aa r X i v : . [ m a t h . DG ] S e p NONNEGATIVE HERMITIAN VECTOR BUNDLES AND CHERNNUMBERS
PING LI
Abstract.
We show in this article that if a holomorphic vector bundle has a nonnegativeHermitian metric in the sense of Bott and Chern, which always exists on globally gener-ated holomorphic vector bundles, then some special linear combinations of Chern forms arestrongly nonnegative. This particularly implies that all the Chern numbers of such a holo-morphic vector bundle are nonnegative and can be bounded below and above respectively bytwo special Chern numbers. As applications, we obtain a family of new results on compactconnected complex manifolds which are homogeneous or can be holomorphically immersedinto complex tori, some of which improve several classical results. Introduction
Throughout this article, all the vector bundles considered are holomorphic and over compact complex manifold.The concept of positivity/nonnegativity has played a central role in complex differentialgeometry and algebraic geometry. Bott and Chern ([BC65]) introduced a notion of nonneg-ativity on Hermitian vector bundles and applied it to initiate the study of high-dimensionalvalue distribution theory. As showed in [BC65], the existence of such a metric implies thatall the Chern forms are strongly nonnegative and on globally generated vector bundles suchmetrics always exist. Later, globally generated vector bundles were investigated in detail byMatsushima and Stoll in [MS73] and they, among other things, related the non-vanishing ofthe Chern classes and Chern numbers to the transcendence degree of meromorphic functionson the base manifolds.Soon after the appearance of [BC65], the notions of ampleness and Griffiths-positivitywere introduced respectively by Hartshorne ([Ha66]) and Griffiths ([Gr69]) and it turns outthat Bott-Chern nonnegativity implies Griffiths-nonnegativity while Griffiths-positivity im-plies ampleness. Griffiths raised in [Gr69] the question of characterizing the polynomialsin the Chern classes/forms which are positive as cohomology classes/differential forms forGriffiths-positive or ample vector bundles. On the class level this was answered completely byFulton and Lazarsfeld ([FL83]), extending an earlier result of Bloch and Gieseker ([BG71]).Indeed Fulton and Lazarsfeld showed that the set of such polynomials in the Chern classesfor ample vector bundles is exactly the cone generated by Schur polynomials of the Chernclasses. These inequalities of Fulton-Lazarsfeld type were showed by Demailly, Peternell and
Mathematics Subject Classification.
Key words and phrases.
Nonnegativity, Hermitian vector bundle, Chern form, Chern class, Chern number,globally generated vector bundle, homogeneous complex manifold, Moishezon manifold, Kodaira dimension,algebraic dimension.The author was partially supported by the National Natural Science Foundation of China (Grant No.11722109).
Schneider ([DPS94]) to remain true for numerically effective (“nef” for short) vector bundlesover compact K¨ahler manifolds.On the other hand, Griffiths’ question on the form level is still largely unknown. Griffithshimself showed in [Gr69, p. 249] that the second Chern form is positive on a Griffiths-positivevector bundle. Note that its proof is purely algebraic, which seems to be difficult to begeneralized to higher dimensions. Recently Guler ([Gu12]) showed that the dual Segre forms,which are formal inverse of the total Chern forms, are positive on Griffiths-positive vectorbundles using some geometric arguments.The main purposes of this article are two-folded. Our first main purpose is to show a usefultechnical result, Proposition 3.1, which says that on a Bott-Chern nonnegative Hermitianvector bundles, the cone generated by the Schur polynomials in the Chern forms are stronglynonnegative, thus providing some positive evidence to the aforementioned Griffiths’ questionon the form level. Our proof of Prop. 3.1 is built on a relationship established in [FL83] thatthe cone generated by the Schur polynomials coincides to the Griffiths cone defined in [Gr69].When taking some special Schur polynomials in Prop. 3.1, we shall see in Theorem 3.2 thatthe self-products of Chern forms and thus all the Chern numbers of such vector bundles can bebounded below and above by two of them respectively. As is well-known globally generatedvector bundles admit Bott-Chern nonnegative metrics (see Example 4.1), this implies thatTheorem 3.2 imposes strong constraints on the Chern numbers of such vector bundles.
Our second main purpose is to apply Theorem 3.2 to such vector bundles and especiallyto compact complex manifolds whose holomorphic tangent or cotangent bundles are globallygenerated to yield a family of new results on them, some of which improve several relatedclassical results. To be more precise, our first application (Thm 5.1, Coro. 5.2) is to givelower and upper bounds for Chern numbers of various nonnegative-type holomorphic vectorbundles on compact complex manifolds, whose nonnegativity has been well-known for severaldecades.
Our second application (Thms 5.3, 5.4 and 5.5) is concerned with the projectivicityof compact complex manifolds equipped with globally generated vector bundles and homo-geneous compact complex manifolds, and its relationship with the non-vanishing of Chernclasses/numbers.
Our third application (Thm 5.9) is to give an in-depth investigation on thestructure of compact complex manifolds holomorphically immersed into complex tori in detail.The rest of this article is organized as follows. We introduce some necessary notation andsymbols in Section 2 and then state in Section 3 our main technical results in this article,Prop. 3.1 and Thm. 3.2. Then we give in Section 4 some examples where the Bott-Chernnonnegativity is satisfied. In Section 5 we apply our established technical results to obtainvarious related consequences, some of which improve several classical results, and the proofof Prop. 3.1 will be given in the last section, Section 6.
Acknowledgements
This paper was initiated during the author’s visit to the Max-Planck Institut f¨ur Mathe-matik in Bonn from September 2016 to February 2017. The author thanks the Institute forhospitality and financial support. The author also thanks Xiaokui Yang for informing himthe related results in [Zha97] and Vamsi Pingali for some crucial comments and pointing outthe reference [BP13].
