of weighted norms ∥·∥c′,ν := ∥·∥Kc′,χνon Kc′is then defined. Let the corresponding inner products, Hilbert spaces and Dom also be distinguished by using the subscripts c′, ν, and the corresponding adjoint of Tq−1by T∗,ν q−1. For any qin the given range, the L2estimate in Proposition 5.2.1 holds under each of the above weighted norms with T∗ q−1replaced by T∗,ν q−1. Since ⟨ζ, Ψeδc′ceχν⟩c′,ν = ⟨ζ, Ψeδc′c⟩Kc′and the right hand side vanishes for all ζ∈kerKc′Tq−1= kerc′,ν Tq−1 by the assumption on Ψ, it follows that Ψeδc′ceχν∈(kerc′,ν Tq−1)⊥= imc′,ν T∗,ν q−1. Given the L2estimate, Theorem 3.1.1 (2) then asserts that there exists e Ξν∈ Domc′,ν T∗,ν q−1such that T∗,ν q−1e Ξν= Ψeδc′ceχν. Therefore, one has ⟨ζ, Ψeδc′ceχν⟩c′,ν =⟨ζ, T∗,ν q−1e Ξν⟩c′,ν =⟨Tq−1ζ, e Ξν⟩c′,ν =⟨Tq−1ζ, e Ξνe−χν⟩Kc′ for all ν∈Nand for all ζ∈Domc′,ν Tq−1= DomKc′Tq−1. By defining Ξν:= e Ξνe−χν, one obtains (∗)⟨ζ, Ψeδc′c⟩Kc′=⟨Tq−1ζ, Ξν⟩Kc′. Moreover, notice that the constant in the L2estimate is independent of ν(which is chosen to be π 4in Proposition 5.2.1). The estimate on the solution e Ξνfrom Theorem 3.1.1 (2) then implies that (∗∗)π 4∫Kc′|Ξν|2 g,ηc′eχν≤∫Kc′Ψeδc′c2 g,ηc′eχν=∫Kc|Ψ|2 g,ηceδc′ceχν, where the last equality is due to the fact that Ψ vanishes on Kc′\Kc. Since eχν(φ) is independent of νwhen φ≤c, the integral on the right hand side is independent of ν, so the left hand side is a bounded sequence in ν. This in turn implies that there exists a subsequence of {Ξν}ν∈Nwhich converges to some Ξ ∈L2 0,q <2>(Kc′;L) (unweighted) in the weak topology. From (∗∗), since eχν(φ)↗+∞for φ > c, it follows that Ξ ≡0 when φ > c, i.e. on Kc′\Kc. Moreover, from (∗) it follows that (†) holds for all ζ∈DomKc′Tq−1. This is what is desired. □ 5.4. Proof of Theorem 1.1.1 for general L First notice that, if q= 0 < s− F, then the L2estimate in Proposition 5.2.1 holds when the metrics are chosen suitably, and thus for any ψ∈H(X;L)∩ker ∂one has 0 = ∂ψ2 Kc≥π 4∥ψ∥2 Kc 5.4. PROOF OF THEOREM 1.1.1 FOR GENERAL L33 (note that T∗ −1ζ= 0 for all ζ∈A(Kc;L)). This means that ψ|Kc= 0 for any c > 0, and thus ψ= 0 on X. Therefore, one has the following Theorem 5.4.1.If s− F>0, one has H0(X, L) = 0. Assume 0 < q < s− For q > m −s+ Fin what follows. The metrics gand ην’s from Corollary 5.2.2 are fixed for this section. Again, write L2 0,(0,q) ην,χ (Kν;L) as L2 0,(0,q) χ(Kν;L) and so on, and notations like L2 