On projective Kähler manifolds of partially positive curvature and rational connectedness
aa r X i v : . [ m a t h . AG ] S e p ON PROJECTIVE K ¨AHLER MANIFOLDS OF PARTIALLYPOSITIVE CURVATURE AND RATIONALCONNECTEDNESS
GORDON HEIER AND BUN WONG
Abstract.
In a previous paper, we proved that a projective K¨ahlermanifold of positive total scalar curvature is uniruled. At the other endof the spectrum, it is a well-known theorem of Campana and Koll´ar-Miyaoka-Mori that a projective K¨ahler manifold of positive Ricci cur-vature is rationally connected. In the present work, we investigate theintermediate notion of k -positive Ricci curvature and prove that for aprojective n -dimensional K¨ahler manifold of k -positive Ricci curvaturethe MRC fibration has generic fibers of dimension at least n − k + 1.We also establish an analogous result for projective K¨ahler manifoldsof semi-positive holomorphic sectional curvature based on an invariantwhich records the largest codimension of maximal subspaces in the tan-gent spaces on which the holomorphic sectional curvature vanishes. Inparticular, the latter result confirms a conjecture of S.-T. Yau in theprojective case. Introduction and statement of the results
In our previous work [HW12], we showed that an n -dimensional complexprojective K¨ahler manifold M of positive total scalar curvature is uniruled ,which means that there exists a dominant rational map from P × N onto M , where N is a complex projective variety of dimension n −
1. Recallthat being uniruled is the same as having a rational curve passing throughevery point. Our proof was based on the important equivalence, establishedby Boucksom-Demailly-Peternell-P˘aun in [BDPP13], of uniruledness to theproperty that the canonical line bundle of M is not pseudo-effective.On the other hand, it is a well-known result of Campana [Cam92] andKoll´ar-Miyaoka-Mori [KMM92] that a projective K¨ahler manifold of positiveRicci curvature (aka a Fano manifold) is rationally connected , i.e., any two Mathematics Subject Classification.
Key words and phrases.
Complex projective manifolds, K¨ahler metrics, positive holo-morphic sectional curvature, k -positive Ricci curvature, rational curves, uniruledness, ra-tional connectedness.The first author is partially supported by the National Security Agency under GrantNumber H98230-12-1-0235. The United States Government is authorized to reproduceand distribute reprints notwithstanding any copyright notation herein. points can be joined by a chain of rational curves or, equivalently, by asingle rational curve. In light of all of this, we establish two main theoremsin the case of partially positive curvature which are natural extensions ofthe previous results. Throughout the paper, we shall work over the field ofcomplex numbers C .The main kinds of partially positive curvatures which we will consider are k -positive Ricci curvature and semi-positive holomorphic sectional curvaturewith a certain amount of positivity captured by the numerical invariant r + M .For the definitions, we refer to Section 2. Our first main theorem is thefollowing. Theorem 1.1.
Let M be a projective manifold with a K¨ahler metric with k -positive Ricci curvature, where k ∈ { , . . . , n := dim M } . Then a genericfiber of the MRC fibration of M has dimension at least n − k + 1 . Remark 1.2.
The case k = 1 in Theorem 1.1 represents the above-mentionedwell-known result of Campana and Koll´ar-Miyaoka-Mori. The case k = n isthe case of positive scalar curvature, which was handled in [HW12] (underthe even weaker assumption of positive total scalar curvature).We remark that at the end of the proof of Theorem 1.1, the symbol ε merely has to denote a semi-positive continuous function which is positiveat at least one point in order for the proof to go through verbatim. Thus,Theorem 1.1 can immediately be generalized to the following theorem. Theorem 1.3.
Let M be a projective manifold with a K¨ahler metric with k -semi-positive Ricci curvature, where k ∈ { , . . . , n := dim M } . Assumethat that there exists at least one point of M at which the K¨ahler metric has k -positive Ricci curvature. Then a generic fiber of the MRC fibration of M has dimension at least n − k + 1 . Since it might be of independent interest, we would also like to point outthat the proof of Theorem 1.1 immediately yields the following corollary.
Corollary 1.4.
