Kähler manifolds of semi-negative holomorphic sectional curvature
aa r X i v : . [ m a t h . AG ] J un K ¨AHLER MANIFOLDS OF SEMI-NEGATIVEHOLOMORPHIC SECTIONAL CURVATURE
GORDON HEIER, STEVEN S. Y. LU, BUN WONG
Abstract.
In an earlier work, we investigated some consequences of the ex-istence of a K¨ahler metric of negative holomorphic sectional curvature on aprojective manifold. In the present work, we extend our results to the caseof semi-negative (i.e., non-positive) holomorphic sectional curvature. In doingso, we define a new invariant that records the largest codimension of maximalsubspaces in the tangent spaces on which the holomorphic sectional curvaturevanishes. Using this invariant, we establish lower bounds for the nef dimensionand, under certain additional assumptions, for the Kodaira dimension of themanifold. In dimension two, a precise structure theorem is obtained. Introduction
One of the most basic questions posed by S.-T. Yau concerning the geometry of aprojective (or compact) K¨ahler manifold is the relationship between its holomorphicsectional curvature and its Ricci curvature. The former plays a key role in classicalcomplex geometry (recall for example the constant holomorphic sectional curvaturecharacterization of quotients of B n , C n and P n ), while the latter is the keystoneof the modern theory of (projective) K¨ahler manifolds. Although it is known thatthe holomorphic sectional curvature completely determines the curvature tensor,there is no direct local link between its sign and that of the Ricci curvature. Inthis paper, we provide results in this direction for projective K¨ahler manifolds ofsemi-negative (i.e., non-positive) holomorphic sectional curvature by analyzing thestructural implications of the curvature assumption with the help of a new invariantthat records the largest codimension of maximal subspaces in the tangent spaceson which the holomorphic sectional curvature vanishes.In our previous paper [HLW10], we investigated the implications of negativeholomorphic sectional curvature for the positivity of the canonical line bundle K M on a projective K¨ahler manifold M . Our main results in that paper can be summedup in the following theorem. In Section 2 below, the reader will find the definitionsof the notions involved. We shall always work over the field of complex numbers. Theorem 1.1 ([HLW10]) . Let M be a projective manifold with a K¨ahler metric ofnegative holomorphic sectional curvature. Then (i) the numerical dimension of M is positive, and (ii) the nef dimension of M is equal to the dimension of M . Mathematics Subject Classification.
It follows from a generalized Schwarz Lemma due to Ahlfors that on a compacthermitian manifold M with negative holomorphic sectional curvature there existsno non-constant holomorphic map from the complex plane into M (i.e., M is Brodyhyperbolic). In particular, there exist no rational curves on M . It thus follows fromMori’s bend and break technique that a projective manifold with a K¨ahler metricof negative holomorphic sectional curvature has nef canonical line bundle.Furthermore, it is a consequence of the Abundance Conjecture that for a pro-jective manifold with nef canonical bundle, the Kodaira dimension equals the nefdimension (see the discussion at the beginning of Section 4). Thus, under the as-sumption of the Abundance Conjecture, Theorem 1.1 implies that M is of generaltype, i.e., its canonical bundle is big. Additionally, due to [Kaw85a], a Brody hy-perbolic projective manifold (or even one that is merely free of rational curves)has ample canonical bundle if it is of general type. Thus, up to the validity ofthe Abundance Conjecture, our work in [HLW10] proves the following conjecture,which the third named author learnt from S.-T. Yau in personal conversations inthe early 1970s. Conjecture 1.2.
Let M be a projective manifold with a K¨ahler metric of negativeholomorphic sectional curvature. Then its canonical line bundle K M is ample.Since the Abundance Conjecture in dimension three is known by the works ofMiyaoka and Kawamata (see [MP97, Lecture IV] for a nice account), our previouswork in particular establishes the three dimensional case of Conjecture 1.2 ([HLW10,Theorem 1.1]).After the publication of our paper [HLW10], another paper on this topic ap-peared, namely [WWY12]. Its main result is as follows. Theorem 1.3 ([WWY12]) . Let M be a projective manifold of Picard numberone. If M admits a K¨ahler metric whose holomorphic sectional curvature is semi-negative everywhere and strictly negative at some point of M , then the canonicalline bundle of M is ample. The proof of this theorem as given in [WWY12] is based on a refined SchwarzLemma. The purpose of the present work is to treat the case of semi-negativeholomorphic sectional curvature more comprehensively and in line with our earlierapproach, but with an integrated form of the Schwarz Lemma (Proposition 2.2, seealso Proposition 1.9). In particular, we recover the above Theorem 1.3.To state our first result, we make the following definitions. For p ∈ M , let η ( p ) be the maximum of those integers k ∈ { , . . . , n := dim M } such that thereexists a k -dimensional subspace L ⊂ T p M with H ( v ) = 0 for all v ∈ L \{ ~ } . Set η M := min p ∈ M η ( p ) and r M := n − η M . Note that by definition r M = 0 if and only if H vanishes identically. Also, r M = dim M if and only there exists at least one point p ∈ M such that H is strictly negative at p . Moreover, η ( p ) is upper-semicontinuousas a function of p , and consequently the set { p ∈ M | η ( p ) = η M } is an open set in M (in the classical topology). Now, the first of our results is thefollowing. Theorem 1.4.
Let M be a projective manifold with a K¨ahler metric of semi-negative holomorphic sectional curvature. Then M contains no rational curves EMI-NEGATIVE HOLOMORPHIC SECTIONAL CURVATURE 3 and the canonical line bundle K M is nef. Moreover, if the holomorphic sectionalcurvature vanishes identically, then M is an abelian variety up to a finite unramifiedcovering. If the holomorphic sectional curvature does not vanish identically, then (i) the numerical dimension of M is strictly positive, and (ii) the nef dimension of M is greater than or equal to r M ≥ . For the proof, we follow the basic strategy used in [HLW10]. We recall that[HLW10] was partially inspired by an earlier work of Peternell [Pet91] on Calabi-Yau and hyperbolic manifolds. Note that Theorem 1.3 is an immediate corollaryof Theorem 1.4(i) due the following simple lemma, applied to the case L = K M . Lemma 1.5.