ONNEGATIVE HERMITIAN VECTOR BUNDLES AND CHERN NUMBERS 3 Preliminaries
Suppose that ( E r , h ) −→ M n is a rank r Hermitian vector bundle over a compact complexmanifold M with dim C M = n . This means that E r is a rank r holomorphic vector bundleover M and equipped with a Hermitian metric h on each fiber varying smoothly. We denoteby ∇ the Chern connection of ( E r , h ) −→ M n , i.e., ∇ is the unique connection compatiblewith both the complex structure and the Hermitian metric h . Then the curvature tensor R of ∇ is given by R := ∇ ∈ A , ( M ; Hom( E, E )) = A , ( M ; E ∗ ⊗ E ) , where A , ( M ; · ) is the set of complex-valued smooth (1 , { e , . . . , e r } is a locally defined frame field of E , then R ( e , . . . , e r ) = ( e , . . . , e r )(Ω ij ) , where Ω := (Ω ij ) ( i row, j column) is the curvature matrix with respect to { e i } whose entriesare (1 , { ˜ e i } is another frame with( ˜ e , . . . , ˜ e r ) = ( e , . . . , e r ) P, then the curvature matrix e Ω with respect to { ˜ e i } is related to Ω by(2.1) e Ω = P − Ω P. It is well-known that the following c i ( E, h ) (0 ≤ i ≤ n ):(2.2) det (cid:0) tI r + √− π (Ω ij ) (cid:1) =: n X i =0 c i ( E, h ) · t n − i , I r : r × r identity matrix , are globally well-defined, real and closed ( i, i )-forms and called the i -th Chern forms of ( E, h ),which represent the Chern classes c i ( E ) of E .The following definition was introduced by Bott and Chern ([BC65, p. 90]). Definition 2.1 (Bott-Chern) . A Hermitian vector bundle ( E r , h ) −→ M n is called Bott-Chern nonnegative , denoted by h ≥ BC
0, if for any point of M , there exist a unitary framefield around it and a matrix A with r rows and whose entries are (1 , A ∧ A t . Here “ t ” denotes the transpose of a matrix. Remark 2.2. (1) This definition is independent of the unitary frame we choose as the transformationmatrix P is a unitary matrix between them and thus from (2 .
1) we have e Ω = P − Ω P = P − ( A ∧ A t ) P = P t ( A ∧ A t ) P = ( P t A ) ∧ ( P t A ) t . (2) Here we don’t have any requirement on the number of columns of the matrix A , whichmay vary along the choice of the points on M .(3) We shall see in Example 4.1 that globally generated vector bundles over compact com-plex manifolds form an important subclass of those admitting Bott-Chern nonnegativeHermitian metrics. PING LI
Before stating our first main result, we need some more notation. Following [FL83], wedenote by Γ( i, r ) the set of partitions λ = ( λ , λ , . . . , λ i ) of weight i by nonnegative integers λ j ≤ r : r ≥ λ ≥ λ ≥ · · · ≥ λ i ≥ , i X j =1 λ j = i. For each partition λ = ( λ , λ , . . . , λ i ) ∈ Γ( i, r ), a Schur polynomial S λ ( c , . . . , c r ) ∈ Z [ c , . . . , c r ]is attached to as follows: S λ ( c , . . . , c r ) :=det( c λ j − j + k ) ≤ j,k ≤ i ( j : row, k : column)= (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c λ c λ +1 · · · c λ + i − c λ − c λ · · · c λ + i − ... ... . . . ... c λ i − i +1 c λ i − i +2 · · · c λ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Here we make the convention that c = 1 and c j = 0 if j < j > r . In particular, we have(2.4) S ( i, ,..., ( c , . . . , c r ) = c i and S ( i − j,j, ,..., ( c , . . . , c r )= (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c i − j c i − j +1 ∗ · · · ∗ c j − c j ∗ · · · ∗ · · · ∗ ... ... ... . . .0 0 0 · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) ≤ j ≤ [ i (cid:1) = c i − j c j − c i − j +1 c j − . (2.5)The expression (2.5) shall play a decisive role in establishing the lower and upper boundsof the Chern classes/numbers in our Theorem 3.2, which was partly inspired by [DPS94] asthe special case S ( i − , , ,..., ( c , . . . , c r ) = c i − c − c i has been observed and applied in [DPS94, Coro. 2.6].Now we recall the notions of nonnegativity and strong nonnegativity for real ( p, p )-forms([De12, Ch. 3, § p, p )-form ϕ (1 ≤ p ≤ n ) on a compact complexmanifold M n is called nonnegative , denoted by ϕ ≥
0, if for any point x ∈ M and any(1 , X , . . . , X p at x , we have( −√− p ϕ ( X , . . . , X p , X , . . . , X p ) ≥ . A real ( p, p )-form ϕ is called strongly nonnegaive , denoted by ϕ ≥ s
0, if it can be written as ϕ = ( √− p X i ψ i ∧ ψ i , where these ψ i are ( p, p, p )-forms ϕ and ϕ are said to be ϕ ≤ ϕ (resp. ϕ ≤ s ϕ ) if ϕ − ϕ ≥ ≥ s ONNEGATIVE HERMITIAN VECTOR BUNDLES AND CHERN NUMBERS 5 be nonnegative (cf. [BP13]). Nevertheless, it is obvious that the product of two stronglynonnegative forms is still strongly nonnegative.3.
Main technical results
Bott and Chern noticed that all the Chern forms c i ( E, h ) (1 ≤ i ≤ n ) of a Bott-Chernnonnegative Hermitian vector bundle ( E r , h ) −→ M n are strongly nonnegative ([BC65, p.91]). The following proposition, which is our first main technical observation in this article,states that indeed many more real forms involving in the Chern forms, including c i ( E, h )themselves, are strongly nonnegative.
Proposition 3.1.
