THE FUNDAMENTAL GROUP, RATIONAL CONNECTEDNESS AND

THE POSITIVITY OF KAHLER MANIFOLDS

LEI NI

Abstract. First a conjecture asserting that any compact Kahler manifold N withRic⊥ > 0 must be simply-connected is confirmed by adapting the comass of (p, 0)-formsinto a maximum principle via the viscosity consideration. Secondly the projectivity andthe rational connectedness of a Kahler manifold of complex dimension n under the con-

dition Rick > 0 (for some k ∈ 1, · · · , n) is proved, generalizing the previous resultof Campana, and Kollar-Miyaoka-Mori independently, for the Fano manifolds. Thirdlywe show that under the assumption of Picard number one a manifold with Ric⊥ > 0 is

Fano. Thirdly via a new curvature notion motivated by Ric⊥ and the classical work ofCalabi-Vesentini, the cohomology vanishing Hq(N,T ′N) = 0 for any 1 ≤ q ≤ n (aswell as a deformation rigidity result) for classical Kahler C-spaces with b2 = 1 is proved.Besides possible curvature characterization of homogenous Kahler manifolds this new

curvature (which is quadratic in terms of linear maps from T ′N to T ′′N) leads to arelated notion of Ricci curvature, Ric+. We also show that a compact Kahler manifoldwith Ric+ > 0 is projective and simply-connected.

1. Introduction

Kahler manifolds bridge the complex manifolds and complex algebraic manifolds, in par-ticular availing analytic and geometric techniques in the study of algebraic manifolds. Theprojectivity (namely the holomorphic embedding of the manifold into a complex projectivespace) of high dimensional Kahler manifolds was originated by Kodaira [21]. The celebratedtheorem of Kodaira asserts that a compact Kahler manifold is projective if it admits an in-tegral Kahler form. This condition on the Kahler form can be satisfied if the Chern-formof a positive holomoporhic line bundle L over the concerned manifold M is positive. Acanonical way of associating a line bundle to a complex manifold M is via its canonical andanti-canonical line bundles. The associated intrinsic curvature is the Ricci curvature ofthe manifold. The compact Kahler manifolds with positive Ricci curvature form a specialclass of smooth algebraic varieties, namely the Fano manifolds, whose study has been oneof central and active focus in last decades.

There also exists at least two well known curvature notions of positivity for Kahler mani-folds. The first one is the bisectional curvature R(X,X, Y, Y ) for any (1, 0) vectors X,Y ∈T ′M (the holomorphic tangent bundle), where R is the curvature tensor of M . (For Her-mitian manifolds there are well known Griffiths’ positivity [14] and Nakano’s positivity [29].The former is the same as the bisectional curvature positivity for Kahler manifolds, the later

2010 Mathematics Subject Classification. 53C55, 53C44, 53C30.Key words and phrases. Positivity of Kahler manifolds, Simply-connectedness, Rational connectedness,

Kahler C-spaces, Orthogonal Ricci, Rick, Cross quadratic curvature.The research is partially supported by “Capacity Building for Sci-Tech Innovation-Fundamental Research

Funds”.

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is same as the curvature operator acting on the holomorphic tangents. On various positivitynotions in algebraic geometry there exist the wonderful books [24].) However the positivity(even the nonnegativity) of bisectional curvature is rather restrictive. It was proved byMori [27], and subsequently Siu-Yau [39] (with mostly Kahler geometric method) that theonly compact Kahler manifold with positive bisectional curvature is the complex projectivespace. In fact Mori’s theorem solves one of Hartshorne’s conjectures and asserts that theampleness of the tangent bundle implies the manifold being a projective space. The secondwell-known notion is the holomorphic sectional curvature, H(x) = R(X,X,X,X), whichmeasures the sectional curvature of complex lines in the holomorphic tangent bundle. Theprojectivity of compact Kahler manifolds with positive holomorphic sectional curvature wasonly confirmed very recently [46]. Beyond this (and the rational connectedness and simply-connectedness described below) not much has been known for the Kahler manifolds withpositive holomorphic sectional curvature unless very strong pinching condition is imposed.

Rational connectedness is a very useful property for algebraic manifolds [11]. For compactKahler manifolds with positive Ricci curvature this property was established by Campana[6], Kollar-Miyaoka-Mori [23] a while ago. For the algebraic manifolds with positive holo-morphic sectional curvature it was only proved recently by Heier-Wong [15]. One result ofthis paper is to prove the projectivity and rational connectedness under any one of inter-polating positivities, namely Rick > 0 for any 1 ≤ k ≤ n, of the Kahler manifolds. Thepositivity of Rick measures the Ricci curvature of the k-dimensional holomorphic subspacesof the holomorphic tangent bundle. Hence it coincides with the holomorphic sectional cur-vature for k = 1, and with the Ricci curvature for k = n, the complex dimension of themanifold. The condition is significantly different from the Riemannian analogue, namelythe so-called q-Ricci (see next section for details) since unlike their Riemannian analogues,Rick > 0 do not monotonely become weaker as k increases. The notion of Rick was ini-tiated in the recent study of the k-hyperbolicity of a compact complex manifold [30] and itclosely related to the degeneracy of holomorphic mappings from Ck into concerned mani-folds. The condition Rick > 0 allows some semi-positivity (even negativity) of holomorphicsectional curvature if k > 1. The detailed definition (and statement of theorem) can befound in the next section. The proof of our result is much different from the proof of theprevious special cases in [6, 23, 15]. The proof here is built upon some recent techniquesof author applying the partial maximum principle via the viscosity consideration developedin [30, 34], as well as the proof for the next result of this paper described in the followingparagraphs. The method first employs the second variation consideration to obtain desiredestimates in the fiber direction of the flags in the tangent bundle. Then these estimates areused in the Bochner type consideration, but only to partial tangent directions instead oftaking the trace. It is in the proof of the projectivity, we have to use the co-mass to localizethe problem via a viscosity consideration.

There exists a big literature on the study (cf. [2]) of the topology and the fundamentalgroup of Kahler manifolds. A result of Kobayashi [19] asserts that a compact Kahler man-ifold with Ric > 0 must be simply-connected. Same conclusion was proved by Tsukamoto[41] for compact Kahler manifold with positive holomorphic sectional curvature. It wasknown via examples of Hitchin [17] (see also [1, 33]) that the two conditions are indepen-dent for complex dimension greater than one. The second purpose of this paper is to studythe fundamental group under another curvature positivity, namely Ric⊥ > 0.

THE FUNDAMENTAL GROUP AND THE POSITIVITY 3

In [33], motivated by the Laplace comparison theorem and the holomorphic Hessian com-

parison theorem, orthogonal Ricci curvature Ric⊥, which is defined by

Ric⊥(X,X) + Ric(X,X)−H(X)/|X|2

for any type (1, 0) tangent vector X, was studied. For a compact Kahler manifold Nn

(n = dimC(N)), with Ric⊥ > 0 everywhere, it was shown in [33] that the manifold isprojective, via a less known normal form for (1, 1)-forms1. It was also proved in [33] that

the manifold has finite π1(N). Further studies of compact Kaher manifolds with Ric⊥ >0 were carried in a recent work [32]. Besides a complete classification for threefolds, apartial classification for fourfolds, a Frankel type result were also obtained for compactKahler manifolds with Ric⊥ > 0 in [32]. Many examples were constructed in [33, 32]

illustrating that Ric⊥, H, and Ric are completely independent except the trivial relationRic(X,X) = Ric⊥(X,X)+H(X)/|X|2. The next result of this paper is to prove an analogue

of Kobayashi’s and Tsukamoto’s theorem for manifolds with Ric⊥ > 0, namely the simply-connectedness of compact Kahler manifolds with Ric⊥ > 0.

As explained in [33, 32], Ric⊥(X,X) does not come from a Hermitian symmetric sesquilin-ear form, and can be viewed as the holomorphic sectional curvature of a Bochner curvatureoperator (namely the curvature operator which arises in the standard Bochner formula com-puting the Laplacian of the square of the norm of two forms). Despite this close connectionwith the holomorphic sectional curvature, our proof follows the general scheme of Kobayashiby reducing it, via a Riemann-Roch-Hirzebruch formula, to a vanishing theorem of Hodgenumbers. However the proof of the needed vanishing theorem (which perhaps has its own in-terest) requires a new idea/technique introduced in this paper. (The situation of Kobayashi[19] is quite different since then the vanishing theorem needed under the positivity of Riccihad been known before due to Kodaira’s earlier work.) The method of proving this newvanishing theorem, namely Theorem 2.2, plays a very important role in the proof of therational connectedness result mentioned above.

The study of Ric⊥ > 0 in [35, 32] is motivated by the so-called generalized Hartshorneconjecture (cf. Conjectures 11.1, 11.2 of [8] and Conjecture 8.23 of [48]): A Fano manifoldhas nef tangent bundle if and only if it is a Kahler C-space. The natural curvature notionassociated with the nefness condition perhaps is the so-called almost nonnegative curvature.It has been proved recently however that the almost nonnegative bisectional curvature [3](even the almost nonnegative orthogonal bisectional curvature [25]) has very strong topo-logical restrictions/consequences if the volume is noncollapsing (namely they all imply thatthe manifolds are essentially Hermitian symmetric spaces, at least differential topologically).About ten years ago, a curvature positivity notion, namely the quadratic orthogonal bisec-tional curvature (again see next section for the definition) was proposed by Wu-Yau-Zheng[44] for the purpose of providing a curvature characterization of the Kahler C-spaces. Thisnotion perhaps is a bit obscure at its first appearance. But more importantly, as shown byChau-Tam [10] it is off the target since only for about eighty percentage of classical KahlerC-spaces (with b2 = 1) the canonical Kahler-Einstein metric has this curvature positive,while for the rest twenty percent this curvature (of the canonical metric) is negative some-

where. Hence a step back, the positivity of Ric⊥ was studied in [35] for the purpose of thecurvature characterization of C-spaces instead. However, except for dimension n = 2, 3,(perhaps even n = 4 if we are optimistic [32]), Ric⊥ > 0 appears a bit too weak for this

1This perhaps explains, to some degree, the fact that since it was first introduced in [26], Ric⊥ remainsdormant for more than fifteen years until the recent works [33], [32], [35].