Let M be a projective manifold of Picard number one. Let L be a nefline bundle on M which is of positive numerical dimension. Then L is ample.Proof. Let A be an ample divisor on M . Then L is numerically equivalent to cA for some rational number c . Since L is nef, we have c ≥
0. If we had c = 0, then L would be numerically trivial, and thus its numerical dimension would be equal tozero (see Remark 2.3) in violation of the assumption. So we have c >
0, and L isample by the Nakai-Moishezon-Kleiman ampleness criterion. (cid:3) Remark 1.6.
It is of course an interesting question if it is ever possible to havea strict inequality in Theorem 1.4(ii). Even when dim M = 2 and r M = 1, it isunclear whether M can be a surface of nef dimension 2, i.e., a surface of general type.Intuitively, a surface with r M = 1 should be a properly elliptic surface ([BPVdV84,p. 189]) at least in the case when the metric is analytic.In view of the Abundance Conjecture (see the discussion at the beginning ofSection 4), Theorem 1.4 suggests that r M should in fact be a lower bound for theKodaira dimension of M , which we denote by kod( M ). We offer the following twotheorems that establish a partial solution to the problem of Abundance under ourcurvature assumption. Theorem 1.7.
Let M be an n -dimensional projective manifold of Albanese di-mension d > n − . Let M possess a K¨ahler metric of semi-negative holomorphicsectional curvature. Then kod( M ) ≥ r M − ( n − d − max { , r M − d } ) . In particular, if M has maximal Albanese dimension (i.e., d = n ), then we have kod( M ) ≥ r M . Theorem 1.8.
Let M be an n -dimensional projective K¨ahler manifold of semi-negative holomorphic sectional curvature. Suppose the Abundance Conjecture holdsup to dimension e (which is currently known for e = 3 ). Suppose kod( M ) ≥ n − e .Then kod( M ) ≥ r M . We remark again that, by [Kaw85a], manifolds of maximal Kodaira dimensionwithout rational curves have ample canonical bundles. Hence the above two theo-rems represent generalizations of the key theorems of [HLW10], see Section 4.Theorem 1.8 follows immediately from applying the subsequent proposition tothe Kodaira-Iitaka map of M . The proposition generalizes Proposition 2.2 andLemma 3.1 and is a general integrated (and non-equidimensional) form of theSchwarz Lemma which should be of strong independent interest. GORDON HEIER, STEVEN S. Y. LU, BUN WONG
Proposition 1.9.
Let M be a projective manifold with a K¨ahler metric of semi-negative holomorphic sectional curvature. Let N be a k -dimensional projective va-riety with at most canonical (or even klt) singularities having pseudo-effective anti-canonical Q -Cartier divisor − K N . Let f : N M be a rational map that isgenerically finite, i.e., df has rank k somewhere. Then K N is numerically trivial,and f is a holomorphic immersion that induces a flat metric on the smooth locus N sm of N and is totally geodesic along N sm . In particular, if N is smooth, then N admits an abelian variety as an unramified covering and f is a totally geodesicholomorphic immersion that induces a flat metric on N . In dimension two, we are able to obtain the precise structure theorem below.Note that by the base point freeness of pluricanonical systems in dimension two, wecan and do take the Kodaira-Iitaka map to be the morphism given by an appropriatepluricanonical map here.
Theorem 1.10.
Let M be a smooth projective surface with a K¨ahler metric ofsemi-negative holomorphic sectional curvature. Then one of the following will holdtrue. (i) kod( M ) = 0 : M is an abelian surface up to a finite unramified covering,i.e., M is an abelian surface or a hyperelliptic surface. (ii) kod( M ) = 1 : The Kodaira-Iitaka map of M is an elliptic fibration whoseonly singular fibers are multiple elliptic curves. The base space is a smoothorbifold curve with ample orbifold canonical divisor. Moreover, M admitsa product of smooth curves C × F as a finite unramified covering and themetric of M pulled back to C × F is the product of a non-flat metric ofsemi-negative curvature on C and a flat metric on F up to the addition ofmixed terms each a product of (anti)holomorphic one forms, one from C and one from F as in Proposition 1.11. (iii) kod( M ) = 2 : The canonical line bundle of M is ample. The proof of this theorem is partially based on the following general metricdecomposition result valid in arbitrary dimension.
Proposition 1.11.
Let M = Y × F , where Y and F are projective manifolds. Let π and p be the projections to Y and F respectively. Let ω be a K¨ahler form on M whose restrictions to the fibers of π yield K¨ahler-Einstein metrics on these fibers.Then these restrictions are in fact the pullback of a K¨ahler-Einstein form ω F on F and ω − p ∗ ω F = π ∗ ω Y + X i ( π ∗ µ i ∧ p ∗ ν i + π ∗ µ i ∧ p ∗ ν i ) for a K¨ahler form ω Y on Y and holomorphic one forms µ i on Y and ν i on F . Inparticular, if Y or F has zero irregularity, then ω corresponds to the product of aK¨ahler-Einstein metric on F and a K¨ahler metric on Y . Remark 1.12.
Under stronger curvature assumptions such as semi-negative sec-tional or bisectional curvature, there is a considerable amount of previous work thatyields structural results stronger than ours. Moreover, our invariant r M is similarto the more standard Ricci rank , which comes with an associated
Ricci kernel fo-liation . Our present results regarding holomorphic sectional curvature can be seenas complementary to those earlier results. We refer the reader to [Zhe95], [WZ02],[Zhe02], [Liu14] for further details.
EMI-NEGATIVE HOLOMORPHIC SECTIONAL CURVATURE 5
The contents of the sections of this paper can be summarized as follows. InSection 2, we recall the key definitions and establish basic properties, in particularthe incompatibility statement Proposition 2.2. In Section 3, we shall prove Theorem1.4. In Section 4, we discuss implications of the Abundance Conjecture, whichincludes the statement of corollaries to Theorem 1.4 in dimension no greater thanthree. In Section 5, we discuss the case of positive Albanese dimension, whereour principal theorem is Theorem 1.7 (repeated as Theorem 5.3). In Section 6,we prove Theorem 1.8 (repeated as Theorem 6.1) and the related Proposition 1.9(repeated as Proposition 6.2) of independent interest. In Section 7, we prove theabove structural theorem on the decomposition of surfaces.
Acknowledgement.