Suppose that ( E r , h ) −→ M n is a Hermitian vector bundles with h ≥ BC i, i )-forms S λ (cid:0) c ( E, h ) , . . . , c r ( E, h ) (cid:1) , ( ∀ λ ∈ Γ( i, r ) , ∀ ≤ i ≤ n )are strongly nonnegative. In particular, by (2.4) and (2.5) the closed real ( i, i )-forms c i ( E, h ) (1 ≤ i ≤ n ) c i − j ( E, h ) c j ( E, h ) − c i − j +1 ( E, h ) c j − ( E, h ) (1 ≤ i ≤ n, ≤ j ≤ [ i ])(3.1)are strongly nonnegative.For later simplicity we denote by c λ ( E, h ) := i Y j =1 c λ j ( E, h ) , ∀ λ = ( λ , . . . , λ i ) ∈ Γ( i, r ) ,c λ [ E ] := Z M n Y j =1 c λ j ( E, h ) ∈ Z , ∀ λ = ( λ , . . . , λ n ) ∈ Γ( n, r ) , and c λ ( E ) := i Y j =1 c λ j ( E ) ∈ H i ( M ; Z ) , ∀ λ = ( λ , . . . , λ i ) ∈ Γ( i, r ) . We are now able to deduce our second technical result, which says that the strongly non-negative forms in (3.1) in fact implies that the self-products of Chern forms and thus theChern numbers can be bounded below and above as follows.
Theorem 3.2.
Suppose that ( E r , h ) −→ M n is a Hermitian vector bundle with h ≥ BC .Then as real ( i, i ) -forms, c λ ( E, h ) ≥ s , and bounded below and above respectively by c i ( E, h ) and c i ( E, h ) : ≤ s c i ( E, h ) ≤ s c λ ( E, h ) ≤ s c ( E, h ) i , ∀ λ ∈ Γ( i, r ) , ∀ ≤ i ≤ n. Consequently the Chern numbers satisfy ≤ c n [ E ] ≤ c λ [ E ] ≤ c n [ E ] , ∀ λ ∈ Γ( n, r ) . PING LI
Proof.
For any λ = ( λ , . . . , λ i ) ∈ Γ( i, r ), repeated use of (3.1) leads to0 ≤ s c i ( E, h ) ≤ s c i − ( E, h ) c ( E, h ) ≤ s c i − ( E, h ) c ( E, h ) ≤ s · · ·≤ s c i − λ ( E, h ) c λ ( E, h ), if λ ≤ [ i ≤ s c i ( E, h ) ≤ s c i − ( E, h ) c ( E, h ) ≤ s · · · ≤ s c λ ( E, h ) c i − λ ( E, h ) , if λ > [ i ≤ s c i ( E, h ) ≤ s c λ ( E, h ) c i − λ ( E, h )Similarly, c i − λ ( E, h ) ≤ s c λ ( E, h ) c i − λ − λ ( E, h ) ,c i − λ − λ ( E, h ) ≤ s c λ ( E, h ) c i − λ − λ − λ ( E, h ) , · · · . (3.3)Note that all these forms discussed here are strongly nonnegative by Prop. 3.1 and thus theirproducts are still strongly nonnegative:(3.4) 0 ≤ s c i ( E, h ) ≤ s i Y j =1 c λ j ( E, h ) = c λ ( E, h ) . On the other hand, similar arguments yield c λ j ( E, h ) ≤ s c λ j − ( E, h ) c ( E, h ) ≤ s (cid:2) c λ j − ( E, h ) c ( E, h ) (cid:3) c ( E, h ) ≤ s · · ·≤ s c ( E, h ) λ j . (3.5)Therefore,(3.6) c λ ( E, h ) = i Y j =1 c λ j ( E, h ) ≤ s i Y j =1 c ( E, h ) λ j = c ( E, h ) i , ∀ λ ∈ Γ( i, r ) . (cid:3) Remark 3.3.
This theorem is indeed partly inspired by an observation in [DPS94, Coro.2.6], where they noticed that S ( i − , , ,..., = c c i − − c i and thus gave the upper bound c i for c λ as we have done in (3.5) and (3.6) in the context of nef vector bundles on compactK¨ahler manifolds. The reason why the deductions in (3.5) and (3.6) are true in their situationis due to the facts that nonnegative real (1 , c ( E, h )) are stronglynonnegative and the product of a strongly nonnegative form with a nonnegative form is stillnonnegative ([De12, p. 132]). Nevertheless, in their context the inequality (3.4) is no longertrue and so their is no respectively lower bound in their situation. Indeed we shall see inSection 5 that it is this lower bound that plays key roles in many related applications. ONNEGATIVE HERMITIAN VECTOR BUNDLES AND CHERN NUMBERS 7 Examples
In this section we shall illustrate by some examples that in many important situations theBott-Chern nonnegativity can be satisfied. Although these examples and facts are well-knownto experts, for the reader’s convenience, we still outline some necessary details and/or pointout some references.Recall that a holomorphic vector bundle E −→ M is called globally generated if theglobal holomorphic sections of E span the fiber over each point of M . If M is compactthen H ( M, E ), the complex vector space consisting of holomorphic sections of M , is finite-dimensional. Then the property of being globally generated implies that the following bundlesequence 0 −→ ker( ϕ ) −→ M × H ( M, E ) ϕ −→ E −→ x, s ) (cid:0) x, s ( x ) (cid:1) . (4.1)is exact. This means that a globally generated vector bundle over a compact complex manifoldcan be realized as a quotient bundle of a trivial vector bundle. Then the fact that any globally generated vector bundle over a compact complex manifoldadmits a Bott-Chern nonnegative Hermitian metric (4.2)follows from the following general result. Example 4.1.