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purpose. This is particularly so in view of the flexible construction of metrics with Ric⊥ > 0via the fiberation in [32]). On the other hand it was verified there that all the classical

C-spaces with b2 = 1 the canonical Kahler-Einstein metrics satisfy Ric⊥ > 0.

As the third positivity of Kahler manifolds, motivated by the classical work of Calabi-Vesentini [5] we introduce in this paper two stronger (than Ric⊥) notions of curvaturepositivity, namely the cross quadratic bisectional curvature (abbreviated as CQB> 0) andits dual dCQB. They are perhaps equally obscure at the first encountering as the quadraticorthogonal bisectional curvature. However, the positivity of CQB and its dual can beverified (has been verified in this paper) for all classical C-spaces with b2 = 1 (cf. [36]for the nonnegativity of CQB and positivity of dCQB for Type A Kahler C-spaces witharbitrarily large b2). Some initial studies of these two notions of curvature are carried out in

this paper as well. In particular, since CQB> 0 implies the Ric⊥ > 0, the projectivity andsimply-connectedness follows by the earlier results of [33] and results of this paper describedabove. Furthermore a deformation rigidity result for manifolds with positive dCQB is provedhere. Hence there is a good chance that one of these two curvature notions will serve thepurpose of the curvature characterization the Kahler C-spaces. Utilizing the Kahler-Ricciflow further study of CQB, dCQB, including the Fanoness under the assumption of CQB≥ 0and finiteness of the fundamental group, has been carried out in the sequel [36].

It is also proved in this paper that any compact Kahler manifolds with Ric⊥ > 0 and ofPicard number one must be Fano, although the general case remains open. Since Ric⊥ > 0allows b2 arbitrarily large (cf. examples in [36]), the implications of Ric⊥ > 0 on thedimension of certain harmonic (1, 1)-forms is proved as well.

It is our hope that this paper serves an introduction and an advocate to three notions ofpositivity, namely Rick, Ric

⊥, CQB, as well as their dual Ric+, dCQB, for Kahler manifolds.One can find many questions/open problems, and examples, in later sections of this paper(see also the survey [35] and the sequel [36]).

2. Definitions and statements of results

The following conjecture was proposed in Conjecture 1.6 of [33].

Conjecture 2.1. Let Nn (n ≥ 2) be a compact Kahler manifold with Ric⊥ > 0 everywhere.Then for any 1 ≤ p ≤ n, there is no non-trivial global holomorphic p-form, namely, theHodge number hp,0 = 0. In particular, Nn is simply-connected.

The conjecture was confirmed for n = 2, 3, 4 in [33] following a general scheme of Kobayashi.The “in particular” part, namely the simply-connectedness of compact Kahler manifolds,would follow from Hirzebruch’s Riemann-Roch formula [16] (noting that by Theorem 1.7 of[33] N is algebraic) as follows: First the Euler characteristic number

χ(ON ) = 1− h1,0 + h2,0 − · · ·+ (−1)nhn,0

where ON is the structure sheaf, satisfies that χ(ON ) = gχ(ON ) by the Riemann-Roch-

Hirzebruch formula, if N is a finite g-sheets covering of N . On the other hand, the vanishingof all Hodge numbers hp,0 for 1 ≤ p ≤ n (which is the main part of the conjecture) asserts

that χ(ON ) = 1 for both N and N , if N is compact and of Ric⊥ > 0 (hence projective),which forces g = 1. The assertion that Nn must be simply-connected (cf. [19]) follows from

that the universal cover N of N is of Ric⊥ > δ > 0, hence is compact and projective by

THE FUNDAMENTAL GROUP AND THE POSITIVITY 5

Theorem 3.2 of [33]. This argument was used in [19] proving the simply-connectedness of aFano manifold. It was also used in [33] for the special case n = 2, 3, 4.

In this paper we prove the above conjecture for all n ≥ 2. In fact we prove a strongerresult which asserts the vanishing of hp,0 under a weaker curvature condition related to p.This condition was first introduced in Section 4 of [33], which we recall below.

Motivated by [34], for any k-subspace Σ ⊂ T ′xN , we define

S⊥k (x,Σ) + k

∫Z∈Σ,|Z|=1

Ric⊥(Z,Z) dθ(Z) (2.1)

where∫f(Z) dθ(Z) denotes 1

V ol(S2k−1)

∫S2k−1 f(Z) dθ(Z). The S⊥

k (x,Σ) interpolate be-

tween Ric⊥(X,X) and n−1n+1S(x) (see Lemma 5.1). We say S⊥

k (x) > 0 if for any k-subspace

Σ ⊂ T ′xM , S⊥

k (x,Σ) > 0. It is easy to see that S⊥l > 0 implies S⊥

k > 0 for k ≥ l. And it isnot hard to prove that

S⊥k (x,Σ) =

(Ric(E1, E1) + Ric(E2, E2) + · · ·+Ric(Ek, Ek)

)− 2

(k + 1)Sk(x,Σ).

Here Sk(x,Σ) is the k-scalar curvature defined in [34] (namely taking the average of holomor-

phic sectional curvature instead of Ric⊥ over the unit sphere of Σ in (2.1)). The collectionof k-scalar curvatures Sk(x,Σ) for k = 1, · · · , n, interpolates between the holomorphicsectional curvature H(X) and the scalar curvature S(x). The above equation in particularimplies that S⊥

n (x) = n−1n+1S(x). The relation (2.1) suggested a question: whether or not

S⊥k (x) > 0 implies hp,0 = 0 for p ≥ k. The first theorem of this paper answers this question

affirmatively, which implies Conjecture 2.1 since Ric⊥ > 0 implies that S⊥k > 0 for any

1 ≤ k ≤ n.

Theorem 2.2. Let (N, g) be a compact Kahler manifolds such that S⊥k (x) > 0 for any

x ∈ N . Then hp,0 = 0 for any p ≥ k. In particular, if Ric⊥ > 0, hp,0(N) = 0 for all1 ≤ p ≤ n, and N is simply-connected.

The case p = 2 was proved in Section 4 of [33]. The proof here is motivated by anidea of [30] in proving a new Schwarz Lemma by the author. We recall that idea firstbefore explaining the related idea here. Starting from the work of Ahlfors, the SchwarzLemma concerns estimating the gradient of a holomorphic map f between two Kahler (orHermitian) manifolds (Mm, h) and (Nn, g). For that it is instrumental to study the pull-back (1, 1)-form f∗ωg, where ωg is the Kahler form of (N, g). The traditional approach(before the work of [30]) is to compute the Laplacian of the trace of f∗ωg. But in [30],the author estimated the largest singular value of df , equivalently the biggest eigenvalue off∗ωg, via the action of the ∂∂-operator acting on the maximum eigenvalue via a viscosityconsideration. It allows the author to prove another natural generalization of Ahlfors’ resultwith a sharp estimate on the largest singular value in terms of the holomorphic sectionalcurvatures of both the domain and target manifolds. This estimate can be viewed as acomplex version of Pogorelov’s estimate for solutions of the Monge-Ampere equation [37].

To prove the vanishing of holomorphic (p, 0)-forms under the assumption of Ric⊥ > 0, theaction of ∂∂-operator on the comass of holomorphic (p, 0)-forms (cf. [12, 42]), through aviscosity consideration with the help of some basic properties of the comass from Whitney’sclassic [42], holds the key. This new idea also allows an alternate proof of the main theoremin [34].

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Combining this new idea with the work of [34] we study the rational connectedness ofcompact Kahler manifolds under the condition Rick > 0. The notion Rick is a variationof Ricci curvature introduced in [30] to prove that any Kahler manifold with Rick < 0uniformly must be k-hyperbolic, a concept generalizing the Kobayashi hyperbolicity (whichamounts to 1-hyperbolic). Simply put Rick is the Ricci curvature of the curvature operatorR of (N,h) restricted to k-dimensional subspaces of the holomorphic tangent space T ′

xN .The condition Rick > 0 for k = 1 is equivalent to that the holomorphic sectional curvatureH > 0. For k = n, Rick is the Ricci curvature. By [17, 1] Rick > 0 is independent fromRicℓ > 0 for k = ℓ (cf. also [45, 33] for more examples), and that the class of manifolds withRick > 0 for k = n contains non-Fano manifolds. However, we prove the following result.

Theorem 2.3. Let (Nn, h) be a compact Kahler manifold with Rick > 0, for some 1 ≤ k ≤n. Then N is projective and rationally connected. In particular, π1(N) = 0.

The projectivity is proved by a vanishing theorem similar to Theorem 2.2. See Theorem4.1. Namely we show that hp,0 = 0 for any 1 ≤ p ≤ n under the assumption that Rick > 0for some 1 ≤ k ≤ n. The rational connectedness is proved by showing another vanishingtheorem, whose validity is a criterion of the rational connectedness proved in [7]. Both thetechniques of [34] and the one utilizing the comass for (p, 0)-forms introduced in Section3 of this paper are crucial in proving these two vanishing theorems. The result abovegeneralizes both the result for Fano manifolds [6, 23] (the case k = n, namely the Fanocase of Campana, Kollar-Miyaoka-Mori), and the more recent result for the compact Kahlermanifolds with positive holomorphic sectional curvature [15] by Heier-Wong (cf. also [46]for the projectivity for the case k = 1), since Ric1 > 0 amounts to H > 0 and Ricn = Ric.It seems that Rick > 0 has nothing to do with that Ricci curvature is k-positive in general.At least when k = 1, Hitchin’s examples show that they are independent. However it isrelated to the notion of q-Ricci studied in Riemannian geometry [43]. In particular, if the2k−1-Ricci is positive in the sense of H. Wu then Rick > 0. The positivity of the 2k−1-Ricciis a much stronger condition than Rick > 0 since it puts the strict positivity requirement onall 2k-dimensional subspaces of the tangent space at x, most of which are neither invariantunder the almost complex structure, nor a subspace of T ′N . Moreover Rick ≥ 0 does notimply Rick+1 ≥ 0, unlike the q-Ricci conditions.

In Section 5 of the paper addresses the question when compact Kahler manifolds withRic⊥ > 0 are Fano. This question was raised in [33]. We give an affirmative answer for aspecial case.

Theorem 2.4. Let (N,h) be a compact Kahler manifold of complex dimension n. Then (i)

if Ric⊥ > 0 and the Picard number ρ(N) = 1, then N must be Fano; (ii) if Ric⊥ < 0 andh1,1(N) = 1, N must be projective with ample canonical line bundle KN . In particular inthe case (i) N admits a Kahler metric with positive Ricci, and in the case of (ii) N admitsa Kahler-Einstein metric with negative Einstein constant.