The first author would like to thank CRM/CIRGET and theD´epartement de Math´ematiques at the Universit´e du Qu´ebec `a Montr´eal for theirhospitality during the preparation of this paper. The second author would like tothank NSERC for its financial support that allowed its write-up. He is also indebtedto Hongnian Huang for discussions related to Proposition 1.11. We thank FangyangZheng for pointing out an issue with an earlier definition of r M in a previous versionof this paper. 2. Basic definitions and properties
Let M be an n -dimensional manifold with local coordinates z , . . . , z n . Let g = n X i,j =1 g i ¯ j dz i ⊗ d ¯ z j be a hermitian metric on M . The components R i ¯ jk ¯ l of the curvature tensor R associated with the metric connection are locally given by the formula R i ¯ jk ¯ l = − ∂ g i ¯ j ∂z k ∂ ¯ z l + n X p,q =1 g p ¯ q ∂g i ¯ p ∂z k ∂g q ¯ j ∂ ¯ z l . If g is K¨ahler, the Ricci curvature takes a particularly nice form. In fact, we candefine the Ricci curvature form to beRic = −√− ∂ ¯ ∂ log det( g i ¯ j ) . By a result of Chern, the class of the form π Ric is equal to c ( M ) = c ( − K M ),where K M is the canonical line bundle of M .In Section 6, we also use the symbol Ric to denote the curvature form of ahermitian metric on a line bundle.The scalar curvature S of g is defined to be the trace of Ric with respect to aunitary frame.It follows from linear algebra and the definition of scalar curvature thatRic ∧ ω n − = n S ω n , where ω = √− P ni,j =1 g i ¯ j d z i ∧ d¯ z j is the (1,1)-form associated to g . In situationswhere there are several spaces, metrics, and associated forms involved, the readershould assume that the unadorned symbols g and ω pertain to M . GORDON HEIER, STEVEN S. Y. LU, BUN WONG If ξ = P ni =1 ξ i ∂∂z i is a non-zero complex tangent vector at p ∈ M , then theholomorphic sectional curvature H ( ξ ) is given by H ( ξ ) = n X i,j,k,l =1 R i ¯ jk ¯ l ( p ) ξ i ¯ ξ j ξ k ¯ ξ l / n X i,j,k,l =1 g i ¯ j g k ¯ l ξ i ¯ ξ j ξ k ¯ ξ l . An important fact about holomorphic sectional curvature is the following. If M ′ is a submanifold of M , then the holomorphic sectional curvature of M ′ does notexceed that of M . To be precise, if ξ is a non-zero tangent vector to M ′ , then H ′ ( ξ ) ≤ H ( ξ ) , where H ′ is the holomorphic sectional curvature associated to the metric on M ′ induced by g . For a short proof of this inequality see [Wu73, Lemma 1]. Basically,the inequality is an immediate consequence of the Gauss-Codazzi equation.We have the following pointwise result due to Berger [Ber66] (see [HM13] for arecent new approach). Theorem 2.1 ([Ber66]) . Let M be a compact manifold with a K¨ahler metric ofsemi-negative holomorphic sectional curvature. Then the scalar curvature function S is also semi-negative everywhere on M . Moreover, let p ∈ M and assume thatthere exists w ∈ T p M \{ ~ } such that H ( w ) < . Then S ( p ) < . Berger’s theorem is proven using a pointwise formula expressing the scalar cur-vature at a point in terms of the average holomorphic sectional curvature on theunit sphere in the tangent space at that point. Based on Berger’s theorem, we havethe following proposition.
Proposition 2.2.
Let M be a projective manifold whose first real Chern class iszero. Let g be a K¨ahler metric on M whose holomorphic sectional curvature issemi-negative. Then the holomorphic sectional curvature of g vanishes identicallyand M is an abelian variety up to a finite unramified covering.Proof. Assume the holomorphic sectional curvature of g does not vanish identically.Then there exists a point p ∈ M and w ∈ T p M \{ ~ } such that H ( w ) <
0. ByTheorem 2.1, the scalar curvature is non-positive everywhere, and S ( p ) <
0. Thus,0 = 2 π Z M c ( − K M ) ∧ ω n − = Z M Ric( g ) ∧ ω n − = Z M n S ω n < , which is a contradiction.Having shown that the holomorphic sectional curvature of g does vanish identi-cally, it is immediate that M is an abelian variety up to a finite unramified covering.Namely, it is a basic fact that the holomorphic sectional curvature of a K¨ahler met-ric completely determines the curvature tensor R ([KN69, Proposition 7.1, p. 166]).In particular, if H vanishes identically, then R vanishes identically. However, dueto [Igu54], a projective K¨ahler manifold with vanishing curvature tensor admits afinite unramified covering by an abelian variety. (cid:3) EMI-NEGATIVE HOLOMORPHIC SECTIONAL CURVATURE 7
We conclude this section by defining the two notions of positivity of the canonicalline bundle that appear in Theorem 1.4. Let L be an arbitrary nef line bundleon M . Then the numerical dimension of L , which we denote ν ( L ) , is max { k ∈{ , , . . . , dim M } : ( c R ( L )) k = [0] } , where c R ( L ) denotes the first real Chern classof L . We write ν ( M ) for ν ( K M ), the numerical dimension of M (aka the numericalKodaira dimension of M ). Remark 2.3.
It is immediate that ν ( L ) = 0, i.e., c R ( L ) = [0], implies that L is numerically trivial. The converse also holds true, which is nicely explained in[Laz04, Remark 1.1.20].The notion of nef dimension is based on the following theorem ([Tsu00],[BCE + Theorem 2.4.
Let L be a nef line bundle on a normal projective variety M . Thenthere exists an almost holomorphic dominant rational map f : M Y with con-nected fibers, called a “reduction map,” such that (i) L is numerically trivial on all compact fibers F of f with dim F = dim M − dim Y , and (ii) for every general point x ∈ M and every irreducible curve C passingthrough x with dim f ( C ) > , we have L.C > .The map f is unique up to birational equivalence of Y . We call dim Y the nef dimension of L . When we apply the above theorem with L = K M , we call n ( M ) := dim Y the nef dimension of M .3. Proof of Theorem 1.4
The non-existence of rational curves and the nefness of the canonicalline bundle. If K M is not nef, then, by the bend and break technique of Mori, M contains a rational curve, i.e., there exists a non-constant holomorphic map P → M . Thus, it only remains to show the non-existence of rational curves, whichhas been known at least since the 1970’s (e.g., via the disk condition in Shiffman’s[Shi71] or by [Roy80, Corollary 2]). We state the absence of rational curves in thefollowing lemma, which actually is a slightly stronger result. Note in particularthat we require the hermitian manifold ( M, h ) to be neither complete nor K¨ahler.The first proof is of an analytic nature and perhaps preferable to some readers,since it avoids the use of saturations of subsheaves. The second proof relies on anintegrated version of the Schwarz Lemma and thus is very much in line with thegeneral theme of this paper.