Let M be a projective manifold with a K¨ahler metric with k -semi-positive Ricci curvature, where k ∈ { , . . . , dim M } . Assume thatthat there exists at least one point of M at which the K¨ahler metric has k -positive Ricci curvature. Let Z be a projective k -dimensional manifold withpseudo-effective canonical line bundle. Then there does not exist a dominantrational map from M to Z . Our second main theorem is the subsequent one. The invariant r + M cap-tures the largest codimension of maximal subspaces in the tangent spaceson which the holomorphic sectional curvature vanishes. Its analog in thesemi-negative curvature case (which we denoted by r M ) was used for thestructure theorems in [HLW15] and it is similar to the established notion of N PROJECTIVE K ¨AHLER MANIFOLDS OF PARTIALLY POSITIVE CURVATURE 3 the Ricci rank. One of our more philosophical points is that in connectionwith holomorphic sectional curvature, r M and r + M are appropriate numericalinvariants to consider. It is thus tempting to call them the HSC rank . Fora precise definition, we refer to Section 2.
Theorem 1.5.
Let M be a projective manifold with a K¨ahler metric ofsemi-positive holomorphic sectional curvature. Then a generic fiber of theMRC fibration of M has dimension at least r + M . Remark 1.6.
In the special case that r + M = dim M , the above theoremyields that M is rationally connected. It is immediate from the definitionthat r + M = dim M is achieved as soon as the holomorphic sectional curvatureis positive at one point of M (and semi-positive on all of M ). In particular,Theorem 1.5 yields that a projective manifold with a K¨ahler metric of posi-tive holomorphic sectional curvature is rationally connected. This had beenconjectured by S.-T. Yau (even in the K¨ahler case) and was included in his1982 list of problems [Yau82, Problem 47].Moreover, recall that Tsukamoto [Tsu57] proved that a compact K¨ahlermanifold of positive holomorphic sectional curvature is simply connected.We remark that the above theorem yields the same conclusion for the case ofa projective K¨ahler manifold M of positive holomorphic sectional curvatureand in fact extends it to the case of semi-positive holomorphic sectionalcurvature with r + M = dim M . The reason is that, due to [Cam91], it isknown that a rationally connected projective manifold is simply connected.Recall furthermore that it is known that a rationally connected projectivemanifold has no global non-zero covariant holomorphic tensor fields and thatthe converse of this statement is a conjecture of Mumford (see [Kol96, p.202]).The proof of the next theorem is essentially identical to that of Theorem1.5. Thus, we simply record the theorem here and omit its proof. Theorem 1.7.
Let M be a projective manifold with a K¨ahler metric ofsemi-positive scalar curvature with respect to k -dimensional subspaces, where k ∈ { , . . . , n := dim M } . Assume that there exists at least one point of M at which the scalar curvature with respect to k -dimensional subspaces ispositive. Then a generic fiber of the MRC fibration of M has dimension atleast n − k + 1 . In this case, the corresponding statement regarding the non-existence ofcertain maps is the following corollary.
Corollary 1.8.
Let M be a projective manifold with a K¨ahler metric ofsemi-positive scalar curvature with respect to k -dimensional subspaces, where k ∈ { , . . . , dim M } . Assume that there exists at least one point of M atwhich the scalar curvature with respect to k -dimensional subspaces is pos-itive. Let Z be a projective k -dimensional manifold with pseudo-effective GORDON HEIER AND BUN WONG canonical line bundle. Then there does not exist a dominant rational mapfrom M to Z . At first glance, it might seem that the kinds of methods used in our proofof Theorems 1.5 and 1.7 require that the generic fibers of the dominantrational map in the above corollary are assumed to be compact. However,this is not the case, as one can show with the following additional arguments.Let us assume that a map ς from M to Z exists as in the corollary. We mayremove the indeterminacy of ς as usual with a holomorphic birational map ρ : M ∗ → M such that ς ◦ ρ : M ∗ → Z is holomorphic. It is immediate fromthe definition that the generic fibers of the respective MRC fibrations of M and M ∗ have the same dimension, which is at least n − k + 1 according toTheorem 1.7. Since Z is assumed to be k -dimensional, the generic fibers of ς ◦ ρ are of the strictly smaller dimension n − k , and it is now clear thatthere exists a rational curve through a generic point of Z , i.e., Z is uniruled.However, this is impossible due to [BDPP13] and the assumption that Z has pseudo-effective canonical line bundle. Remark 1.9.
We would like to point out that in our theorems, the Riccicurvature may have negative eigenvalues and our theorems will still applyas long as the assumed positivity conditions hold, while other works in thisdirection seem to require that the Ricci curvature is semi-definite.
Remark 1.10.