0,(0,q)(Kc;L) or ∥·∥Kcare understood as unweighted objects, i.e. χ= 0. For every integer ν≥1, as δν+1 −δνis smooth on Xand Kν+1 is compact, there exists a constant M′ ν+1 ≥1 such that (eq 5.5) ∥ζ∥Kν≤M′ ν+1 ∥ζ∥Kν+1 for all ζ∈L2 0,(0,q)(Kν+1;L). Define also M1:= 1 and Mν:= ∏ν k=2 M′ kfor ν≥2. Proposition 5.3.1 is used to complete the proof of Theorem 1.1.1. The following argument is adopted from [GR, Ch. IV, §1, Thm. 7]. Theorem 5.4.2.Suppose 0< q < s− For q > m−s+ F. Then one has Hq(X, L) = 0 for any qin the given range. Proof. Given any ψ∈H0,q(X;L)∩ker ∂, Corollary 5.2.2 provides a sequence of local solutions {ξ′ ν}ν≥1such that ξ′ ν∈L2 0,(0,q−1)(Kν;L) and ∂ξ′ ν=ψ|Kνfor all integers ν≥1. First a sequence of local solutions {ξν}ν≥1such that ξν∈L2 0,(0,q−1)(Kν;L), ∂ξν=ψ|Kνand (∗)∥ξν+1 −ξν∥Kν<1 Mν2ν for all ν≥1 is defined inductively as follows. Set ξ1:= ξ′ 1. Suppose ξ1, . . . , ξν are defined for some ν≥1. Let γ′ ν:= ξ′ ν+1|Kν−ξν. Notice that γ′ ν∈kerKνTq−1⊂ L2 0,(0,q−1)(Kν;L). Proposition 5.3.1 then implies that there exists γν∈kerKν+1 Tq−1⊂ L2 0,(0,q−1)(Kν+1;L) such that ∥γ′ ν−γν∥Kν<1 Mν2ν. Set ξν+1 := ξ′ ν+1 −γν. Then one has ∂ξν+1 =∂ξ′ ν+1 =ψ|Kν+1 and the inequality (∗) is satisfied. The required sequence {ξν}ν≥1is therefore defined. Notice that, for every ν≥1, the sequence {ξµ|Kν}µ≥νconverges in L2 0,(0,q−1)(Kν;L). Indeed, for any µ≥ν≥1 and for any integer k > 0, ∥ξµ+k−ξµ∥Kν≤ k−1 ∑ r=0 ∥ξµ+r+1 −ξµ+r∥Kν ≤ k−1 ∑ r=0 Mµ+r Mν∥ξµ+r+1 −ξµ+r∥Kµ+rby (eq 5.5) , ≤1 Mν k−1 ∑ r=0 1 2µ+rby (∗), ≤1 Mν2µ−1, 34 5. THE NON-LINEARIZABLE CASE which tends to 0 as µ→ ∞, so {ξµ|Kν}µ≥νis a Cauchy sequence in L2 0,(0,q−1)(Kν;L). Let ξ(ν)be the limit of {ξµ|Kν}µ≥νin L2 0,(0,q−1)(Kν;L). Since ∂ξµ|Kν=ψ|Kνfor all µ≥ν, and ∂is a closed operator, one has ∂ξ(ν)=ψ|Kνfor all ν≥1. Now notice that restriction from Kν+1 to Kνis continuous by (eq 5.5), so ξ(ν+1)|Kν−ξ(ν)= lim µ≥ν+1 µ→∞ (ξµ|Kν−ξµ|Kν) = 0 in L2 0,(0,q−1)(Kν;L). On every Kν, different choices of δν∈H(X) yield equivalent norms. Therefore, by fixing one δ∈H(X), one can consider L2 0,q−1(X;L; loc), the space of locally L2L-valued (0, q −1)-forms on X, and there exists ξ′∈ L2 0,q−1(X;L; loc) such that ξ′|Kν=ξ(ν)for all ν≥1,and ∂ξ′=ψin L2 0,q−1(X;L; loc) . Remark 3.1.6 then assures that there exists ξ∈H0,q−1(X;L) such that ∂ξ =ψon X. Since ψ∈H0,q(X;L)∩ker ∂is arbitrary, this shows that Hq(X, L) = 0. This completes the proof. □