Lemma 3.1.
Let N be a compact Riemann surface of genus γ and ( M, h ) a (notnecessarily complete) hermitian manifold of semi-negative holomorphic sectionalcurvature. If f : N → M is a non-constant holomorphic map, then γ ≥ and γ = 1 if and only if f is a totally geodesic immersion inducing a flat metric on N .Proof of Lemma 3.1 using analytic methods. We write ω for the (1 , M, h ) and ω KE for the (1 , N with constant scalar curvature S = 2 − γ . Then u := f ∗ ωω KE GORDON HEIER, STEVEN S. Y. LU, BUN WONG is a non-negative smooth function on N that is not identically vanishing (otherwise df would be identically zero and thus f would be constant). Consequently, df hasat most a finite number of zeros. Outside of these zeros, u is positive, and we have √− ∂∂ log u = Sω KE − kf ∗ ω, where k is the holomorphic sectional curvature of the K¨ahler form f ∗ ω induced by h . Due to the curvature decreasing property of subbundles and submanifolds, wehave k ≤
0. We may assume that S ≥ u is a subharmonic function and hence is constant since N iscompact. This means that u is a positive constant function and hence that df hasno zeros. It also implies that S − ku = 0, forcing both S = 0 and the everywherevanishing of k . We have thus shown that γ = 1 and that f is a totally geodesicimmersion inducing a flat metric f ∗ ω on N . Since the converse of the last statementof the lemma is clear, the proof is now complete. (cid:3) Proof of Lemma 3.1 using an integrated version of the Schwarz Lemma.
We abusenotation and denote (holomorphic) vector bundles and their sheaves of (holomor-phic) sections by the same symbols. Let L be the saturation of the rank one subsheafof f − T M over N given by the image of T N via the differential of f naturally givenas section df of Hom N ( T N, f − T M ). Then L is a subbundle of f − T M with aninduced hermitian metric h L and df identifies with a section s of the line bundle Hom N ( T N, L ). Since L is holomorphically identified with T N via s over the denseopen subset N of N where s = 0 and since, over N , h L = f ∗ h is the inducedmetric on T N , the metric h L has semi-negative curvature on N by the curvaturedecreasing property. Hence, the result follows from2 − γ = deg T N ≤ deg T N + deg( s ) = deg L = Z N c ( L, h L ) ≤ γ = 1) and from the Gauss-Codazzi equation. (cid:3) The case of vanishing holomorphic sectional curvature.
If the holo-morphic sectional curvature of M vanishes, then the argument at the end of theproof of Proposition 2.2 shows that M must be an abelian variety up to a finiteunramified covering.3.3. Positivity of the numerical dimension when H does not vanish iden-tically.
Next, we show that the numerical dimension of M is positive when H doesnot vanish identically. To this end, let us assume that the numerical dimension of M is zero. By definition, this means that the first real Chern class of F is trivial.By Proposition 2.2, this implies that the holomorphic sectional curvature vanishesidentically, which is a contradiction to our assumption.3.4. The bound on the nef dimension.
Now, we prove that the nef dimension n ( M ) is greater than or equal to r M . To this end, let us denote by f : M Y anef reduction map with respect to K M , and let I ⊂ M denote the set of points ofindeterminacy of f . We write f h for the holomorphic map f | M \ I .Now, let us assume that n ( M ) < r M and derive a contradiction. Since M issmooth, we can apply the generic smoothness theorem [Har77, Corollary III.10.7]and conclude that there exists an open and dense subset V ⊂ Y such that f h : EMI-NEGATIVE HOLOMORPHIC SECTIONAL CURVATURE 9 f − h ( V ) → V is a smooth submersion. Since a smooth submersion is an open map,the set ˜ V := f h ( { p ∈ M | η ( p ) = η M }\ I )is a non-empty open subset (in the classical topology) of V . We pick a point y ∈ ˜ V such that the fiber F of f h over y is compact and has the expected dimension δ := n − n ( M ) > n − r M .By the adjunction formula, K M | F = K F , so K F is numerically trivial. By[Laz04, Remark 1.1.20], this implies that c R ( K F ) = [0]. Now, let p ∈ F be suchthat η ( p ) = η M . Due to δ > n − r M = η M , there exists a nonzero vector w in T p F ⊂ T p M with H ( w ) <
0. Due to the curvature decreasing property mentionedin Section 2, the holomorphic sectional curvature H ′ of the K¨ahler metric g ′ inducedon F by g satisfies H ′ ( w ) <
0. Thus, H ′ is semi-negative and does not vanishidentically. By Proposition 2.2, we have obtained a contradiction.4. Remarks related to the Abundance Conjecture
On a projective manifold M with nef canonical line bundle K M , the followingchain of inequalities holds:(1) kod( M ) ≤ ν ( M ) ≤ n ( M ) . The first inequality was established by Kawamata in [Kaw85b, Proposition 2.2]and the second inequality is in [BCE +
02, Proposition 2.8]. The name AbundanceConjecture is commonly used to refer to the claim that the first inequality is actuallyan equality, i.e., that Kodaira dimension and numerical dimension of M agree([Kaw85b, Conjecture 7.2]).Furthermore, it is known that the Abundance Conjecture actually implieskod( M ) = n ( M ), making (1) an all-around equality. The argument for this goesas follows.If n ( M ) = 0, then 0 ≤ ν ( M ) ≤ n ( M ) = 0 implies ν ( M ) = 0. Due to theAbundance Conjecture (which is actually [Kaw85a, Theorem 8.2] in this case),kod( M ) = ν ( M ) = 0 = n ( M ).If n ( M ) >
0, then we observe first of all that ν ( M ) > M ) = ν ( M ) >
0. For ourpurposes, it is convenient to think of the Kodaira-Iitaka map as represented by themap furnished by a large enough multiple of K M as described in [Laz04, Theorem2.1.33]. Due to the semi-ampleness of K M established in [Kaw85b, Theorem 1.1],this map is holomorphic. A generic fiber F has kod( F ) = 0 and thus, again by theAbundance Conjecture, ν ( F ) = 0. Therefore, by construction of the nef reductionmap, kod( M ) ≥ n ( M ). Together with (1), we obtain the desired equality. In fact,what we have seen is that the map furnished by a large enough multiple of K M canserve as a representative of both the nef reduction map with respect to K M and theKodaira-Iitaka map (both of which are only defined up to birational equivalence)if the Abundance Conjecture is valid.Due to the above, an immediate corollary to Theorem 1.4 is the following. Corollary 4.1.