If ( E r , h ) −→ M is a Hermitian vector bundle over a compact complex man-ifold M and S is an l -dimensional holomorphic subbundle of E ( l < r ) and thus Q := E/S is a ( r − l )-dimensional holomorphic quotient bundle of E , i.e., we have the following shortbundle exact sequence: 0 −→ S −→ E −→ Q −→ . The Hermitian metric h on E naturally induces a metric on S and on Q . Bott and Chernnoticed that ([BC65, p. 91]), under any unitary frame { e , . . . , e r } of E such that { e , . . . , e l } and { e l +1 , . . . , e r } are unitary frames of S and Q respectively, the curvature matrices of E and Q , denoted by Ω E and Ω Q , are related by (cf. [GH78, p. 79], [De12, p. 275])Ω E (cid:12)(cid:12) Q = Ω Q − A ∧ A t , where ( · ) (cid:12)(cid:12) Q denotes the restriction to Q and A is a matrix whose entries are (1 , S in E . This property is usuallycalled “ curvature increases in holomorphic quotient bundles ”. This means that if E can beequipped with a flat Hermitian metric as in (4.1), which implies that Ω E (cid:12)(cid:12) Q = 0, then thequotient bundle Q admits a Bott-Chern nonnegative Hermitian metric. Thus the fact (4.2)follows.Our principal concerns in this article are compact complex manifolds whose holomorphictangent or cotangent bundles, denoted for simplicity in the sequel by T and T ∗ respectively,are globally generated. To each kind of them there is a well-known result, which we record inthe following example for our later purpose. Example 4.2. (1) Recall that a compact complex manifold M is called homogeneous ifthe holomorphic automorphism group of M , denoted by Aut( M ), acts transitivelyon it. It is well-known that a compact connected complex manifold M has globally PING LI generated T M if and only if it is homogeneous . This particularly implies that the holo-morphic tangent bundles of Hermitian symmetric spaces, projective-rational manifoldsand tori are globally generated.(2) A compact connected K¨ahler manifold has globally generated T ∗ if and only if it canbe holomorphically immersed into some complex torus . This result should be due toMatsushima and Stoll ([MS73, p. 100] or [Ma92, p. 606]), at least to the author’sbest knowledge. The basic idea is to consider the Albanese map related to this K¨ahlermanifold. A more compact proof can be found in [Sm76, p. 271]. Note that the term“globally generated” was called “ample” and “weakly ample” respectively in [MS73]and [Sm76]. Also note that a compact complex manifold holomorphically immersedinto some torus is automatically K¨ahler as the restriction of the obvious K¨ahler metricon the complex torus is still K¨ahler. This implies that any compact connected complexmanifold holomorphically immersed into some complex torus has globally generated T ∗ .The notion of nefness of line/vector bundles over projective algebraic manifolds is well-known and has been extended to general compact complex manifolds by Demailly, whichcan be viewed as a limiting case of ampleness. For its definition and basic properties werefer to [De92, §
1] or [DPS94, § Example 4.3.
We have the following implications:(4.3) Bott-Chern nonnegativity = ⇒ Griffiths nonnegativity = ⇒ Nefness . Proof.
We show the first implication in (4.3). Under a local local coordinate ( z , . . . , z n ) , theentries Ω ij in the curvature matrix Ω with respect to some unitary frame field can be writtenas Ω ij = R ijp ¯ q d z p ∧ d z q . The Hermitian metric h on E is called Griffiths nonnegative ([Gr69, p. 181]) if, at any pointof M , we have X i,j,p,q R ijp ¯ q ξ j η p ξ i η q ≥ ξ = ( ξ , . . . , ξ r ) ∈ C r , and any η = ( η , . . . , η n ) ∈ C n . Now we assume that h ≥ BC ij ) = A ∧ A t , A = ( X p T ( p ) ij d z p ) . This implies that R ijp ¯ q = X k T ( p ) ik T ( q ) jk ONNEGATIVE HERMITIAN VECTOR BUNDLES AND CHERN NUMBERS 9 and thus X i,j,p,q R ijpq ξ j η p ξ i η q = X k,i,j,p,q T ( p ) ik T ( q ) jk ξ j η p ξ i η q = X k,i,j,p,q ( T ( p ) ik ξ i η p )( T ( q ) jk ξ j η q )= X k | X i,p T ( p ) ik ξ i η p | ≥ . (cid:3) Applications
In this section we give three kinds of related applications, whose main contents have beenbriefly described in the Introduction.5.1.
The first application.
Bott and Chern observed that ([BC65, p. 91]) the Chern formsof Bott-Chern nonnegative Hermitian vector bundles are all strongly nonnegative and thus itis immediate that their Chern numbers are all nonnegative.Our first direct application of our Theorem 3.2 yields the following improvements of theabove-mentioned nonnegativity by giving lower and upper bounds.
Theorem 5.1.
Suppose ( E r , h ) −→ M n is a Bott-Chern nonnegative Hermitian vector bun-dle. Then all their Chern numbers are nonnegative and bounded below and above by c n [ E ] and c n [ E ] : (5.1) 0 ≤ c n [ E ] ≤ c λ [ E ] ≤ c n [ E ] , ∀ λ ∈ Γ( n, r ) . This particularly holds for globally generated vector bundles. In particular, if the Chern number c n [ E ] = 0 , then all the Chern numbers vanish. Compact connected homogeneous complex manifolds have been well-studied and it is alsoa classical result that their Chern numbers are nonnegative ([Go54],[BR62],[GR62], [MS73]).The nonnegativity of the signed Chern numbers of compact connected complex manifoldsholomorphically immersed into complex tori has also been obtained by Matsushima and Stoll([MS73, Thm 6.5], [Ma92, p. 607]). So the following corollary improves the nonnegativityresults for Chern numbers/signed Chern numbers of these manifolds by giving lower and upperbounds respectively.