Since it was proved in [33] that N is projective and h1,0(N) = h2,0(N) = 0 = h0,2(N) =

h0,1(N) under the assumption that Ric⊥ > 0, the assumption of ρ(N) = 1 for case (i)is equivalent to the assumption that the second Betti number b2 = 1. In [32], it has beenshown that for all Kahler C-spaces of classical type with b2 = 1 the canonical Kahler-Einsteinmetric satisfies Ric⊥ > 0.

To put Theorem 2.4 into perspectives perhaps it is appropriate to recall some earlierworks. First related to Ric⊥ ≥ 0 there exists a stronger condition called the nonnegative

THE FUNDAMENTAL GROUP AND THE POSITIVITY 7

quadratic orthogonal bisectional sectional curvature, studied by various people includingauthors of [44] and [9], etc. Quadratic orthogonal bisectional curvature (abbreviated asQB), is defined for any real vector a = (a1, · · · , an)tr and any unitary frame Ei of T ′N ,QB(a) =

∑i,j Riijj(ai−aj)2. Invariantly it can be formulated as a quadratic form (in terms

of a curvature operator R) acting on Hermitian symmetric tensors A (at any given pointon the manifold) as

QBR(A) + ⟨R,A2∧id−A∧A⟩.Interested readers can refer to [31] for the notations involved. Another formulation of QBcurvature is to view it as a Hermitian quadratic form defined for Hermitian symmetrictensors A : T ′N → T ′N , defined as

QBR(A) =

n∑α,β=1

R(A(Eα), A(Eα), Eβ , Eβ)−R(Eα, Eβ , A(Eβ), A(Eα)) (2.2)

for any unitary orthogonal frame Eα of T ′N . Its nonnegativity, abbreviated as (NQOB), isequivalent to that QB(a) ≥ 0 for any a and any unitary frame Ei. This curvature conditionwas formally introduced in [44] (perhaps appeared implicitly in the work of Bishop-Goldbergin 1960s). It is easy to see that QB> 0 implies Ric ⊥> 0.2 In [9] the following was provedby Chau-Tam (cf. [9], Theorem 4.1):

Theorem 2.5. Let (N,h) be a compact Kahler manifold with (NQOB) with h1,1(N) = 1.Assume further that N is locally irreducible then c1(M) > 0.

In this regard, Theorem 2.4 has the following corollary.

Corollary 2.6. Let (N,h) be a compact Kahler manifold of complex dimension n with

Ric⊥ ≥ 0. Assume further that h1,1(N) = 1 and N is locally irreducible. Then c1(N) > 0,

namely N is Fano. A similar result holds under the assumption Ric⊥ ≤ 0.

Since (NQOB) implies that Ric⊥ ≥ 0 (cf. [9, 33]), one can view the above corollary as ageneralization of Theorem 2.5 of Chau-Tam. There are certainly compact Kahler manifoldswith b2 > 1 (cf. construction in [32] via projectivized bundles) and Ric⊥ > 0. It remains aninteresting question whether or not the same conclusion of Theorem 2.4 (i) holds withoutthe assumption h1,1 = 1. Since QB> 0 implies that h1,1 = 1, as a consequence we havethat any compact Kahler manifold with QB> 0 must be Fano. Whether or not the sameconclusion of part (ii) holds without assuming that h1,1 = 1 remains open.

Even though the above result toward N being Fano (assuming Ric⊥ > 0) is with a simple

proof, and far from the final one, the investigation of the relation between QB and Ric⊥

naturally leads to some new results concerning the cohomology vanishing theorem of (0, 1)-forms valued in the holomorphic tangent bundle. By combining the techniques and resultsof [5], [10], [18], and [32] we obtain the deformation rigidity for classical Kahler C-spaces,as a consequence of the criterion of Frolicher and Nijenhuis [13, 22].

Theorem 2.7. Let Nn be a classical Kahler C-space with n ≥ 2 and b2 = 1 (or moregenerally a compact Kahler manifolds with dCQB> 0). Then Hq(N,T ′N) = 0, for 1 ≤q ≤ n, and N is deformation rigid in the sense that it does not admit nontrivial infinitesimalholomorphic deformation.

2In Section 5 of this paper, motivated by the work of Calabi-Vesentini in 1960s, we introduce the so-called

cross quadratic bisectional curvature (abbreviated as CQB), another (quadratic form type) curvature, whose

positivity also implies Ric⊥ > 0.

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The result was proved via a Kodaira-Bochner formula (cf. [5]) and the role of a curvaturenotion dCQB (dual-cross quadratic bisectional curvature) played in such a Kodaira-Bochnerformula. The dual cross quadratic bisectional curvature (dCQB) is defined as a Hermitianquadratic form on linear maps A : T ′M → T ′′M :

dCQB(A) =n∑

α,β=1

R(A(Eα), A(Eα), Eβ , Eβ) +R(Eα, Eβ , A(Eα), A(Eβ)) (2.3)

where Eα is a unitary frame of T ′M .

The particular result on Kahler C-spaces perhaps could be implied by Bott’s earlier result[4]. But we derive it as a general vanishing theorem for manifolds with dCQB> 0. Hencethe above result in any case is a more general result, at least before one can obtain aclassification of Kaher manifolds with dCQB> 0 (cf. [36] for a precise conjecture relatedto this). The dual-cross quadratic bisectional curvature dCQB is related to the quadraticorthogonal bisectional QB. One can refer Sections 5 and 6 for motivations and detaileddiscussion on this new curvature.

The new dual-cross quadratic bisectional curvature dCQB naturally induces a Ricci typecurvature (in a similar fashion as QB is related to Ric⊥, which is explained in Section 5).It is denoted by Ric+, and is defined, for any X ∈ T ′

xN , as

Ric+(X,X) = Ric(X,X) +H(X)/|X|2.

This notion of Ricci curvature is not as natural as Ric⊥. However for the compact Kahlermanifolds with Ric+ > 0 we have the following result similar to the Ric⊥ > 0 case.

Theorem 2.8. Let (N,h) be a compact Kahler manifold with Ric+ > 0. Then hp,0 = 0 forall n ≥ p ≥ 1. In particular, N is simply-connected and N is projective.

This implies that the manifolds with a uniform positive lower bound on dCQB must becompact, projective, and simply-connected. The proof of the above result makes use of thecomass and viscosity ideas introduced in Section 3 and follows a similar line of argumentas the proof of Theorem 2.2. In Section 6 we also prove a diameter estimate and a resultsimilar to Corollary 2.6 for Ric+.

The cross quadratic bisectional curvature (abbreviated as CQB which is introduced first inSection 5) and its dual dCQB (studied in Section 6) are all positive on the compact classicalKahler C-spaces with b2 = 1 and on some exceptional ones. Since CQB> 0 (as QB> 0)

implies Ric⊥ > 0, this generalizes the result of [32]. On the other hand it was shown byChau-Tam [10] that QB> 0 fails to hold for all Kahler C-spaces with b2 = 1, and it wasshown in [32] that there exists a non-homogenous example compact Kahler manifold with

Ric⊥ > 0. Hence it is perhaps reasonable to expect that one of these two new curvaturenotions possibly provides a curvature characterization of the compact Kahler C-spaces withb2 = 1. Towards this direction we prove (in Theorem 5.3) that a compact Kahler manifoldwith CQB> 0 must be rationally connected. More recently it has been proved to be Fano(cf. [36]) under either the assumption CQB> 0 or dCQB> 0. More ambitious expectationis that they perhaps shed some lights on the generalized Hartshorne conjecture concerningthe Fano manifolds with a nef tangent bundle (cf. conjectures formulated in [36]).

In the appendix, we study the gap between QB> 0 and Ric⊥ > 0. Most results in thispaper can be adapted to Hermitian manifolds without much difficulty, if the notions ofinvolved curvatures are properly extended.

THE FUNDAMENTAL GROUP AND THE POSITIVITY 9

3. Comass and the proof of Theorem 2.2

Let V be a Euclidean space. For a r-covector ω the comass is defined in [42] as

∥ω∥0 + sup|ω(a)| : a is a simple r-vector, ∥a∥ = 1.Here the norm ∥ · ∥ is the norm (a L2-norm in some sense) induced by the inner productdefined for simple vectors a = x1 ∧ · · · ∧ xr,b = y1 ∧ · · · ∧ yr, with xi, yj ∈ V , as

⟨a,b⟩ + det(⟨xi, yj⟩)and then extended bi-linearly to all r-covectors a and b which are linear combination ofsimple vectors. The following results concerning the comass are well-known. The interestedreaders can find their proof in Whitney’s classics [42] or Federer’s [12].

Proposition 3.1. (i) ∥ω∥0 = sup|ω(a)| : ∥a∥0 = 1, where ∥a∥0 is the mass of a definedas

∥a∥0 + inf∑

∥ai∥ : α =∑

ai, the ai simple.

(ii) For each ω there exists a r-vector b such that ∥ω∥0 = |ω(b)|, b is simple, and ∥b∥ = 1.

(iii) If ω is simple, ∥ω∥0 = ∥ω∥.

(iv) ∥ω∥ ≥ ∥ω∥0 ≥ k!(n−k)!n! ∥ω∥.

We shall prove the theorem via an argument by contradiction. Assume that ϕ is a harmonic(p, 0)-form which is not zero. It is well-known that it is holomorphic. Let ∥ϕ∥0(x) be itscomass at x. Then its maximum (nonzero) must be attained somewhere at x0 ∈ N . Weshall exam ϕ more closely in a coordinate chart (to be specified later) of x0. By the aboveproposition, at x0, there exits a simple p-vector b with ∥b∥ = 1, which we may assumeto be ∂

∂z1∧ · · · ∧ ∂

∂zpfor a unitary frame ∂

∂zkk=1,··· ,n at x0, such that maxx∈N ∥ϕ∥(x) =

∥ϕ∥0(x0) = |ϕ(b)|. If we denote ϕ = 1p!