Let M be a projective manifold with a K¨ahler metric of semi-negative holomorphic sectional curvature. If M satisfies the Abundance Conjec-ture, which is the case if its dimension is no greater than three, then the Kodairadimension of M satisfies kod( M ) ≥ r M . In particular, a projective manifold M of dimension n ≤ r M = n is of generaltype. Moreover, as we saw in Section 3.1, M contains no rational curves, so thatbased on [Kaw85a], we obtain the following improvement of [HLW10, Theorem 1.1]. Corollary 4.2.
Let M be a projective manifold of dimension n = r M satisfyingthe same hypotheses as in the above corollary. Then K M is ample. In light of this, it is rather clear that Conjecture 1.2 should be generalized asfollows.
Conjecture 4.3.
Let M be a projective manifold with a K¨ahler metric of semi-negative holomorphic sectional curvature. Then the Kodaira dimension of M sat-isfies kod( M ) ≥ r M . In particular, if r M = dim M , then the canonical line bundle K M is ample.We will see in the next section that if M is of high enough Albanese dimension,then Conjecture 4.3 holds true for M (without any use of the Abundance Conjec-ture). In Section 6, we will prove the conjecture for the case when kod( M ) is highenough.We conclude this section by mentioning that a proof of the Abundance Conjec-ture was announced in [Siu11], although complete details of this proof seem to benot yet available.5. Manifolds of positive Albanese dimension
For projective manifolds of positive Albanese dimension, we actually can provesome versions of Conjecture 4.3. Our results are as follows.
Theorem 5.1.
Let M be an n -dimensional projective manifold whose Albanesedimension is maximal, i.e., equal to n . Let M possess a K¨ahler metric of semi-negative holomorphic sectional curvature. Then kod( M ) ≥ r M . Proof.
By the definition of M being of maximal Albanese dimension, M possessesa generically finite map to an abelian variety A , namely its Albanese map a . Thus,the Kodaira dimension kod( M ) ≥
0, since one can pull back a not identicallyzero holomorphic n -form from the abelian variety under a , which will yield a notidentically zero holomorphic n -form on M .To warm up (and avoid trivialities in the treatment of the general case) we firstdeal with the case when kod( M ) = 0. By [Kaw81, Theorem 1], the Albanese map a : M → A is a fiber space. Since the Albanese dimension of M is maximal, themap a is generically finite. Thus, we have so far established that a is a birationalholomorphic map. Since the Albanese torus A is smooth, the exceptional set of a is covered by rational curves due to [Abh56]. Since there are no rational curves on M , a is injective. As an injective and onto holomorphic map between manifolds, a is an isomorphism and M is an abelian variety. By Proposition 2.2, r M = 0, andthe theorem is proven in the case kod( M ) = 0.We now treat the general case kod( M ) >
0. Let π : M ∗ → Y ∗ be a holomor-phic version of the Kodaira-Iitaka map of M and σ : M ∗ → M the pertaining EMI-NEGATIVE HOLOMORPHIC SECTIONAL CURVATURE 11 modification of M . We have a diagram M ∗ π −−−−→ Y ∗ σ y M a −−−−→ A .
Let G be a general fiber of π . Such a G is a submanifold of M ∗ of dimension n − kod( M ) with kod( G ) = 0. The map a ◦ σ | G : G → ( a ◦ σ )( G ) is a genericallyfinite holomorphic map. We let B := ( a ◦ σ )( G ). Due to [Kaw81, Corollary 9],0 = kod( G ) ≥ kod( B ). Moreover, [Uen75, Lemma 10.1] yields kod( B ) ≥
0, sokod( B ) = 0. By [Uen75, Theorem 10.3], B is the translate of an abelian subvariety A of A . Since there are only countably many abelian subvarieties of A , we canassume that A does not depend on G . We write p : A → A/A for the canonicalprojection.Next, we observe that the map σ | G : G → σ ( G ) is a birational holomorphicmap. Now, note that A/A is again an abelian variety and consider the map p ◦ a : M → A/A . An irreducible component of a general fiber of p ◦ a will be ofthe form σ ( G ), where G is a general fiber of π . Due to generic smoothness, σ ( G )will be smooth, and due to birational invariance, kod( σ ( G )) = 0. By definitionof the Kodaira-Iitaka map, the dimension of σ ( G ) is n − kod( M ). In fact, σ ( G )is an abelian variety for the following reason. In our situation, it follows from[KV80, Main Theorem] that σ ( G ) is birational to an abelian variety. Since itsimage ( a ◦ σ )( G ) is smooth, the same argument as above based on [Abh56] showsthat σ ( G ) is isomorphic to an abelian variety.To conclude the proof, assume that the general fiber G is chosen such that theabelian variety σ ( G ) has non-empty intersection with the open set { p ∈ M | η ( p ) = η M } . By the curvature decreasing property and Proposition 2.2, dim σ ( G ) = n − kod( M ) can be no greater than η M , i.e., kod( M ) ≥ n − η M = r M . (cid:3) The above Theorem 5.1 states in particular that a projective manifold M ofmaximal Albanese dimension with a K¨ahler metric of semi-negative holomorphicsectional curvature satisfying r M = dim M is of general type. The establishedabsence of rational curves, together with [Kaw85a], then yields the following specialcase of Conjecture 4.3. Corollary 5.2.