Corollary 5.2. (1) The Chern numbers of any compact connected homogeneous complexmanifold M n are nonnegative and bounded below and above by c n [ M ] and c n [ M ]:(5.2) 0 ≤ c n [ M ] ≤ c λ [ M ] ≤ c n [ M ] , ∀ λ ∈ Γ( n, n ) . (2) The signed Chern numbers of any compact connected complex manifold M n whoseholomorphic cotangent bundle is globally generated are nonnegative and boundedbelow and above by ( − n c n [ M ] and ( − n c n [ M ]:(5.3) 0 ≤ ( − n c n [ M ] ≤ ( − n c λ [ M ] ≤ ( − n c n [ M ] , ∀ λ ∈ Γ( n, n ) . This particularly holds for compact connected complex manifolds holomorphicallyimmersed into complex tori.In particular, in these two cases, if the Chern number c n [ M ] = 0, then all the Chernnumbers of M vanish. The second application.
Now we come to our second application. First let us recallsome more notation. For a compact connected complex manifold M n , the algebraic dimension of M , denoted by a ( M ), is defined to be the maximal number of meromorphic functionson M that can be algebraically independent. In other words, a ( M ) is the transcendentaldegree of function field of M over C . a ( M ) satisfies 0 ≤ a ( M n ) ≤ n . A compact complexmanifold M n is called Moishezon if the algebraic dimension a ( M n ) = n , in which case it wasinvestigated in detail by Moishezon ([Mo66]). We refer the reader to [MM07, Ch. 2] for adetailed presentation of the materials related to Moishezon manifolds. The starting point ofour second application is to recall several basic results due to Matsushima-Stoll, Siu-Demaillyand Demailly respectively.A major result of Matsushima-Stoll in [MS73] says that ([MS73, Thms 5,5, 5.6] or [Ma92,p. 600]) if a globally generated vector bundle E r −→ M n satisfies 0 = c λ ( E ) ∈ H i ( M, Z ) forsome λ ∈ Γ( i, r ), then the algebraic dimension a ( M ) ≥ i. In particular, if some Chern numberof a globally generated vector bundle over M is nonzero, then M is Moishezon. This result,together with Moishezon’s result that (cf. [MM07, p. 95])(5.4) a Moishezon manifold is K¨ahler if and only if it is projective algebraic, leads to another main theorem of [MS73] ([MS73, p. 94, Main Thm] or [Ma92, p. 600])saying that if some Chern number of a globally generated vector bundle over a compact K¨ahlermanifold is nonzero, then this manifold is projective algebraic .A basic result due to Siu-Demailly (cf. [MM07, p. 96]) solving the Grauert-Riemenschneiderconjecture gives another effective sufficient condition for a compact connected complex man-ifold to be Moishezon, which says a compact connected complex manifold M is Moishezon provided that there exists aquasi-positive element in H , ( M ; Z ) := H ( M ; Z ) ∩ H , ( M ; R ) . (5.5)Here an element in H , ( M ; Z ) is called quasi-positive if there exists a real (1 , a Moishezon manifold with nef tangent bundle is projective algebraic. Our strating observation in this second application is to give a new proof of the above-mentioned result due to Matsushima-Stoll by combining our Prop. 3.1 and Thm. 5.1 withSiu-Demailly’s result (5.5).
Theorem 5.3 (Matsushima-Stoll) . Suppose E r −→ M n is a globally generated vector bundleover a compact complex (resp. K¨ahler) manifold . If some Chern number of E −→ M isnonzero, then M is a Moishezon (resp. projective algebraic) manifold.Proof. By (4.2) there exists a Hermitian metric h on E such that h ≥ BC
0. Therefore Prop.3.1 says that the first Chern form c ( E, h ) is nonnegative as a real (1 , M is nonzero, then the Chern number c n [ E ] = Z M c ( E, h ) n > c ( E, h ) must be positive somewhere and hence the first Chernclass [ c ( E, h )] ∈ H , ( M ; Z ) is quasi-positive. So the desired conclusion follows from theSiu-Demailly criterion (5.5). (cid:3) ONNEGATIVE HERMITIAN VECTOR BUNDLES AND CHERN NUMBERS 11
Compact connected homogeneous complex manifolds have been well-studied and a well-known related fact, which should be due to H.C. Wang, states that (cf. [MS73, Thm 6.2]or [Ma92, p. 601]) they are projective algebraic provided that their Euler characteristic arenonzero. Our following result is a refinement of this classical fact, and, moreover its proof isbuilt on the aforementioned two basic results due to Matsushima-Stoll and Demailly and thusdifferent from the original one.
Theorem 5.4.
Suppose M n is a compact connected homogeneous complex manifold. Then M is a projective algebraic manifold if some Chern number of M is nonzero.Proof. If some Chern number is nonzerothen M is Moishezon due to Theorem 5.3 Now theproperty of being globally generated on T implies the existence of a Bott-Chern nonnegativeHermitian metric on T and thus implies nefness by (4.3). Then the fact of M being projectivealgebraic follows from Demailly’s result (5.6). (cid:3) Note that in Theorem 5.4 the condition of some Chern number be nonzero is only sufficientand its converse part is not true in general, i.e., even if a compact connected homogeneouscomplex manifold is projective algebraic, it may happen that its all Chern numbers vanish.For instance, the product of an abelian variety over C and a projective-rational manifold isprojective algebraic but its Chern numbers are all zero. Indeed, a classical result of Borel andRemmert tells us that ([BR62]) any compact connected homogeneous K¨ahler manifold is the product of a complextorus and a projective-rational manifold. (5.7)This result (5.7) implies that the condition of some Chern number being nonzero is alsonecessary in Theorem 5.4 if we further impose simple-connectedness on the manifolds inquestion and thus we have the following result. Theorem 5.5.