∑IpaIpdz

i1 ∧ · · · ∧ dzip , where Ip = (i1, · · · , ip) runsall p-tuples with is = it if s = t,

∥ϕ∥0(x0) = |a12···p|(x0).Extends the frame to a normal complex coordinate chart U centered at x0. This means thatat x0, the metric tensor gαβ satisfies (cf, [40])

gαβ = δαβ , dgαβ = 0,∂2gαβ∂zγ∂zδ

= 0.

Now ϕ = 1p!

∑IpaIpdz

i1 ∧ · · · ∧ dzip for x ∈ U with aIp(x) being holomorphic.

Let ϕ(x) = a12···p(x)dz1 ∧ · · · ∧ dzp locally. Clearly it is also holomorphic in U . Let b(x)

be the extended p-vector ∂∂z1

∧ · · · ∧ ∂∂zp

(which not necessarily of norm 1) at x. For any

a = v1 ∧ · · · ∧ vp we denote aT = P (v1) ∧ · · · ∧ P (vp) with P being the unitary projection

to the p-dimensional subspace spanned by ∂∂z1 , · · · , ∂

∂zp (namely b).

Lemma 3.1. For x ∈ U , ∥ϕ∥0(x) ≤ ∥ϕ∥0(x).

Proof. Pick a simple p-vector a so that aT = 0. Then at x we have

|ϕ(a)|∥a∥

=|ϕ(aT )|∥a∥

≤ |ϕ(aT )|∥aT ∥

=|ϕ(aT )|∥aT ∥

≤ ∥ϕ∥0(x).

This proves that ∥ϕ∥0(x) ≤ ∥ϕ∥0(x).

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As a consequence since ∥ϕ∥0(x) ≤ ∥ϕ∥0(x0), we have that

∥ϕ∥(x) = ∥ϕ∥0(x) ≤ ∥ϕ∥0(x) ≤ ∥ϕ∥0(x0) = |a1···p(x0)| = ∥ϕ∥0(x0) = ∥ϕ∥(x0).

In summary, we have constructed a simple holomorphic (p, 0)-form ϕ(x) in the neighborhoodof x0 such that its norm attains its maximum value at x0. Now we recall that the ∂∂-Bochnerformula (cf. [20]) for a holomorphic (p, 0)−form ϕ = 1

p!

∑IpaIpdz

i1 ∧ · · · ∧ dzip yields for

any v ∈ T ′x0N

0 ≥ ⟨√−1∂∂|ϕ|2, 1√

−1v ∧ v⟩ = ⟨∇vϕ, ∇vϕ⟩+

1

p!

∑Ip

p∑k=1

n∑l=1

⟨Rvvik laIp , ai1···(l)k···ip⟩. (3.1)

Given that ϕ is simple, namely only nonzero aIp is the one with Ip = (1, 2, · · · , p) or itspermutations, then the above implies that at x0

0 ≥p∑

j=1

Rvvjj . (3.2)

Now we are essentially at the same position of the proof in [33]. For the sake of thecompleteness we include the argument below. Let Σ = span ∂

∂z1, · · · , ∂

∂zp. It is easy to see

from (3.2) that Sp(x0,Σ) ≤ 0, where Sp(x0,Σ) denotes the scalar curvature of the curvatureR restricted to Σ. In fact S(x0,Σ) =

∑pi,j=1Riijj .

On the other hand as in [33]

1

pS⊥p (x0,Σ) =

∫Z∈Σ,|Z|=1

Ric⊥(Z,Z) dθ(Z) =

∫Z∈Σ,|Z|=1

(Ric(Z,Z)−H(Z)

)dθ(Z)

=

∫1

V ol(S2n−1)

(∫S2n−1

(nR(Z,Z,W,W )−H(Z)

)dθ(W )

)dθ(Z)

=1

V ol(S2n−1)

∫S2n−1

(∫ (nR(Z,Z,W,W )−H(Z)

)dθ(Z)

)dθ(W )

=1

p(Ric11 +Ric22 + · · ·+Ricpp)−

2

p(p+ 1)Sp(x0,Σ). (3.3)

Applying (3.2) to v = ∂∂zi

for i = p + 1, · · · , n, and summing the obtained inequalities wehave that

Ric11 +Ric22 + · · ·+Ricpp = Sp(x0,Σ) +n∑

ℓ=p+1

p∑j=1

Rℓℓjj ≤ Sp(x0,Σ). (3.4)

Combining (3.3) and (3.4) we have that

0 < S⊥k (x0) ≤ S⊥

p (x0,Σ) ≤ Sp(x0,Σ)−2

p+ 1Sp(x0,Σ) =

p− 1

p+ 1Sp(x0,Σ). (3.5)

This, for p ≥ 2, implies Sp(x0,Σ) > 0, a contradiction, since we have shown that a conse-quence of (3.2) is Sp(x0,Σ) ≤ 0.

From the definition of S⊥k it is easy to see that Ric⊥ > 0 implies that S⊥

k > 0 for all

k ∈ 1, · · · , n. Hence hp,0 = 0 for all p ≥ 2 by the above under the assumption Ric⊥ > 0.On the other hand π1 is finite by the result of [33]. This in particular implies that b1 =2h1,0 = 0. The simply-connectedness claimed in Theorem 2.2 follows from the argument of[19] illustrated in the introduction.

THE FUNDAMENTAL GROUP AND THE POSITIVITY 11

Remark 3.1. The argument here also provides a more direct proof of the vanishing theoremin [34]. It is clear that the Kahlerity is not used essentially except in the estimation onS⊥(x,Σ). Hence one easily formulate a corresponding result for Hermitian manifolds.Weleave this to interested readers. The concepts of Sk(x,Σ) and S⊥

k (x0,Σ) were conceived in[33, 34]. Moreover the argument of [34] can be adapted to prove more general vanishingtheorem related to the rational connectedness. See the next section for more details.

4. Rational connectedness and Rick

A complex manifold N is called rationally connected if any two points of N can be joinedby a chain of rational curves. Various criterion on the rational connectedness have beenestablished by various authors. In particular the following was prove in [7]:

Proposition 4.1. Let N be a projective algebraic manifold of complex dimension n. ThenN being rationally connected if and only if for any ample line bundle L, there exist C(L)such that

H0(N, ((T ′N)∗)⊗p ⊗ L⊗ℓ) = 0 (4.1)

for any p ≥ C(L)ℓ, with ℓ being any positive integer.

It was proved in [15] that a compact projective manifold with positive holomorphic sectionalcurvature must be rationally connected. The projectivity was proved in [46] afterwards (analternate proof of the rational connectedness was also given there). In [30], the conceptRick was introduced, which interpolates between the holomorphic sectional curvature andthe Ricci curvature. Precisely for any k dimensional subspace Σ ⊂ T ′

xN , Rick(x,Σ) is theRicci curvature of R|Σ. Under Rick < 0, the k-hyperbolicity was proved in [30].

We say Rick(x) > λ(x) if Rick(x,Σ)(v, v) > λ|v|2, for any v ∈ Σ and for every k-dimensional subspace Σ. Similarly Rick > 0 means that Rick(x) > 0 everywhere. Thecondition Rick > 0 does not become weaker as k increases since more v needs to be tested.In fact Hitchin [17] illustrated examples of Kahler metrics (on surfaces) with Ric1 > 0, butdoes not have Ric2 > 0. More examples can be found in [1, 33]. But it is easy to see thatSk > 0 does follows from Rick > 0, and Sk > 0 becomes weaker as k increases with S1 beingthe same as the holomorphic sectional curvature and Sn being the scalar curvature. Thefollowing result is obvious by the vanishing theorem of [34].

Lemma 4.1. For any λ, Rick(x) ≥ λ implies that Sk ≥ kλ. In particular, for a compactKahler manifold with Rick > 0, hp,0 = 0 for p ≥ k.

Hence if Ric2 > 0, N is also projective by the result of [34]. Naturally one would askwhether or not a compact Kahler manifold with Rick > 0 for some k ∈ 3, · · · , n − 1 isprojective since the projectivity has been known for the case for k = 1 and the case k = n.We first provide an affirmative answer to this question.

Theorem 4.1. Let (Nn, h) be a compact Kahler manifold with Rick > 0 for some 1 ≤ k ≤ n.Then hp,0 = 0. In particular, N must be projective.

Proof. By the above lemma we have that hp,0 = 0 for p ≥ k. Hence we only need to focuson the case p < k. The first part of proof of Theorem 2.2 asserts that if ϕ is a holomorphic(p, 0)-form, which is non-trivial, then (3.2) holds. Namely there exists x0 ∈ N , and a unitary

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normal coordinate centered at x0 such that at x0:p∑

j=1

Rvvjj ≤ 0 (4.2)

for any v ∈ T ′x0N .

Now we pick a k-subspace Σ ⊂ T ′x0N such that it contains the p-dimensional subspace

spanned by ∂∂z1 , · · · , ∂

∂zp . Then by the assumption Rick > 0,∫v∈S2k−1⊂Σ

Rvvjj dθ(v) =1

kRick

(∂

∂zj,∂

∂zj

)> 0

for every j ∈ 1, · · · , p. Thus we have that∫v∈S2k−1⊂Σ

p∑j=1

Rvvjj dθ(v) > 0.

This is a contradiction to (4.2). The contradiction proves that hp,0 = 0 for p < k. Theprojectivity follows from h2,0 = 0 and a theorem of Kodaira (cf. [28], Theorem 8.3 ofChapter 3).

For k = 1, 2, n, the result for these cases are previously known except the case that k =2, p = 1.

The argument above proves a bit more.

Definition 4.2. We call the curvature operator R is BC-p positive at x0 (BC stands forthe bisectional curvature) if for any p-unitary orthogonal vectors E1, · · · , Ep, there existsa v ∈ T ′

x0N such that

p∑i=1

RvvEiEi> 0. (4.3)

We say that (N,h) is BC-p positive if it holds all x0 ∈ N . This can be easily adapted toHermitian bundle (V, h) over Hermitian manifolds since condition (4.3) makes for v ∈ Vx0 .

It is easy to see that BC-1 positivity is the same as RC-positivity for the tangent bundledefined in [46]. For general p, BC-p positivity is considerably weaker than RC-positivity. Ffwe endow a compact complex manifold Nn with a Hermitian metric. Let R be its curvature,which can be viewed as section of

∧1,1(End(Tx0N)). Then BC-p positivity defined for any

Hermitian vector bundles specializes to V = T ′N .