Let M be a projective manifold with a K¨ahler metric of semi-negative holomorphic sectional curvature. Assume that M is of maximal Albanesedimension and that r M = dim M . Then the canonical line bundle of M is ample. The following theorem represents a generalization of [HLW10, Theorem 1.2]. Itcontains Theorem 5.1 as the special case d = n , but since the bound below may besomewhat hard to parse on a first reading and since fewer deep results had to becited in the earlier proof, we thought it best to isolate the more concise Theorem5.1 at the beginning of this section. Theorem 5.3.
Let M be an n -dimensional projective manifold of Albanese di-mension d > n − . Let M possess a K¨ahler metric of semi-negative holomorphicsectional curvature. Then kod( M ) ≥ r M − ( n − d − max { , r M − d } ) . Proof.
Under the assumption d > n −
4, the Albanese map a : M → a ( M ) isa holomorphic map such that an irreducible component F of a general fiber hasdimension n − d ≤
3. Since the Iitaka Conjecture holds in the case of fibers ofdimension no greater than three ([Kaw85a], see also [Bir09] for some expositorycomments), we have kod( M ) ≥
0. Moreover, by Corollary 4.1 and the curvaturedecreasing property, kod( F ) ≥ r F . Again by the curvature decreasing property, r F ≥ max { , r M − d } .We again consider the diagram as in the proof of Theorem 5.1. We denote by˜ F the strict transform of F under σ , which satisfies kod( ˜ F ) = kod( F ) ≥ r F ≥ max { , r M − d } .The restriction of π to ˜ F gives a holomorphic map π : ˜ F → π ( ˜ F ). If we denote anirreducible component of a general fiber of this map by S , then the Easy AdditionFormula (applied after a Stein factorization) yields(2) max { , r M − d } ≤ r F ≤ kod( ˜ F ) ≤ kod( S ) + dim π ( ˜ F ) . Moreover, it is clear that n − d = dim ˜ F = dim S + dim π ( ˜ F ).On the other hand, let G be the fiber of π : M ∗ → Y ∗ such that S is a componentof G ∩ ˜ F . By the standard properties of the Kodaira-Iitaka map, when F and S are appropriately chosen, G is a projective manifold with kod( G ) = 0. Due to theresolved Iitaka Conjecture in the case of fibers of dimension no greater than three,0 = kod( G ) ≥ kod( S ) + kod( a ( σ ( G ))) . By [Uen75, Lemma 10.1], we know that kod(( a ◦ σ )( G )) is at least 0. From (2), it isalso clear that kod( S ) ≥
0. Hence, 0 = kod( S ) = kod(( a ◦ σ )( G )), and by [Uen75,Theorem 10.3], ( a ◦ σ )( G ) is the translate of an abelian subvariety. Again from (2),we infer dim π ( ˜ F ) ≥ max { , r M − d } and thus dim( S ) ≤ n − d − max { , r M − d } .Next, observe that dim( a ( σ ( G ))) is bounded below bydim( G ) − dim( S ) ≥ dim( G ) − ( n − d − max { , r M − d } )= n − kod( M ) − ( n − d − max { , r M − d } ) . We have now established that a ( σ ( G )) is the translate of an abelian subvariety A of dimension at least n − kod( M ) − ( n − d − max { , r M − d } ). We write p : A → A/A for the canonical projection. As in the proof of Theorem 5.1, this yields n − kod( M ) − ( n − d − max { , r M − d } ) ≤ n − r M , i.e., kod( M ) ≥ r M − ( n − d − max { , r M − d } ). (cid:3) Note that if n = r M , then r M − ( n − d − max { , r M − d } ) = r M in the statementof Theorem 5.3. Therefore, Corollary 5.2 can be strengthened to the following. Corollary 5.4.
Let M be a projective manifold with a K¨ahler metric of semi-negative holomorphic sectional curvature. Assume that r M = dim M and that theAlbanese dimension of M is greater than dim M − . Then the canonical line bundleof M is ample. The following final theorem of this section applies in the case of arbitrary pos-itive Albanese dimension. The assumption of non-negative Kodaira dimension isnecessary, because based on the other assumptions, we cannot prove the existenceof any pluricanonical sections. Without at least one pluricanonical section, there isno Kodaira-Iitaka map, and our argument does not work.
EMI-NEGATIVE HOLOMORPHIC SECTIONAL CURVATURE 13
Theorem 5.5.
Let M be an n -dimensional projective manifold of Kodaira dimen-sion at least zero and Albanese dimension d > . Let M possess a K¨ahler metricof semi-negative holomorphic sectional curvature. Contingent on the validity ofthe Iitaka Conjecture for fibrations with fiber dimension no greater than n − d , thefollowing holds: kod( M ) ≥ r M − ( n − d ) . Proof.
The proof is a simplified version of the proof of Theorem 5.3. We let F ,˜ F , S , G be as above. In the present situation, we cannot rule out kod( ˜ F ) = −∞ (although it is ruled out conjecturally by Conjecture 4.3), so the Easy AdditionFormula (2) becomes vacuous. Instead, we apply the Easy Addition Formula to a : σ ( G ) → ( a ◦ σ )( G ), which yields0 = kod( G ) ≤ kod( S ) + dim( a ( σ ( G ))) . Thus, kod( S ) ≥
0. Moreover, due to the Iitaka Conjecture in the case of fibers ofdimension no greater than n − d , we have0 = kod( G ) ≥ kod( S ) + kod( a ( σ ( G ))) . Again, we conclude kod( a ( σ ( G ))) = 0. It remains to observe that dim( a ( σ ( G ))) isbounded below by dim G − dim S ≥ dim G − dim F = n − kod( M ) − ( n − d ) . Asbefore, we conclude n − kod( M ) − ( n − d ) ≤ n − r M , i.e., kod( M ) ≥ r M − ( n − d ). (cid:3) Manifolds of high Kodaira dimension
If we assume the validity of the Abundance Conjecture up to some dimension e with 1 ≤ e ≤ n , then we can prove the desired inequality kod( M ) ≥ r M providedwe may additionally assume that kod( M ) ≥ n − e (which is of course a non-vacuousstatement only if r M > n − e ). Theorem 6.1.
Let M be an n -dimensional projective K¨ahler manifold of semi-negative holomorphic sectional curvature. Suppose the Abundance Conjecture holdsup to dimension e (which is currently known for e = 3 ). Suppose kod( M ) ≥ n − e .Then kod( M ) ≥ r M . The theorem follows readily from the following proposition applied to the generalfibers of the Kodaira-Iitaka map of M . Proposition 6.2.