Suppose M n is a simply-connected compact connected homogeneous complexmanifold. Then the following four conditions are equivalent: (1) The Chern number c n [ M ] = 0 . (2) Some Chern number of M is nonzero. (3) M is a projective algebraic manifold. (4) M is a K¨ahler manifold. Remark 5.6. (1) The manifolds considered in Theorem 5.5 were called C -spaces byWang ([Wa54]) and investigated in detail by many authors ([BR62], [Go54], [GR62],[Wa54] etc.). However, the contents in our Theorem 5.5 should be completely new, atleast to the author’s best knowledge.(2) There are homogeneous complex structures on the product of odd-dimensional spheres: M p,q := S p +1 × S q +1 (cid:0) ( p, q ) = (0 , (cid:1) , which are called Calabi-Eckmann manifolds([CE53]) and whose Chern numbers are all zero. M p,q are indeed non-K¨ahler as itssecond Betti number is zero and so these examples match Theorem 5.5 very well.5.3. The third application.
Our third application is to give an in-depth investigation on thestructure of compact connected complex manifolds holomorphically immersed into complextori in detail. As we have seen in Example 4.2, these manifolds are precisely those compactconnected K¨ahler manifolds whose holomorphic cotangent bundles are globally generated.The structure of these manifolds was first investigated by Matsushima and his coauthors ([HM75], [Ma74], [MS73]), Yau ([Ya74, Chapter 3]) and Smyth ([Sm76]) etc around the sametime. Later in [Zha97] Zhang related the ampleness of the canonical line bundles of thesemanifolds to their Todd genera.Before continuing, let us digress to recall the notions of
Kodaira dimension κ ( M ) for com-pact complex manifolds M ([Zhe00, p. 132]), which has several equivalent definitions. Herefor our later purpose we adopt the following one. Denote by K M the canonical line bundle of M . It turns out that dim C H ( M, K ⊗ mM ), the complex dimension of the holomorphic sectionsof K ⊗ mM , has the following property: either H ( M, K ⊗ mM ) = 0 for all m ≥ ≤ κ ( M ) ≤ dim C ( M ) and constants 0 ≤ C < C such that(5.8) C m κ ( M ) ≤ dim C H ( M, K ⊗ mM ) ≤ C m κ ( M ) , m >> , which means dim C H ( M, K ⊗ mM ) grows at a rate of m κ ( M ) . In the former case κ ( M ) := −∞ . M is called of general type if κ ( · ) attains its maximum: κ ( M ) = dim C ( M ). It is an elementaryfact between the Kodaira dimension κ ( M ) and the algebraic dimension a ( M ) that ([Zhe00,p. 135]):(5.9) κ ( M ) ≤ a ( M ) ≤ dim C ( M ) . Denote by Aut ( M ) the identity component of the holomorphic automorphism group ofa compact complex manifold M , which is a connected complex Lie group. The most fun-damental structure of compact connected complex manifolds holomorphically immersed intocomplex tori is the following result. Theorem 5.7. ([Ma74, Prop. 1] , [Ya74, Th. 5] , [Sm76, Th. 1]) Suppose M n is a com-pact connected complex manifold holomorphically immersed into some complex torus withAut ( M ) = { } . Then Aut ( M ) is a complex torus and acts freely on M . Moreover, the quo-tient manifold N := M/ Aut ( M ) is also a compact connected complex manifold and can beholomorphically immersed into some complex torus and Aut ( N ) = { } . Consequently, M isa holomorphic principal complex torus bundle over some compact connected complex manifold N also holomorphically immersed into some complex torus with Aut ( N ) = { } . This structure theorem has several applications in [Ma74], [Ya74] and [Sm76]. Let us recordseveral of them and some results in [Zha97] related to our next result in the following theorem.
Theorem 5.8.
Suppose M n is a compact connected complex manifold holomorphically im-mersed into some complex torus. Then (1) ([Ya74, p. 238]) M is a (possibly trivial) torus bundle over a compact K¨ahler manifoldof general type.(2) ([Sm76, p. 278]) The following three conditions are equivalent:(5.10) Aut ( M ) = { } ⇐⇒ c n [ M ] = 0 ⇐⇒ c n [ M ] = 0 . (3) ([Zha97, Thm 3, Coro. 3]) The following three conditions are equivalent:(5.11) K M is ample ⇐⇒ ( − n td( M ) > ⇐⇒ ( − n c n [ M ] > , where td( M ) is the Todd genus of M , which by definition is the alternating sum ofthe Hodge numbers td( M ) = X q ( − q h ,q ( M )and the Hirzebruch-Riemann-Roch theorem says that it is a ratinally linear combina-tion of Chern numbers. ONNEGATIVE HERMITIAN VECTOR BUNDLES AND CHERN NUMBERS 13
We can now give our third application, which relates Aut ( M ), all the Chern numbers of M ,the Kodaira dimension κ ( M ), the canonical line bundle K M , and its torus bundle structureand particularly improves various results summarized in Theorem 5.8. Theorem 5.9.
Suppose M n is a compact connected complex manifold holomorphically im-mersed into some complex torus, or equivalently, M n is a compact connected K¨ahler manifoldwith globally generated holomorphic cotangent bundle. Then the following six conditions areequivalent: (1) Aut ( M ) = { } , (2) all the signed Chern numbers ( − n c λ [ M ] are strictly positive, (3) M is of general type, (4) the canonical line bundle K M is ample, (5) the signed Todd genus is positive: ( − n td ( M ) > , (6) M cannot be realized as a total space of some nontrivial torus bundle;and the following six conditions are equivalent: (7) Aut ( M ) = { } , (8) all the Chern numbers of M vanish, (9) the Kodaira dimension κ ( M ) < n , (10) the canonical line bundle K M is not ample, (11) the Todd genus td ( M ) vanishes, (12) M is a nontrivial holomorphic principal complex torus bundle over a compact con-nected K¨ahler manifold with ample canonical line bundle.Moreover, under ( any one of ) the first six equivalent conditions, M is projective algebraic. A direct corollary of Theorem 5.9 is the following result, which tells us that whether or notthe Chern numbers and Todd genus vanish is simultaneous.