Corollary 4.3. If the curvature of a Hermitian manifold (Nn, h) satisfies that BC-p positivefor some 1 ≤ p ≤ n, then hp,0 = 0. Hence any Kahler manifold with BC-2 positive curvaturemust be projective. In particular, the 2-positivity of Rick (of course for some k ≥ 2 to bemeaningful) is sufficient for the projectivity of N .

Proof. By the above proof of theorem, we only need to focus on the last statement. The2-positivity of Rick implies that for any k-dimensional Σ ⊂ T ′

x0N and any two unitary

orthogonal E1, E2 ∈ Σ

Rick(x0,Σ)(E1, E1) + Rick(x0,Σ)(E2, E2) > 0.

This clearly implies BC-2 positivity since for any given unitary orthogonal E1, E2 onecan apply the above to some Σ containing Eii=1,2.

THE FUNDAMENTAL GROUP AND THE POSITIVITY 13

Proposition 4.2. For a Kahler manifold (N,h), Sk(x0) > 0 implies BC-p positive for anyp ≥ k, and Rick(x0) > 0 implies BC-p positive for any 1 ≤ p ≤ n.

Proof. The first claim follows from the simple observation that Sk > 0 implies that Sp > 0for any p ≥ k. The second one follows from the proof of the theorem above.

One can also extend the definition of Rick to a Hermitian vector bundle over Hermitianmanifolds. Let R = R j

αβidzα ∧ dzβ ⊗ e∗i ⊗ ej be the curvature of a Hermitian vector bundle

(V, h) over a Hermitian manifold.

Definition 4.4. Let Σ ⊂ T ′x0N and σ ⊂ Vx0 be two k-dimensional subspaces. Define for

X ∈ T ′x0N , v = viei ∈ Vx0 with ekLk=1 being a unitary frame of Vx0 , L = dim(Vx0), the

first and second Rick as follows:

Ric1k(x0, σ)(X,X) =k∑

i=1

R rXX s

asiatihrt; Ric2k(x0,Σ)(v, v) =

k∑α=1

R j

EαEα ivivlhjl,

with Eαkα=1 being a unitary frame of Σ, and eiki=1 being a unitary frame of σ. Here

ei =∑L

k=1 aki ek.

Note that Ric1 is a (1, 1)-form of N , and it coincides with the first Chern-Ricci of aHermitian manifold if k = n and V = T ′N . Observe that for V = T ′N , Ric1k

∣∣σis Rick(x0, σ)

when N is Kahler, and generalizes Rick for N being just Hermitian.

The Corollary 4.3 provides a generalization of the projective embedding theorem provedin [34]. Towards the rational connectedness we prove the following result.

Theorem 4.5. Let (Nn, h) be a compact projective manifold with Rick > 0 for some k ∈1, · · · , n. Then (4.1) holds, and N must be rationally connected.

Proof. Before the general case, we start with a proof for the special case k = 1 by provingthe above criterion in Proposition 4.1 directly via the ∂∂-Bochner formula. Let s be aholomorphic section in H0(N, ((T ′N)∗)⊗p ⊗ L⊗ℓ). Locally it can be expressed as

s =∑Ip

aIp,ℓdzi1 ⊗ · · · ⊗ dzip ⊗ eℓ

with Ip = (i1, · · · , ip) ∈ Np, and e being a local holomorphic section of L and eℓ = e⊗· · ·⊗ebeing the ℓ power of e. Equip L with a Hermitian metric a and let Ca be the correspondingcurvature form. The point-wise norm |s|2 is with respect to the induced metric of ((T ′N)∗)⊗p

and L⊗ℓ). The ∂∂-Bochner formula implies that for any v ∈ T ′xN :

∂v∂v|s|2 = |∇vs|2 +∑Ip

m∑t=1

p∑α=1

⟨aIp,ℓRvviα tdzi1 ⊗ · · · ⊗ dziα−1 ⊗ dzt ⊗ · · · ⊗ dzip ⊗ eℓ, s⟩

−∑Ip

⟨aIp,ℓℓCa(v, v)dzi1 ⊗ · · · ⊗ dzip ⊗ eℓ, s⟩. (4.4)

Applying the above equation at the point x0, where |s|2 attains its maximum, with respectto a normal coordinate centered at x0. Pick a unit vector v such that H(v) attains itsminimum on S2n−1 ⊂ T ′

x0N . By the assumption H > 0, there exists a δ > 0 such that

H(v) ≥ δ for any unit vector and any x ∈ N . Diagonalize Rvv(·)(·) by a suitable chosen

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unitary frame ∂∂z1 , · · · , ∂

∂zn . Applying the first and second derivative tests, it shows that

if at v ∈ S2n−1, H(v) attains its minimum, then Rvvww ≥ δ2 , and Rvvvw = 0, for any w with

|w| = 1, and ⟨w, v⟩ = 0. This implies that

Rvviα iα = |µ1|2Rvvvv + |β1|2Rvvww ≥ δ

2

where we write ∂∂ziα = µ1v + β1w with |µ1|2 + |β1|2 = 1, w ∈ v⊥ and |w| = 1. (This

perhaps goes back to the work of Berger. See also for example [46] or Corollary 2.1 of [33].)If A is the upper bound of Ca(v, v), we have that

0 ≥ ∂v∂v|s|2 ≥(pδ

2− ℓA

)|s|2.

This is a contradiction for p ≥ 3Aℓδ if s = 0. Hence we can conclude that for any p ≥ C(L)ℓ

with C(L) = 3Aδ , H0(N, ((T ′N)∗)⊗p ⊗ L⊗ℓ) = 0.

For the general case, namely Rick > 0 for some k ∈ 1, · · · , n, we combine the argumentabove with the proof of the vanishing theorem in [34]. At the point x0 where the maximumof |s|2 is attained, we pick Σ such that Sk(x0,Σ) attains its minimum δ1 > 0. For simplicityof the notations, we denote the average of a function f(X) over the unit sphere S2k−1 in Σas∫f(X). The second variation consideration in [34] gives the following useful estimates.

Proposition 4.3 (Proposition 3.1 of [34]). Let E1, . . . , Em be a unitary frame at x0 suchthat Ei1≤i≤k spans Σ. Let I be any non-empty subset of 1, 2, . . . , k. Then for anyE ∈ Σ, E′ ⊥ Σ, and any k + 1 ≤ p ≤ m, we have∫

R(E,E′, Z, Z)dθ(Z) =

∫R(E′, E, Z, Z)dθ(Z) = 0, (4.5)

∫ R(Ep, Ep, Z, Z) +∑j∈I

R(Ej , Ej , Z, Z)

dθ(Z) ≥ Sk(x0,Σ)

k(k + 1), (4.6)

∫R(Ep, Ep, Z, Z) dθ(Z) ≥ Sk(x0,Σ)

k(k + 1). (4.7)

As [34], we may choose the frame so that∫R

vv(·)(·) is diagonal. Integrating (4.4) over the

unit sphere S2k−1 ⊂ Σ we have that

0 ≥∫∂v∂v|s|2 dθ(v) ≥

∑Ip

|aIp,ℓ|2∫ ( p∑

α=1

Rvviα iα − ℓCa(v, v)

)dθ(v).

Here we have chosen a unitary frame ∂∂z1 , · · · , ∂

∂zn so that∫R

vv(·)(·) dθ(v) is diagonal.

As in [34], decompose ∂∂zi into the sum of µiEi ∈ Σ and βiE

′i ∈ Σ⊥ with |Ei| = |E′

i| = 1

and |µi|2 + |βi|2 = 1. If we denote the lower bound of Rick by δ2 > 0, by (4.5) and (4.7)∫Rvv11 dθ(v) = |µ1|2

∫RvvE1E1

dθ(v) + |β1|2∫RvvE′

1E′1dθ(v)

=|µ1|2

kRick(E1, E1) + |β1|2

∫RvvE′

1E′1dθ(v) ≥ |µ1|2

kδ2 +

|β1|2

k(k + 1)δ1

≥ min (δ1, δ2)

k(k + 1).

THE FUNDAMENTAL GROUP AND THE POSITIVITY 15

The above estimate holds for any ∂∂ziα as well. Hence combining two estimates above we

have that

0 ≥∫∂v∂v|s|2 dθ(v) ≥

(pmin (δ1, δ2)

k(k + 1)− ℓA

)|s|2.

The same argument as the special case k = 1 leads to a contradiction, if p ≥ C(L)ℓ forsuitable chosen C(L), provided that s = 0. This proves the vanishing theorem claimed in(4.1) for Rick > 0.

The simply-connectedness part of Theorem 2.3 follows from Theorem 4.1 and the argumentof [19] (recalled in the introduction) via Hirzebruch’s Riemann-Roch theorem. It can alsobe inferred from the rational connectedness and Corollary 4.29 of [11]. It is expected thatthe construction via the projectivization in [32, 45] would give more examples of Kahlermanifolds with Rick > 0.

Since the boundedness of smooth Fano varieties (namely there are finitely many deforma-tion types) was also proved in [23], it is natural to ask whether or not the family of Kahlermanifolds with Rick > 0 (for some k, particularly for n large and n − k = 0 small) isbounded. The result fails for H > 0 given Hirzebruch’s examples (cf. also [1]). Before one

proves that every Kahler manifold with Ric⊥ > 0 is Fano, it remains an interesting futureproject to investigate the rational connectedness of compact Kahler manifols with Ric⊥ > 0.If under QB> 0, as a simple consequence of the results in the next section and the result of[6, 23] we have the following corollary.

Corollary 4.6. Any compact Kahler manifold (N,h) with QB> 0, or more generally with

Ric⊥ > 0 and ρ(N) = 1, must be rationally connected.

The same conclusion holds if Ric⊥ ≥ 0, (Nn, h) is locally irreducible and ρ(N) = 1.

5. Compact Kahler manifolds with h1,1 = 1 and CQB

Recall the following result from [33], which is a consequence of a formula of Berger.

Lemma 5.1. Let (Nn, h) be a Kahler manifold of complex dimension n. At any pointp ∈ N ,

n− 1

n(n+ 1)S(p) =

1

V ol(S2n−1)

∫|Z|=1,Z∈T ′

pN

Ric⊥(Z,Z) dθ(Z) (5.1)

where S(p) =∑n

i=1 Ric(Ei, Ei) (with respect to any unitary frame Ei) denotes the scalarcurvature at p

Note that the first Chern form c1(N) =√−12π rijdz

i ∧ dzj , with rij = Ric( ∂∂zi ,

∂∂zj ). Let

ωh =√−12π hij be the Kahler form (the normalization is to make the Kahler and Riemannian

settings coincide). A direct computation via a unitary frame gives

c1(N)(y) ∧ ωn−1h (y) =

1

nS(y)ωn

h(y). (5.2)

We also let V (N) =∫Nωnh . The normalization above makes sure that the volume of an

algebraic subvariety has its volume being an integer.