Let M be a projective manifold with a K¨ahler metric of semi-negative holomorphic sectional curvature. Let N be a k -dimensional projective va-riety with at most canonical (or even klt) singularities having pseudo-effective anti-canonical Q -Cartier divisor − K N . Let f : N M be a rational map that isgenerically finite, i.e., df has rank k somewhere. Then K N is numerically trivial,and f is a holomorphic immersion that induces a flat metric on the smooth locus N sm of N and is totally geodesic along N sm . In particular, if N is smooth, then N admits an abelian variety as an unramified covering and f is a totally geodesicholomorphic immersion that induces a flat metric on N . Proof.
We will abuse notation and denote Cartier divisors and their associatedinvertible sheaves as well as bundles and their sheaves of sections with the samesymbols, respectively. All metrics on complex bundles are understood to be hermit-ian. For simplicity, we will not distinguish a metric from its associated (1 , M has no rational curves, f is in fact a holomorphic map (see, for example,[KM98]). By the hypothesis on f , it has rank k on a dense Zariski open set of N sm . Hence, det( df ) gives rise to a nontrivial section s N of the locally free sheaf Hom N ( K ∨ N , f − (Λ k T M )) on every Zariski open subset N of N sm , where f = f | N .Let τ : X → N be a resolution of the singularities of N and F = f ◦ τ : X → M .To avoid heavy notation, on the open subset N ⊂ N sm where the birational map F − is holomorphic, we identify f | N with F | F − ( N ) . As before, det( dF ) gives riseto a nontrivial section s X over X of Hom X ( K ∨ X , F − (Λ k T M )) and s X | N = s N under the identification of N with F − ( N ). Here, the symbol F − (Λ k T M ) simplydenotes the pull-back of the vector bundle Λ k T M .Since K X is invertible, after replacing X by some further blowup of X if neces-sary, the same proof as in the resolution of the base locus of a linear system intoonly divisorial parts (for example by blowing up the ideal sheaf given by the imageby s X of the vector sheaf Hom X ( K ∨ X , F − (Λ k T M )) ∨ in O X ) allows us to assumethat the subscheme defined by s X = 0 is of pure codimension one, i.e., a divisor D .This means that the saturation of the subsheaf s X ( K ∨ X ) is given by a line subbun-dle L of F − (Λ k T M ) and s X can be identified with a section s of the line bundle K X ⊗ L = Hom X ( K ∨ X , L ) over X . Clearly ( s ) = D on X by construction and s | N = s N . Note that K X = τ ∗ K N + E for an effective Q -divisor E supported on the exceptional locus of τ and that both E and K N are Cartier outside the singular locus of N .Let N ⊂ N be the Zariski open dense subset on which df : T N → T M has maximal rank, i.e., on which f is an immersion. Theorem 2.1 applied to theinduced metric ω = f ∗ ω on N where f = f | N together with the Gauss-Codazzi equation shows that the scalar curvature S ω of ω is semi-negative on N . Moreover, it vanishes identically there if and only if f : ( N , ω ) → ( M, ω )is totally geodesic and ω is flat. We now proceed to show that not only the latteris the case but that in fact df has maximal rank over N sm so that f is a totallygeodesic immersion with the induced flat metric there and in particular N = N .Since L is a subbundle of F − (Λ k T M ) and Λ k ω is a metric on Λ k T M , wesee that L has an induced metric h which restricts to det ω on N and thusRic( h ) = Ric( ω ) on N . Here, det ω is a metric on det T N identified with L | N via s . As the holomorphic sectional curvature decreases on subvarieties and as ω is K¨ahler on N , ( N , ω ) has scalar curvature S ω ≤ Z X c ( L ) ∧ F ∗ ω k − = 12 π Z X Ric( h ) ∧ F ∗ ω k − = 1 nπ Z N S ω ω k ≤ , with equality in the inequality if and only if ω is flat and f totally geodesic.But the first integral above is the sum of the following two integrals: Z X c ( K X ⊗ L ) ∧ F ∗ ω k − = Z X c ( D ) ∧ F ∗ ω k − = Z D i ∗ F ∗ ω k − , EMI-NEGATIVE HOLOMORPHIC SECTIONAL CURVATURE 15 where i is the inclusion of D red in X , and (as E is τ -exceptional) Z X c ( − K X ) ∧ F ∗ ω k − = Z X c ( − τ ∗ K N − E ) ∧ τ ∗ f ∗ ω k − = Z N c ( − K N ) ∧ f ∗ ω k − , both of which are semi-positive since D is effective and − K N pseudo-effective.This forces all of the above integrals to vanish. In particular, ω is flat on N andtherefore, since Ric( h ) = Ric( ω ) = 0 on N , Ric( h ) = 0 on X . We then have, − K N being pseudo-effective, that for a generic curve C cut out by hyperplanes on N : 0 ≤ D.τ ∗ C = K N .C ≤ , forcing equality. Hence D is τ -exceptional and s is nowhere zero on N . By [Kle66,Ch. I, §
4, Prop. 3], it also follows that K N is numerically trivial.Finally, to see that f is totally geodesic along N sm , we argue as follows. Since theexceptional divisor E of τ is rationally connected, the condition c ( L ) = 0 impliesthat L is trivial on E and that its inclusion into the trivial bundle F − (Λ k T M ) | E is constant. Hence the subbundle L of F − (Λ k T M ) is the pullback of a subbundle˜ L of f − (Λ k T M ) on N . This means that the inclusion of K ∨ X in L over τ − ( N sm )factors through the inclusion ˜ s of K ∨ N sm in ˜ L | N sm given by the section det( df sm ) of Hom ( K ∨ N sm , f − sm (Λ k T M )), where f sm = f | N sm . Since ˜ s = s on N , where we haveshown it is nowhere zero, and since the complement of N in N sm has codimensiontwo or higher, the Cartier divisor (˜ s | N sm ) must be zero on N sm and hence det( df )is nowhere zero on N sm . We may now conclude that f | N sm is a totally geodesicimmersion as before. (cid:3) Remark 6.3.
The above proposition generalizes Proposition 2.2 and Lemma 3.1.Observe that the case of kod( M ) ≥ n − r M = n can be obtained directlyjust from Lemma 3.1 since Kodaira dimension zero minimal surfaces are dominatedeither by an abelian surface or by families of elliptic curves by [MM83]. Remark 6.4.