Corollary 5.10.
Suppose M n is a compact connected complex manifold holomorphicallyimmersed into some complex torus. Then(1) if the Todd genus td( M ) = 0, then all the Chern numbers vanish;(2) if the Todd genus td( M ) = 0, then it has sign ( − n and all the Chern numbers arenonzero and have the same sign ( − n . Proof of Theorem 5.9.Proof.
First we show that the condition (2) implies that M is projective algebraic and thusprove the last conclusion in Theorem 5.9. Indeed, Theorem 5.3 says that M is Moishezon.Then M being projective algebraic follows from (5.4) and the fact that M be K¨ahler.“(1) ⇔ (2)” follows from (5.10) and (5.3).“(2) ⇔ (6)”: If M can be realized as a total space of some nontrival torus bundle, then c n [ M ] = 0 as the Euler characteristic is multiplicative for fiber bundles, which implies that“(2) ⇒ (6)”. “(6) ⇒ (1)” follows from Theorem 5.7.“(2) ⇒ (3)”: The proof is a combination of a Kodaira-type vanishing theorem and theHirzebruch-Riemann-Roch theorem. By definition we can choose a Hermitian metric h on T ∗ M such that ( h, T ∗ M ) ≥ BC
0. Then (3.1) tells us that c ( T ∗ M , h ) ≥
0. This means that the Chernclass(5.12) c ( K ⊗ mM ) = mc ( K M ) = m [ c ( T ∗ M , h )] ≥ , m ≥ . Condition (2) implies that(5.13) Z M (cid:0) c ( K ⊗ mM ) (cid:1) n = m n ( − n c n [ M ] > , m ≥ . We have shown above that M is projective algebraic under the condition (2). Then a com-bination of (5.12), (5.13), M being projective algebraic and the Kodaira-Kawamata-Viehwegvanishing theorem for projective algebraic manifolds ([Ko87, p. 74, (3.10)]) yields(5.14) H q ( M, K ⊗ mM ) = 0 , q ≥ , m ≥ . Therefore the Hirzebruch-Riemann-Roch theorem tells us that ([Hi66])dim C H ( M, K ⊗ mM ) = n X q =0 ( − q dim C H q ( M, K ⊗ mM ) (cid:0) (5 . (cid:1) = Z M (cid:2) Td( M ) · Ch( K ⊗ mM ) (cid:3) (Td: Todd class, Ch: Chern character)= Z M n(cid:2) c ( M ) + · · · (cid:3) · exp (cid:2) − mc ( M ) (cid:3)o = ( − n c n [ M ] n ! m n + O ( m n − ) , ( m >> − n c n [ M ] > κ ( M ) = n , i.e., M is of general type.“(3) ⇒ (4)”: This follows from a result of Ran ([Ra84, p. 176, Coro. 3]), which says thata compact connected complex manifold of general type holomorphically immersed into somecomplex torus must have ample canonical line bundle.“(4) ⇔ (5)”: this follows from (5.11).“(4) ⇒ (2)”: First note that the ampleness of K M implies that M is projective algebraic. Ifon the contrary the Euler characteristic c n [ M ] = 0, then a result of Howard and Matsushima([HM75, Th. 6] or [Ma92, p. 655]) says that M admits a holomorphic one-form whose zero-locus is empty. However, another major result of Zhang in [Zha97, Thm 1] tells us thatthe zero-locus of any holomorphic one-form on a projective algebraic manifold with amplecanonical line bundle is non-empty, which leads to a contradiction. This means that c n [ M ] = 0and thus proves condition (2) by (5.3).“(7) ⇔ (8)” follows from (5.3) and (5.10).“(8) ⇔ (11)”: Note that the Todd genus td( M ) is a rationally linear combination of Chernnumbers and thus “(8) ⇒ (11)”. “(11) ⇒ (8)” follows from “(2) ⇔ (5)”.“(7) ⇒ (12)”: Theorem 5.7 tells us that, under the condition Aut ( M ) = { } , M isa nontrivial holomorphic principal complex torus bundle over a compact connected com-plex manifold N , where N can also be holomrophically immersed some complex torus withAut ( N ) = { } . Then “(1) ⇔ (4)” implies that K N is ample. ONNEGATIVE HERMITIAN VECTOR BUNDLES AND CHERN NUMBERS 15 “(12) ⇒ (7)”: Condition (12) implies that the Euler characteristic c n [ M ] = 0 and thusAut ( M ) = { } follows from “(1) ⇔ (2)”.Other equivalent relations among (7) − (12) are direct via those among (1) − (6). (cid:3) Remark 5.11.