Recall that for any line bundle L its degree d(L) is defined as

d(L) =

∫N

c1(L) ∧ ωn−1h . (5.3)

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When h1,1(N) = 1, it implies that [c1(N)] = ℓ[ωh] for some constant ℓ. Hence we have thatd(K−1

N ) = ℓV (N).

Under the assumption (i) of Theorem 2.4, we know that S(y) > 0 point-wisely by Lemma5.1, which then implies that d(K−1

N ) > 0, hence ℓ > 0. This shows that [c1(N)] > 0. NowYau’s solution to the Calabi’s conjecture [47, 40] implies that N admits a Kahler metricsuch that its Ricci curvature is ℓωh > 0.

The proof for statement (ii) is similar. The existence of negative Kahler-Einstein metricfollows from the Aubin-Yau theorem [47, 40].

To prove Corollary 2.6 we observe that if ℓ = 0 in the above argument, it implies thatS(y) ≡ 0. Hence by Lemma 5.1 we have that Ric⊥ ≡ 0. By Theorem 6.1 of [32] it impliesthat N is flat for n ≥ 3, or n = 2 and N is either flat or locally a product. This contradictsto the assumption of local irreducibility.

Note that the same argument can be applied to conclude the same result for holomorphicsectional curvature.

Proposition 5.1. Let (N,h) be a compact Kahler manifold of complex dimension n. As-sume further that h1,1(N) = 1. Then (i) if H > 0, then N must be Fano; (ii) if H < 0, Nmust be projective with ample canonical line bundle KN . In particular in the case (i) N ad-mits a Kahler metric with positive Ricci, and in the case of (ii) N admits a Kahler-Einsteinmetric with negative Einstein constant.

Before introducing the other curvatures related to QB we first observe that in (2.2) if we

replace A by its traceless part A = A− λ id with λ = trace(A)n , it remains the same. Namely

QB(A)=QB(A). Hence QB is defined on the quotient space S2(Cn)/C id, with S2(Cn)being the space of Hermitian symmetric transformations of Cn. Now QB> 0 means thatQB(A) > 0 for all A = 0 as an equivalence class. This suggests a refined positivity QBk > 0,for any 1 ≤ k ≤ n, defined as QB(A) > 0 for any A /∈ C id of rank not greater than k.Clearly for k < n, a nonzero Hermitian symmetric matrix with rank no greater than k cannot be in C id. It is easy to see QB1 > 0 is equivalent to Ric⊥ > 0 and QBn > 0 is

equivalent to QB> 0. Naturally a possible approach towards the classification of Ric⊥ > 0is through the family of Kahler manifolds with QB> 0 and QBk > 0.

Now we introduce the first of two associated curvatures. We call the first one the crossquadratic bisectional curvature CQB, defined as a Hermitian quadratic form on linear mapsA : T ′′N → T ′N :

CQBR(A) =

n∑α,β=1

R(A(Eα), A(Eα), Eβ , Eβ)−R(Eα, Eβ , A(Eα), A(Eβ)) (5.4)

for any unitary frame Eα of T ′M . This is similar to (2.2). But here we allow A to beany linear maps. We say R has CQB> 0 if CQB(A) > 0 for any A = 0. For any X = 0,if we choose Eα with E1 = X

|X| , and let A be the linear map satisfying A(E1) = E1

and A(Eα) = 0 for any α ≥ 2, it is easy to see that CQBR(A) = Ric⊥(X,X)/|X|2. Hence

CQB> 0 implies that Ric⊥ > 0 as well. However as shown in Theorem 3.3 CQB> 0 holds forall classical Kahler C-spaces with b2 = 1, unlike QB, which fails to be positive on about 20%of Kahler C-spaces with b2 = 1. The expression CQB was implicit in the work of Calabi-Vesentini [5] where the authors studied the deformation rigidity of compact quotients of

THE FUNDAMENTAL GROUP AND THE POSITIVITY 17

Hermitian symmetric spaces of noncompact type. We can introduce the concept CQBk > 0(or CQBk < 0), defined as CQB(A) > 0 for any A with rank not greater than k.

Proposition 5.2. (i)The condition CQB1 > 0 implies Ric⊥ > 0, in particular N satisfieshp,0 = 0, π1(N) = 0, and N is projective.

(ii) If N is compact with n ≥ 2, and CQB2 > 0, then Ricci curvature is 2-positive.

Proof. Part (i) is proved in the paragraph above together with Theorem 2.2. For part (ii),for any unitary frame Eα, let A be the map defined as A(E1) = E2 and A(E2) = −E1,and A(Eα) = 0 for all α > 2. Then the direct checking shows that CQB> 0 is equivalent to

Ric(E1, E1) + Ric(E2, E2) > 0.

Since this holds for any unitary frame we have the 2-positivity of the Ricci curvature.

Tracing the argument in [5], which is essentially based on the Akizuki-Nakano formula, wehave the following result.

Theorem 5.1. Let (N,h) be a compact Kahler manifold with quasi-negative CQB (namelyCQB≤ 0 and < 0 at least at one point). Then

H1(N,T ′N) = 0.In particular, N is deformation rigid in the sense that it does not admit nontrivial infini-tesimal holomorphic deformation.

Proof. Let ϕ =∑n

i,α=1 ϕiαdz

α ⊗ Ei be a (0, 1)-form taking value in T ′N with Ei being a

local holomorphic basis of T ′N . The Arizuki-Nakano formula gives

(∆∂ϕ−∆∂ϕ)iβ = R iτ

j βϕjτ − Ricij ϕ

jτ . (5.5)

Under a normal coordinate we have that

⟨∆∂ϕ−∆∂ϕ, ϕ⟩ = −(Ricji ϕ

j

βϕiβ−Rjiτ βϕ

jτϕ

iβ

).

Hence if ∆∂ϕ = 0, we then have

0 =

∫N

|∂ϕ|2 +∫N

|∂∗ϕ|2 −∫N

(Ricij ϕ

iβϕ

j

β−Rjiτ βϕ

jτϕ

iβ

).

Letting A(Eβ) = ϕiβEi, the assumption amounts to that the expression in the third integral

above is negative over the open subset where CQB< 0 if (ϕiτ ) = 0. This implies (ϕiβ) ≡ 0

over this open subset, hence ϕ = 0 by the unique continuation and the harmonic equations.

It has been proved in [33] that if Ric⊥ < 0, then H0(N,T ′N) = 0. By Table 1 of [5]and the proof of Theorem 5.4 below, all locally Hermitian symmetric spaces of noncompacttype satisfy CQB< 0. Hence the above theorem generalizes Calabi-Vesentini’s result. It isdesirable to have new examples beyond the locally Hermitian symmetric ones.

The results above naturally lead to the following questions (Q1): Does H1(N,T ′N) = 0hold under the weaker assumption that Ric⊥ < 0? Do all Kahler C-spaces (the canonicalKahler metric) with b2 = 1 satisfy CQB> 0 (below we provide a partial answer to this)?These remain to be interesting projects for future investigations with the ultimate goal ofa classification of compact Kahler manifolds with CQB> 0. In a recent preprint [36] it wasshown that b2 can be arbitrarily large under CQB> 0 condition. We should point out that

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in [10] it was shown that not all Kahler C-spaces with b2 = 1 satisfy QB> 0. By flippingthe sign we have the following corollary.

Corollary 5.2. Let (N,h) be a compact Kahler manifold with quasi-positive CQB. Then

H0,1∂ (N,T ′N) = H1,0

∂(N,Ω) = H0(N,Ω1(Ω)) = 0,

where Ω = (T ′N)∗. If only Ric⊥ > 0 is assumed, then H0(N,Ω) = 0.

In fact we can strengthen the argument to prove the following result.

Theorem 5.3. Assume that (N,h) is a compact Kahler manifold with CQB> 0. Then forany ample line bundle L, there exist C(L) such that

H0(N, ((T ′N)∗)⊗p ⊗ L⊗ℓ) = 0 (5.6)

for any p ≥ C(L)ℓ, with ℓ being any positive integer. In particular N is rationally connected.

Proof. First observe that a holomorphic section of ((T ′N)∗)⊗(p+1) ⊗ L⊗ℓ can be viewed as

a holomorphic (1, 0) form valued in ((T ′N)∗)⊗p ⊗ L⊗ℓ. Write it as φ = φIpα dzα ⊗ dzi1 ⊗

dzi2 ⊗ · · · ⊗ dzip ⊗ eℓ. Applying the Arizuki-Nakano formula to the ∂-harmonic φ as above,using the formula for the curvature of the tensor products, and under a normal coordinate,we have that

0 ≤ ⟨∂φ, φ⟩ ≤∫M

(ΩI

JφIαφ

Jα − ΩI

J γαφIαφ

Jγ

)+Aℓ|φ|2 ≤

∫M

(−pδ|φ|2 +Aℓ|φ|2

)where ΩI

J γαdzγ ∧ zα is the curvature of ((T ′N)∗)⊗p and ΩI

J is the corresponding meancurvature, δ > 0 is the lower bound of CQB, A is an upper bound of the scalar curvature ofL (equipped with a Hermitian metric of positive curvature). This implies that φ = 0 if p/ℓis sufficiently large, hence the result.

Recently it was proved that CQB> 0 implies that M is Fano, which gives an alternateproof of the above result. Putting the proofs of [10], [18] and [32] together we have thefollowing result.

Theorem 5.4. Let Nn be a compact Hermitian symmetric space (n ≥ 2), or classical KahlerC-space with n ≥ 2 and b2 = 1. Then the (unique up to constant multiple) Kahler-Einsteinmetric has CQB> 0.

Proof. If we write A(Eβ) = AiβEi, it is easy to see if we change to a different unitary frame

Eα = BβαEβ , the effect on A is BABtr with B being a unitary transformation. Now

CQB(A) = Ricij AiβA

jβ −Rjiτ βA

jτA

iβ .