Although we do not need it in this paper, with a little further work, N can be shown to be smooth. Also, it is clear from the above proof that theprojectivity assumption on M is unnecessary and the singularity assumption on N is made only to guarantee that K N is Q -Cartier if f is a morphism. Implicationsfor M and its pluricanonical systems will be discussed elsewhere.7. Structural decomposition theorem in dimension two
In this section, we prove Theorem 1.10. Under its assumptions, the nef fibrationis known to be a morphism given by a pluricanonical map, which we take to be theKodaira-Iitaka map. Out of the three possibilities for the Kodaira dimension, thecase kod( M ) = 1 is the key one to treat. In this case, we denote the Kodaira-Iitakamap by π : M → Y , where Y is a smooth curve. Now, π induces an orbifoldstructure on Y by assigning the multiplicity m ( q ) to a point q on Y given by themultiplicity of the generic fiber of π above q . The orbifold Y ∂ so endowed has thecanonical Q -divisor K Y ∂ = K Y + P i (1 − m ( q i ) ) q i following the notation of [Lu02]. Proof of Theorem 1.10.
It follows from Theorem 1.4 and the validity of the Abun-dance Conjecture on surfaces that kod( M ) ≥
0. In case kod( M ) = 0, Theorem 1.4 states that M has a finite unramified cover by an abelian surface, so there is nothingto prove. Also, in case kod( M ) = 2, we have already seen that K M is ample. So itremains to treat the case kod( M ) = 1.Under the present assumptions, π is an elliptic fibration whose only degeneratefibers are multiples of elliptic curves, as any other type of degenerate fiber wouldcontain rational curves (see [BPVdV84, p. 150, Table 3. Kodaira’s table of singularelliptic fibers]). It follows that the canonical divisor K M is the pullback of theorbifold canonical Q -divisor K Y ∂ of the base, see for example [BL00]. This meansthat K Y ∂ is ample and that Y ∂ is the quotient of the unit disk by a Fuchsian group.As such a group has finite index torsion free subgroups by Selberg’s lemma (see[Rat06]), there is a finite cover of Y by a smooth curve C of genus at least twobranched over Y to precisely the same multiplicity as that given by π . After thebase change to C and a normalization, we obtain a fibration over C whose totalspace ˜ M is an unramified covering of M and a holomorphic fiber bundle over C (the j -invariant of the fibers gives rise to a holomorphic map j : C → C and thisforces j to be constant). Replacing C by a finite unramified covering if necessary,we then have ˜ M = C × F where F is an elliptic curve and ˜ M is a finite unramifiedcovering of M . The existence of such an unramified cover of C is proven in [Bea96,Prop. VI.8] based on the existence and the affineness of the fine moduli spaces forelliptic curves with level structure.Proposition 1.11 now shows that the pull-back metric ˜ g on ˜ M = C × F of g is the product of a flat metric on F with a metric g C on C up to adding a termcorresponding to a (1 , P i ( p ∗ µ i ∧ p ∗ ν i + p ∗ µ i ∧ p ∗ ν i ) where p , p are the projections and µ i , ν i holomorphic one forms on C and F , respectively.But this additional term vanishes if we pull back by a constant section of p (afiber of p ) so that g C corresponds to the induced metric by ˜ g on this fiber. Bythe curvature decreasing properties on subvarieties, it follows that the holomorphicsectional curvature of g C is semi-negative. (cid:3) Before we prove Proposition 1.11, we note that its statement (and proof) bearsresemblance to Zheng’s theorem in [Zhe93, p. 672], which comes with an assumptionof semi-negative bi sectional curvature. This assumption is not present in our case,but instead we can invoke the uniqueness of K¨ahler-Einstein metrics in a givenK¨ahler class at a crucial point of the proof. Proof of Proposition 1.11.
Since ω is K¨ahler, the K¨unneth formula for (1 , ∂ ¯ ∂ -lemma show that there exist a real C ∞ function φ on M , real (1 , ω F on F and ω Y on Y such that ω − p ∗ ω F − √− ∂ ¯ ∂φ = π ∗ ω Y + X i ( π ∗ µ i ∧ p ∗ ν i + π ∗ µ i ∧ p ∗ ν i )for holomorphic one forms ν i on F and µ i on Y . Now, the right hand side pulls backto zero by each constant section s y : F → M since it factors through the inclusion i y : M y ֒ → M of the fiber M y = π − ( y ). It follows that s ∗ y ω is cohomologousto ω F . Since the former is Einstein, by the uniqueness of such (1 , i ∗ y √− ∂ ¯ ∂φ = 0 for all y ∈ Y ifwe choose ω F to be this unique K¨ahler form. With this choice, φ is a harmonicfunction on M y and therefore constant on M y for all y ∈ Y . This means that φ isa function of y only and thus the term √− ∂ ¯ ∂φ above may be absorbed into ω Y . EMI-NEGATIVE HOLOMORPHIC SECTIONAL CURVATURE 17
The (1 , Y is necessarily a K¨ahler form since its pullback to ahorizontal fiber (a fiber of p ) is also the pullback of the K¨ahler form ω . (cid:3) Remark 7.1.
A brief computation shows that if ω pulls back to flat metrics onthe fibers of π , then these fibers with the induced metric are totally geodesic. Itshould also be noted that the projectivity assumption in Proposition 1.11 is merelymade to make the proposition appear in line with the overall setting of the paper.Its proof does not use it and in fact works for K¨ahler manifolds. Remark 7.2.
After this paper had been completed and gone to press at the Journalof Differential Geometry, Wu and Yau gave a proof of Conjecture 1.2 in [WY15].Subsequently, Tosatti and Yang in [TY15] extended that proof to the K¨ahler case.However, since these new proofs, which are based on the complex Monge-Amp`ereequation and a refined Schwarz Lemma, seem to require strict negativity of theholomorphic sectional curvature, the relationship of these works with our resultsregarding semi-negative holomorphic sectional curvature remains unclear.
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Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston,TX 77204, USA,
E-mail address : [email protected] D´epartment de Math´ematiques, Universit´e du Qu´ebec `a Montr´eal, C.P. 8888, Suc-cursale Centre-Ville, Montr´eal, Qc H3C 3P8, Canada,
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