It is clear that all we really require in terms of positivityassumptions is the positivity of the integrals appearing in our proofs. It is forthis same reason that our result in [HW12] is stated in terms of total scalarcurvature, which is the integral of the scalar curvature function. Therefore,making pointwise positivity assumptions as we do in our theorems is actuallyoverkill and basically just due to our desire to formulate iconic theorems.
Remark 1.11.
In the remainder of this paper, we will use the symbol k aspart of the set of indices i, j, k, l , and thus speak of κ -positive Ricci curvatureetc., again with κ ∈ { , . . . , n := dim M } .The outline of this paper is as follows. In Section 2, we provide the basicdefinitions and technical background. In Section 3, our two main theoremsare proven. The proofs completely coincide except at their very end, wherethe respective specific positivity assumptions are used. In the subsectionentitled Proof of Theorem 1.5, we provide the part of the proof that differsfrom the proof of Theorem 1.1. Acknowledgement.
The first author thanks Brian Lehmann for a helpfuldiscussion about pseudo-effective line bundles.
N PROJECTIVE K ¨AHLER MANIFOLDS OF PARTIALLY POSITIVE CURVATURE 5 Differential and algebraic geometric background material
In this section, we will give all the relevant definitions used throughoutthe paper. They are mostly well-known, but we hope the reader will find ituseful to have all of them gathered neatly in one place.2.1.
Notions of curvature.
Let M be an n -dimensional complex mani-fold. If V is a hermitian vector bundle on M of rank ρ , then we denote by θ αβ ( α, β = 1 , . . . , ρ ) the connection matrix of the metric connection withrespect to a local frame f , . . . , f ρ . The corresponding curvature tensor Θ isdetermined by Θ αβ = dθ αβ − ρ X γ =1 θ αγ ∧ θ γβ . One of the most interesting cases of the above is when V is the holomor-phic tangent bundle of M with a local frame ∂∂z , . . . , ∂∂z n , and the hermitianmetric is a K¨ahler metric. Let us recall the basics of this case.Let z , . . . , z n be local coordinates on M . Let g = n X i,j =1 g i ¯ j dz i ⊗ d ¯ z j be a hermitian metric on M and ω = √− n X i,j =1 g i ¯ j dz i ∧ d ¯ z j the (1 , g . Then g is called K¨ahler if and only if locallythere exists a real-valued function f such that g i ¯ j = ∂ f∂z i ∂ ¯ z j . An equivalentcharacterization of the K¨ahler property is that dω = 0.For holomorphic tangent vectors u = P ni =1 u i ∂∂z i , v = P ni =1 v i ∂∂z i , wedefine the (1 , u ¯ v to beΘ u ¯ v = n X i,j,k =1 Θ ik g k ¯ j u i ¯ v j . Moreover, the curvature 4-tensor is given by R ( ∂∂z i , ∂∂ ¯ z j , ∂∂z k , ∂∂ ¯ z l ) = Θ ∂∂zk ∂∂ ¯ zl ( ∂∂z i , ∂∂ ¯ z j ) . Since we assume the metric g to be K¨ahler, the curvature 4-tensor satisfiesthe K¨ahler symmetry R ( ∂∂z i , ∂∂ ¯ z j , ∂∂z k , ∂∂ ¯ z l ) = R ( ∂∂z k , ∂∂ ¯ z l , ∂∂z i , ∂∂ ¯ z j ) . GORDON HEIER AND BUN WONG
Now, we define the
Ricci curvature form to be the (1 , Ric = √− T r (Θ)( · , · ) = √− n X i,j,k =1 R ( ∂∂z i , ∂∂ ¯ z j , e k , ¯ e k ) dz i ∧ d ¯ z j , where e , . . . , e n is a unitary frame. If we let R i ¯ j = n X k =1 R ( ∂∂z i , ∂∂ ¯ z j , e k , ¯ e k ) , then Ric = √− n X i,j =1 R i ¯ j dz i ∧ d ¯ z j . We say that the Ricci curvature is κ -(semi-)positive at the point p ∈ M if the eigenvalues of the hermitian n × n matrix R i ¯ j at p have the propertythat any sum of κ of them is (semi-)positive. Note that if κ ≤ κ ′ , thenbeing κ -(semi-)positive implies being κ ′ -(semi-)positive. Moreover, by defi-nition, being n -(semi-)positive is the same as having (semi-)positive scalarcurvature, and being 1-(semi-)positive is the same as having (semi-)positiveRicci curvature. We say that the Ricci curvature is κ -(semi-)positive if it is κ -(semi-)positive at all points p ∈ M .The scalar curvature with respect to a κ -dimensional subspace Σ ⊂ T p M is defined to be κ X i,j =1 R ( e i , ¯ e i , e j , ¯ e j ) , where e , . . . , e κ is a unitary basis for Σ. When Σ = T p M , we simplyspeak of the scalar curvature . We say that the scalar curvature with re-spect to