Given that for A symmetric or skew symmetric one can put it into the corresponding normalform via the unitary frame transformations, it suggests that it is useful to write A into sumof the symmetric and skew-symmetric parts. For the special case Ric = λh, namely themetric is Kahler-Einstein with λ > 0, if we decompose A into the symmetric part A1 andthe skew-symmetric part A2, noting that Rjiτ β is symmetric in i, τ and j, β we have

CQB(A) = λ|A1|2 + λ|A2|2 −Rjiτ β(A1)jτ (A1)iβ ≥ λ|A1|2 −Rjiτ β(A1)

jτ (A1)iβ .

Now note that the term Rjiτ β(A1)jτ (A1)iβ is the Hermitian symmetric action Q on the

symmetric tensor (matrix) A considered in [18] and [5]. Let ν denotes the biggest eigenvalue

THE FUNDAMENTAL GROUP AND THE POSITIVITY 19

of Q. As in [32], to verify the result we just need to compare λ and ν. This can be done for allHermitian symmetric spaces by Table 2 in [5]. Note that λ here is R

2n in Calabi-Vesentini’spaper [5]. For the classical homogeneous examples which are not Hermitian symmetric wecan use the comparison done in [32] with the data supplied by [18] and [10]. If we use thenotation of [18] and [10], only the three types below need to be checked:

(Br, αi)r≥3,1<i<r; (Cr, αi)r≥3,1<i<r; (Dr, αi)r≥4,1<i<r−1.

The verification in Section 2 of [32] applies verbatim.

The above result strengthens the one in [32] since CQB> 0 implies Ric⊥ > 0. Note thatthe result also holds for the exceptional (non-Hermitian symmetric) Kahler C-space (F4, α4)since for such a space λ = 11/2 and the biggest eigenvalue of Q is 1. A natural projectafterwards is to classify all the compact Kahler manifolds with CQB> 0 hoping a curvaturecharacterization of the Kahler C-spaces, after which one perhaps can attempt the Ric⊥ > 0classification through CQBk > 0. Given the example in [32], there certainly are compact

Kahler manifolds with Ric⊥ > 0, but not homogeneous.

The second related curvature is a dual version of CQB, which appeared implicitly whenconsidering the compact dual of the noncompact Hermitian symmetric spaces in [5]. Wedenote it by dCQB. It is defined as a quadratic Hermitian form of maps A : T ′N → T ′′N ,defined as

dCQBR(A) + R(A(Ei), A(Ei), Ek, Ek) +R(Ei, Ek, A(Ei), A(Ek)).

Similarly we can introduce the concept dCQBk > 0. The analogy of Ric⊥ is

Ric+(X,X) + Ric(X,X) +H(X)/|X|2.Once fixing a unitary frame of T ′N (hence its dual) one can decompose the dCQB(A) intothe sum of dCQB(A1)+

dCQB(A2) with A1 be the symmetric part and A2 being the skew-symmetric part of A. We say dCQBk > 0 defined as dCQB(A) > 0 for any A = 0 with rankno greater than k. It is easy to see that dCQB1 > 0 implies that Ric+ > 0 if we let A be themap satisfying A(E1) = E1 and A(Ei) = 0 for all i ≥ 2. We discuss geometric implicationsof these two curvature notions in details in the next section.

6. Positive dCQB, deformation rigidity of Kahler C-spaces and Ric+

Properly formulated, results proved in [33] for manifolds with Ric⊥ can be extended toRic+. The argument via the second variational formulae in the proof of Bonnet-Meyertheorem proves the compactness of the Kahler manifolds if the Ric+ is uniformly boundedfrom below by a positive constant.

Theorem 6.1. Let (Nn, h) be a Kahler manifold with Ric+(X,X) ≥ (n + 3)λ|X|2 with

λ > 0. Then N is compact with diameter bounded from the above by√

2n(n+3)λ ·π. Moreover,

for any geodesic γ(η) : [0, ℓ] → N with length ℓ >√

2n(n+3)λ · π, the index i(γ) ≥ 1.

Note that the result is slightly better than√

2n−1n+1 λπ, the one predicted by the Bonnet-

Meyer estimate assuming Ric(X,X) ≥ (n+ 1)λ|X|2 for n ≥ 2. But it is roughly about√2

times the one predicted by the Tsukamoto’s theorem in terms of the lower bound of theholomorphic sectional curvature. Let N = P1 × · · · × P1, namely the product of n copies

20 LEI NI

of P1, its diameter is√

n2π. An easy computation shows that it has Ric = 2 and H ≥ 2

n .This shows that the upper bound provided by Tsukamoto’s theorem holds equality on bothPn and N = P1 × · · · × P1. The product of n-copies of P1 also illustrates a compact Kahlermanifold (after proper scaling) with Ric = n + 1, but its diameter is roughly about

√2

times of that of Pn. The product example and Pn indicate that the above estimate on thediameter is far from being sharp.

We prove Theorem 2.8 via a vanishing theorem with weaker assumptions. For that weintroduce the scalar curvatures S+(x,Σ) which is defined as

S+k (x,Σ) = k

∫Z∈Σ,|Z|=1

Ric+(Z,Z) dθ(Z)

for any k-dimensional subspace Σ ⊂ TxN . Similarly we say S+k > 0 if S+

k (x,Σ) > 0 for anyx and Σ.

Theorem 6.2. Assume that S+k > 0, then hp,0 = 0 for k ≤ p ≤ n.

Proof. The first part of proof follows similarly as in that of Theorem 2.2. Assuming theexistence of a nonzero holomorphic (p, 0)-form ϕ leads to the conclusion that at the pointx0 where the maximum of the comass ∥ϕ∥0 is attained we have that

0 ≥p∑

j=1

Rvvjj

for any v ∈ T ′x0N , for a particularly chosen frame ∂

∂zℓℓ=1,··· ,n with Σ = span ∂

∂z1, · · · , ∂

∂zp.

This implies that Sp(x0,Σ) ≤ 0, by applying the above to v = ∂∂zi

1≤i≤p.

Now a similar calculation as that of Section 2 shows that

1

pS+p (x0,Σ) =

∫Z∈Σ,|Z|=1

Ric+(Z,Z) dθ(Z) =

∫Z∈Σ,|Z|=1

(Ric(Z,Z) +H(Z)

)dθ(Z)

=

∫1

V ol(S2n−1)

(∫S2n−1

(nR(Z,Z,W,W ) +H(Z)

)dθ(W )

)dθ(Z)

=1

V ol(S2n−1)

∫S2n−1

(∫ (nR(Z,Z,W,W ) +H(Z)

)dθ(Z)

)dθ(W )

=1

p(Ric11 +Ric22 + · · ·+Ricpp) +

2

p(p+ 1)Sp(x0,Σ). (6.1)

Using the estimate (3.2) similarly as in Section 2 (cf. (3.4)) we have that

Ric11 +Ric22 + · · ·+Ricpp ≤ Sp(x0,Σ).

Thus together with (6.1) it implies that

0 < S+k (x0) ≤ S+

p (x0,Σ) ≤p+ 3

p+ 1Sp(x0,Σ).

This is a contradiction.

Theorem 2.8 follows from the above theorem since Ric+ > 0 implies that S+p > 0 for all

1 ≤ p ≤ n. Applying the similar argument as that of the last section we also have thefollowing result.

THE FUNDAMENTAL GROUP AND THE POSITIVITY 21

Proposition 6.1. Let (Nn, h) be a compact Kahler manifold of complex dimension n withRic+ > 0. Assume further that h1,1(N) = 1 (or ρ(N) = 1). Then c1(N) > 0, namely N isFano.

This follows from the lemma below, which is the analogue of Lemma 5.1, and the proof inlast section verbatim.

Lemma 6.1. Let (Nn, h) be a Kahler manifold of complex dimension n. At any pointp ∈ N ,

n+ 3

n(n+ 1)S(p) =

1

V ol(S2n−1)

∫|Z|=1,Z∈T ′

pN

Ric+(Z,Z) dθ(Z) (6.2)

where S(p) =∑n

i=1 Ric(Ei, Ei) (with respect to any unitary frame Ei) denotes the scalarcurvature at p.

Following the argument in the Appendix of [32] we also have that a Ric+-Einstein Kahlermetric must be of constant curvature. In particular, the one with zero scalar curvature mustbe flat. Hence we have the same result as Corollary 2.6 if we replace Ric⊥ by Ric+.

Corollary 6.3. Let (N,h) be a compact Kahler manifold of complex dimension n withRic+ ≥ 0. Assume further that h1,1(N) = 1 and N is locally irreducible. Then c1(N) > 0,namely N is Fano. Similar result holds under the assumption Ric+ ≤ 0.

The result similar to Corollary 4.6 holds for Ric+ > 0 and ρ(N) = 1, in view of Theorem2.8, Proposition 6.1 and Corollary 6.3.

Corollary 6.4. Any compact Kahler manifold (N,h) with Ric+ > 0 and ρ(N) = 1, mustbe rationally connected.

The same holds if Ric+ > 0 is replaced with Ric+ ≥ 0 and (Nn, h) is locally irreducible.For compact Kahler manifolds with Ric+ < 0, we have the result below.

Proposition 6.2. Let (N,h) be a compact Kahler manifold with Ric+ < 0. Then N doesnot admits any nonzero holomorphic vector field.

The proof is the same as that of [33]. A dual version of Theorem 5.1 is the following result.

Theorem 6.5. (i) For (N,h) a compact Kahler manifold with quasi-positive dCQB,

H1(N,T ′N) = 0.In particular, N is deformation rigid in the sense that it does not admit nontrivial infini-tesimal holomorphic deformation.

(ii) If compact Kahler manifold (N,h) has dCQB2 > 0, then its Ricci curvature is 2-positive.

(iii) If (N,h) is compact with dCQB1 > 0, then N is projective and simply-connected.

Proof. For (i) one may use the conjugate operator # : A0,1(T ′N) → A1,0((T ′N)∗) which isdefined for ϕ = ϕiαdz

α ⊗ Ei, with Ei being a unitary frame of T ′N , as

#ϕ = ϕiαdzα ⊗ Ei.