A recent result of Popa and Schnell ([PS14]) refined [Zha97, Thm 1] byshowing that the zero-locus of any holomorphic one-form on a projective algebraic manifoldof general type is non-empty. Thus we can also apply it to deduce “(3) ⇒ (2)”.6. Proof of Proposition 3.1
In this last section we shall prove Proposition 3.1 and complete this article.Let us begin by recalling some well-known facts about GL r ( C )-invariant polynomial func-tions. In what follows GL r ( C ) and M r ( C ) denote respectively the general linear group of order r and the r × r matrix group, both over C . A map f : M r ( C ) → C is called a GL r ( C ) -invariantpolynomial function of homogeneous degree i if f can be written as(6.1) f (cid:0) A = ( T αβ ) (cid:1) = X ≤ α j ,β j ≤ r p α ··· α i ,β ··· β i T α β · · · T α i β i for p α ··· α i ,β ··· β i ∈ C and satisfies(6.2) f ( BAB − ) = f ( A ) , ∀ A ∈ M r ( C ) , ∀ B ∈ GL r ( C ) . Denote by I i the complex linear space consisting of all GL r ( C )-invariant polynomial func-tion of homogeneous degree i : I i := { f : M r ( C ) → C | f satisfy (6.1) and (6.2) } and the graded ring I := M i ≥ I i . If we set det( tI r + A ) =: r X i =0 c i ( A ) · t r − i , then these c i ( · ) are GL r ( C )-invariant polynomial functions of homogeneous degree i . Inparticular, c ( A ) = 1 , c ( A ) = trace( A ) and c n ( A ) = det( A ). It is well-known that the gradedring I is multiplicatively generated by c , . . . , c r , i.e., I = C [ c , · · · , c r ] . Now if ( E r , h ) −→ M is a Hermitian holomorphic vector bundle and (Ω ij ) is the curvaturematrix of its Chern connection, then (cid:0) recall (2.1) and (2.2) (cid:1) c i (cid:0) √− π (Ω ij ) (cid:1) =: c i ( E, h ) , (0 ≤ i ≤ r )are globally defined, closed, real-valued ( i, i )-forms over M and represent the i -th Chern classesof E −→ M . Thus we have a natural graded ring homomorphism ϕ sending c i to c i ( E, h ): I = M i ≥ I i = C [ c , · · · , c r ] ϕ −−→ C [ c ( E, h ) , · · · , c r ( E, h )] f f (cid:0) √− π (Ω ij ) (cid:1) . (6.3) Griffiths first showed that ([Gr69, p. 242, (5.6)]) any f ∈ I i can be written, which may notbe unique, in the form f ( A = (cid:0) T ij ) (cid:1) = X π,τ ∈ S i ,ρ =( ρ ,...,ρ i ) ∈ [1 ,r ] i p ρ,π,τ T ρ π (1) ρ τ (1) · · · T ρ π ( i ) ρ τ ( i ) , p ρ,π,τ ∈ C , (6.4)where S i denotes the permutation group on i objects.The following definition was introduced by Griffiths ([Gr69, p. 242, (5.9)]). Definition 6.1 (Griffiths) . A polynomial function f ∈ I i is called Griffiths positive if it canbe expressed in the form (6.4) with(6.5) p ρ,π,τ = X j ∈ I λ ρ,j q ρ,j,π ¯ q ρ,j,τ , ∀ ρ, π, τ, for some real numbers λ ρ,j ≥
0, some complex numbers q ρ,j,π and some finite set J . Denoteby Π i := { f ∈ I i | f are Griffiths positive } . The following key fact relating Griffiths positive polynomials and Schur polynomials wasobserved in [FL83, p. 54, Prop. A.3], whose proof is built on the representation theory and apreviously related result in [UT77].
Proposition 6.2 (Fulton-Lazarsfeld) . (6.6) n X λ ∈ Γ( i,r ) a λ S λ ( c , . . . , c r ) (cid:12)(cid:12) all a λ ≥ o = Π i . Now we are in the position to prove Proposition 3.1.
Proof.
Assume that the Hermitian holomorphic vector bundle ( E r , h ) −→ M n satisfies h ≥ BC
0, i.e., under the local coordinates ( z , . . . , z n ), the curvature matrix (Ω ij ) with respect tosome unitary frame field can be written in the following form(Ω ij ) = A ∧ A t , A = ( X p T ( p ) ij d z p ) . This implies that(6.7) Ω ij = X k,p,q T ( p ) ik T ( q ) jk d z p ∧ d z q . ONNEGATIVE HERMITIAN VECTOR BUNDLES AND CHERN NUMBERS 17
Therefore, for each λ ∈ Γ( i, r ) and 1 ≤ i ≤ n , we have S λ (cid:0) c ( E, h ) , . . . , c r ( E, h ) (cid:1) = ϕ (cid:0) S λ ( c , . . . , c r ) (cid:1) (cid:0) (6 . (cid:1) = S λ (cid:16) c (cid:0) √− π (Ω ij ) (cid:1) , . . . , c r ( (cid:0) √− π (Ω ij ) (cid:1)(cid:17) (cid:0) (6 . (cid:1) =( √− π ) i X λ ρ,j q ρ,j,π ¯ q ρ,j,τ Ω ρ π (1) ρ τ (1) · · · Ω ρ π ( i ) ρ τ ( i ) (cid:0) (6 . , (6 . , (6 . (cid:1) =( √− π ) i X λ ρ,j q ρ,j,π ¯ q ρ,j,τ T ( r ) ρ π (1) k T ( s ) ρ τ (1) k d z r ∧ d z s · · · T ( r i ) ρ π ( i ) k i T ( s i ) ρ τ ( i ) k i d z r i ∧ d z s i (cid:0) (6 . (cid:1) =( √− π ) i ( − i ( i − X ρ,j λ ρ,j n(cid:2) X π,k,r q ρ,j,π T ( r ) ρ π (1) k · · · T ( r i ) ρ π ( i ) k i d z r ∧ · · · ∧ d z r i (cid:3) ∧ (cid:2) X τ,k,s q ρ,j,τ T ( s ) ρ τ (1) k · · · T ( s i ) ρ τ ( i ) k i d z s ∧ · · · ∧ d z s i (cid:3)o =: ( √− i (2 π ) i X ρ,j λ ρ,j ψ ρ,j ∧ ψ ρ,j . Now ψ ρ,j are ( i, λ ρ,j ≥ (cid:0) recall (6.5) (cid:1) , which means that thelast expression is a strongly nonnegative ( i, i )-form. This gives the desired proof and thuscompletes this article. (cid:3) References [BG71] S. Bloch, D. Gieseker:
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