κ -dimensional subspaces is (semi-)positive at the point p ∈ M if thescalar curvature with respect to all κ -dimensional subspaces Σ ⊂ T p M is(semi-)positive. We say that the scalar curvature with respect to κ -dimen-sional subspaces is (semi-)positive if it is (semi-)positive at all points p ∈ M .Note again that these (semi-)positivity properties are preserved under in-creasing the value of κ .If ξ = P ni =1 ξ i ∂∂z i is a non-zero complex tangent vector at p ∈ M , thenthe holomorphic 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 . We say that the holomorphic sectional curvature is (semi-)positive if H ( ξ ) > ≥ ∀ ξ ∀ p ∈ M. Now, let us assume that H is semi-positive. We define the invariant r + M as follows. For p ∈ M , let n ( p ) be the maximum of those integers ℓ ∈ N PROJECTIVE K ¨AHLER MANIFOLDS OF PARTIALLY POSITIVE CURVATURE 7 { , . . . , dim M } such that there exists a ℓ -dimensional subspace L ⊂ T p M with H ( ξ ) = 0 for all ξ ∈ L \{ ~ } . Set r + M := n − min p ∈ M n ( p ) . Note that bydefinition r + M = 0 if and only if H vanishes identically. Also, r + M = dim M ifand only there exists at least one point p ∈ M such that H is positive at p .Moreover, n ( p ) is upper-semicontinuous as a function of p , and consequentlythe set { p ∈ M | n − n ( p ) = r + M } is an open set in M (in the classical topology).We conclude this subsection with some hopefully useful historical remarks,in particular about the relationship of Ricci and holomorphic sectional cur-vature.The existence of a K¨ahler metric of negative holomorphic sectional curva-ture implies the existence of a (different) metric of negative Ricci curvature,as was very recently established by [WY15] in the projective case and, in thegeneral case, by [TY15]. Previously, the three-dimensional projective case ofthis statement had been proven in [HLW10]. The paper [HLW15] containspartial positivity results for the canonical line bundle in the semi-negativeprojective case. Conversely, it is easy to show via a Schwarz Lemma thatthere are many K¨ahler manifolds of negative Ricci curvature which do notadmit metrics of negative holomorphic sectional curvature.Under the assumption of positive scalar curvature on a hermitian man-ifold, in the pioneering paper [Yau74], S.-T. Yau proved that the Kodairadimension is negative. At the other end of the positive curvature spectrum,Siu and Yau [SY80] proved the Frankel conjecture, which states that a com-pact K¨ahler manifold of positive bisectional curvature is biholomorphic toprojective space. At approximately the same time, Mori [Mor79] estab-lished a more general conjecture of Hartshorne, which states that a compactcomplex manifold with ample tangent bundle is biholomorphic to projectivespace.In light of the above, it may come as a bit of a surprise that positiveholomorphic sectional curvature does in general not imply the existence ofa metric of positive Ricci curvature, as pointed out by Hitchin [Hit75] inhis discussion of the Hirzebruch surfaces P ( O P ( a ) ⊕ O P ), a ∈ { , , , . . . } .Moreover, we do not know if a compact K¨ahler manifold of positive holo-morphic sectional curvature is projective. Conversely, the question of theimplications of positive Ricci curvature for the existence of a metric of posi-tive holomorphic sectional curvature seems to be open at this point as well.2.2. The MRC fibration.
On a smooth projective variety X it is quitenatural to consider the equivalence relation of being connected by a rationalcurve, i.e., two points x, y ∈ X are considered to be equivalent if and only ifthere exists a rational curve containing both x and y . The problem is that GORDON HEIER AND BUN WONG the map to the quotient under this equivalence relation is in general not agood map. This is the case, for example, when X is a very general projectiveK3 surface because such an X possesses countably infinitely many rationalcurves. The question of how to obtain a good map based on this equivalencerelation has been answered by Campana [Cam92] and Koll´ar-Miyaoka-Mori[KMM92]. The following theorem is [KMM92, Theorem 2.7]. Theorem 2.1.