Since #(∂ϕ) = ∂(#(ϕ)), it implies that ∂∗(#(ϕ)) = #(∂∗ϕ). Together # induces anisomorphism between Hp,q

∂(N,T ′N) and Hq,p

∂ (N, (T ′N)∗). To prove the result, it suffices to

22 LEI NI

show that any ψ ∈ H1,0∂ (N, (T ′N)∗), ψ = 0. Now we apply the Kodaira-Bochner formula

for ∆∂ operator, and get for ψ = ψiαdz

α ⊗ Ei

(∆∂ψ)iγ = −hαβ∇β∇αψ

iγ +Ri σ

jγ ψjσ + (Ric)σγψ

iσ. (6.3)

Taking product with ψ, as before under the unitary frame, if ∆∂ψ = 0 we have that

0 =

∫N

|∇ψ|2 +∫N

[(Ric)ασψ

iσψ

iα +Rijασψ

jσψ

iα

].

The claimed result follows in the similar way as in the proof of Theorem 5.1.

For part (ii), for any unitary frame Ei, let A be the rank 2 skew-symmetric trans-formation: A(E1) = E2, A(E2) = −E1, and A(Ek) = 0 for all k ≥ 3. Then as in theCQB> 0 case, the second part in the expression of dCQB vanishes and the first part yieldsRic(E1, E1) + Ric(E2, E2).

Part (iii) follows from that dCQB1 > 0 is the same as Ric+ > 0 and Theorem 2.8.

By a similar argument (comparing the Einstein constant with the smallest eigenvalue ofthe symmetric curvature Q obtained in tables of [18]) as in the proof of Theorem 5.4 wealso have the following corollary concerning Kahler C-spaces.

Corollary 6.6. Let Nn be a classical Kahler C-space with n ≥ 2 and b2 = 1, or a compactexceptional Hermitian symmetric space with n ≥ 2. Then the (unique up to constant multi-ple) Kahler-Einstein metric has dCQB> 0. In particular, for a classical Kahler C-space Nwith b2 = 1, Hq(N,T ′N) = 0 with 1 ≤ q ≤ n, and N is deformation rigid in the sensethat it does not admit nontrivial infinitesimal holomorphic deformation.

Proof. To check dCQB> 0, writing A(Ei) = AtiEt, we apply the similar argument as the

case of CQB to see that if we decompose A into A1 + A2, symmetric and skewsymmetricparts, then

dCQB(A) ≥ λ|A1|2 +Rikst(A1)si (A1)tk.

Here λ being the Einstein constant of the canonical metric. Then we reduce the problemto check that λ + ν1 > 0 with ν1 being the smallest eigenvalue of Q. Recall that Q is theself-adjoint linear operator defined as (via extension)

Q(X · Y, Z ·W ) = RXZYW

for X · Y = 12 (X ⊗ Y + Y ⊗ X). This quadratic curvature was considered previously in

[5, 18]. We apply their results below. The Hermitian symmetric case again follows fromTable 2 of [5]. For the nonsymmetric classical Kahler C-spaces, we check them as follows.Note that in [10] and [18] the same normalization for the canonical metric was used. For(Br, αi)r≥3,1<i<r, λ = 2r − i. According to Table 4 of [18] ν1 = −2(r − i) + 1 or −2. Since2r ≥ 2i+2, clearly 2r− i > 2. Also 2r− i− 2r+2i+1 = i+1 > 0. These verify the resultfor both cases of ν1.

For (Cr, αi)r≥3,1<i<r, λ = 2r − i + 1. According to Table 7 of [18], ν1 = −2(r − i + 1).Hence λ+ ν1 = i− 1 > 0 for i ≥ 2. This verifies the result.

For (Dr, αi)r≥4,1<i<r−1, λ = 2r − i− 1. According to Table 10 of [18] ν1 = −2(r − i) + 2or −2. Since 2r− i− 3 ≥ i− 1 > 0 and 2r− i− 1− 2r+2i+2 = i+1 > 0, this also verifiesthe result.

THE FUNDAMENTAL GROUP AND THE POSITIVITY 23

This proved the H1(N,T ′N) = 0. For q > 1, the argument of [5] implies that one onlyneeds to check that λ+ q+1

2q ν1 > 0. This is a consequence of the q = 1 case above.

For the exceptional space (F4, α4) since λ = 11/2 and ν1 = −5, the above result also holds.Hence it should not be surprising that the result in the corollary holds for the rest (22 ofthem total) exceptional Kahler C-spaces. The deformation rigidity result above only holdsinfinitesimally. It does not implies that for any deformation each fiber is biholomorphic tothe central fiber as the main theorem of [38].

Concerning the curvature dCQB one may ask questions similar to those collected in (Q1).For example, one may study the following question (Q2): Whether or not CQBk > 0, anddCQBk > 0 are preserved under the Kahler-Ricci flow? Whether or not there exist compactKahler manifolds with dCQB< 0 (or CQB< 0) which are not compact quotients of someHermitian symmetric spaces of noncompact type? It is also desirable to construct moreexamples of compact Kahler manifolds with Ric+ > 0.

7. Appendix-Estimates on the harmonic (1, 1)-forms of low rank

Here we prove a vanishing theorem for harmonic (1, 1)-forms of low rank related to thecondition QBk > 0 introduced earlier. This is particularly relevant given that in [36] ex-

amples of arbitrary large b2 was constructed with CQB> 0 (in particular with Ric⊥ > 0).First recall that

QBR(A) =n∑

α,β=1

R(A(Eα), A(Eα), Eβ , Eβ)−R(Eα, Eβ , A(Eβ), A(Eα))

vanishes for A = λ id. Hence when define QBk(A) > 0 we require the above expressionpositive for A in S2(Cn) \ λ id, and that A has rank not greater than k. The space of

harmonic (1, 1)-forms H1,1

∂can be decomposed further. First we observe that an (1, 1)-form

Ω =√−1Aijdz

i ∧ dzj can be decomposed as

Ω = Ω1 −√−1Ω2 =

√−1

2Bijdz

i ∧ dzj −√−1(

√−1

2Cijdz

i ∧ dzj)

with

Bij = Aij +Aji; Cij =√−1(Aij −Aji

).

If Ω is harmonic, then ∂Ω = ∂Ω = 0. It can be verified that Ω1 and Ω2 are both harmonic(cf. Theorem 5.4 in Chapter 3 of [28]). This shows that Ω can be decomposed into the sumof a Hermitian symmetric one with −

√−1 times another Hermitian symmetric one. Namely

H1,1

∂= H1,1

∂,s−√−1H1,1

∂,s, where H1,1

∂,sis the spaces of harmonic Ω with (Aij) being Hermitian

symmetric. Within H1,1

∂,swe consider H1,1

∂,s\ Cω. To prove b2 = 1 under the assumption

QB > 0, it suffices to show that H1,1

∂,s\ Cω = 0. We can stratify the space into ones

with rank bounded from above. Let H1,1s,k denote the subspace of H1,1

∂,swhich consists of

Ω =√−12 Aijdz

i ∧ dzj with (Aij) being Hermitian symmetric and of rank no greater than keverywhere on N . The following result can be shown.

Theorem 7.1. Assume that (Nn, g) is a compact Kahler manifold with quasi-positive QBk

with k < n. Then H1,1s,k(N) = 0. In particular, Ric⊥ > 0 implies that H1,1

s,1(N) = 0.

24 LEI NI

Proof. Assume that Ω is a nonzero element in H1,1s,k(N). Applying the ∆ operator to ∥Ω∥2,

by Kodaira-Bochner formula we have that

1

2(∇γ∇γ +∇γ∇γ) ∥Ω∥2(x) = ∥∇γΩ∥2(x) + ∥∇γΩ∥2(x) + 2QB(Ω)(x).

Integrating on N we have that

0 =

∫N

[∥∇γΩ∥2(x) + ∥∇γΩ∥2(x)

]dµ(x) + 2

∫N

QB(Ω)(x) dµ(x) > 0.

The last strictly inequality due to that by the unique continuation we know at a neigh-borhood U where QBk > 0, Ω can not be identically zero. The contradiction implies thatΩ ≡ 0.

For any holomorphic line bundle L over N with a Hermitian metric a, its first Chern

form c1(L, a) = −√−12 ∂∂ log a is a Hermitian symmetric (1, 1)-form. If η is the harmonic

representative of c1(L, a), then η is Hermitian symmetric by the uniqueness of the Hodgedecomposition and Kahler identities (cf. [28], Chapter 3). The following is a simple obser-vation towards possible topological meanings of the rank of η (the minimum k such that

η ∈ H1,1

∂,k, denoted as rk(L)).

Proposition 7.1. Recall that the numerical dimension of L is defined as

nd(L) = maxk = 0, · · · , n : c1(L)k = 0.

Then rk(L) ≥ nd(L).

The proof of the above theorem also shows that if QBk ≥ 0, then any element in H1,1s,k(N)

must be parallel. Thus we have the dimension estimate:

dim(H1,1s,k(N)) ≤ k2.

In fact the existence of a non-vanishing (1, 1)-form of rank at most k has a strong implicationdue to the De Rham decomposition.

Corollary 7.2. Assume that QBk ≥ 0 and H1,1s,k(N) = 0. Then N must be locally

reducible. In particular, if N is locally irreducible and Ric⊥ ≥ 0, then H1,1s,1(N) = 0.

Proof. By the above, we know that the nonzero Ω ∈ H1,1s,k(N) must be parallel. Its null space

is invariant under the parallel transport. This provides a nontrivial parallel distribution,hence the local splitting.

The product example P2 × P2, which satisfies Ric⊥ > 0 and supports non-trivial rank2 harmonic (1, 1)-forms, shows that the above result is sharp for Ric⊥ > 0. Irreducibleexamples of dimension greater than 4 were constructed via the projectivized bundles in [32].

Acknowledgments

The author would like to thank James McKernan for helpful discussions, L.-F. Tam andF. Zheng and for their interests. F. Zheng read the draft carefully and made several helpfulcomments and suggestions. The conjecture 1.1 was formulated in a joint work with him.The author also thank Burkhard Wilking for suggesting the possible connection betweennotions of (2k− 1)-Ricci and Rick, and CUHK, Fuzhou Normal Univ., Peking Univ., SUST

THE FUNDAMENTAL GROUP AND THE POSITIVITY 25

and Xiamen Univ., for their hospitality during his visit in December 2018 when the firstversion of this paper was completed.

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Lei Ni. Department of Mathematics, University of California, San Diego, La Jolla, CA 92093,USA

E-mail address: lni@math.ucsd.edu