Let X be a smooth proper algebraic variety over an alge-braically closed field of characteristic . Then there exist an open densesubset U ⊂ X and a proper smooth morphism f : U → Z with the followingproperties: • Every fiber of f is rationally connected. • For a sufficiently general z ∈ Z , there are no rational curves D ⊂ X such that dim D ∩ f − ( z ) = 0 . The morphism f is called the maximally rationally connected fibration of X (or MRC fibration for short).Equivalently, one can think of f as being a dominant rational map whichis holomorphic and proper on a dense open subset of X (aka “almost holo-morphic”). Its general fiber is rationally connected. Moreover, the fibersof f are maximal in the sense that f factors through any other map withrationally connected fibers.Furthermore, we may assume that the base Z is smooth, because if it isnot, we can compose f with a birational map Z Z ′ which resolves thesingularities of Z . The rational map X Z ′ still has all the properties ofthe original f . In general, it is clear that the MRC fibration is well-definedonly up to birational equivalence.Finally, it will be central to our argument that the base Z of the MRCfibration is not uniruled. This statement can be obtained as an immedi-ate corollary ([GHS03, Corollary 1.3]) to the following theorem of Graber-Harris-Starr ([GHS03, Theorem 1.1]). Theorem 2.2.
Let f : X → B be a proper morphism of complex varietieswith B a smooth curve. If the general fiber of f is rationally connected, then f has a section. It is then a consequence of [BDPP13] that the canonical line bundle K Z of the base Z is pseudo-effective.3. Proof of Theorems 1.1 and 1.5
In this section, we prove Theorems 1.1 and 1.5. The proofs are initiallythe same, but there are some subtle differences towards the end in how the
N PROJECTIVE K ¨AHLER MANIFOLDS OF PARTIALLY POSITIVE CURVATURE 9 contradiction is obtained. In particular, the use of the positivity conditionin the proof of Theorem 1.1 is a matter of linear algebra, whereas the proofof Theorem 1.5 requires a lemma of Berger and a global argument involvingthe Divergence Theorem.3.1.
Proof of Theorem 1.1.
We start with some general observationsabout MRC fibrations for which we could not find a reference in the lit-erature. We thus hope that this part is of independent interest for theoverall understanding of this important map.Let V ⊂ M denote the indeterminacy locus of our MRC fibration f . On M \ V , the pullback of the tangent bundle of Z , denoted f ∗ T Z , is a rank m := dim Z holomorphic vector bundle. Since the codimension of V is atleast two, this vector bundle can be extended to all of M as a reflexive sheafin a unique way, and we denote this extension with the symbol E . Thecanonical line bundle K Z and its dual K ∗ Z can be pulled back under f andextended to all of M as line bundles. We denote these extensions by f ∗ K Z and f ∗ K ∗ Z and note that they agree with det E ∗ and det E , respectively.On M \ V , there is an exact sequence of coherent sheaves0 → T ( M \ V/f ( M \ V )) → T M df → f ∗ T Z → N → , where N is a coherent sheaf supported on the locus B where f : M \ V → f ( M \ V ) is not smooth (see [Ser06, Definition 3.4.5]). Let W = V ∪ B .We will now prove that Z M c ( E ) ∧ ω n − is non-positive, based on the following proposition. The proof of our the-orems will then be finished by establishing that this integral can also beshown to be positive under the assumption that generic fibers of the MRCfibration are of dimension no greater than n − κ and, respectively, that thesefibers are of dimension no greater than r + M −
1. We suspect the statementof the proposition is known to experts, but for lack of a suitable reference,we provide a proof.
Proposition 3.1.
Let ν : X Y be a dominant rational map betweencomplex projective manifolds X, Y . Let L be a pseudo-effective line bundleon Y . Then the pull-back line bundle ν ∗ L is a pseudo-effective line bundleon X .Proof. Among the various equivalent definitions of pseudo-effectivity for aline bundle (see [Dem01]), a convenient one to use in this context is thefollowing. The line bundle L is pseudo-effective if its numerical class liesin the pseudo-effective cone PEff( Y ), i.e., in the closure of the convex cone generated by Chern classes of effective line bundles in the real N´eron-Severivector space N ( X ) R .Furthermore, we observe that for an arbitrary holomorphic map ν ′ : X ′ → Y from a complex projective manifold X ′ , the pullback map ν ′∗ : N ( Y ) R → N ( X ′ ) R is an injective linear map of finite dimensional real vector spaces.Moreover, since PEff( Y ) is generated by the classes of line bundles with asection, there is an induced injective map ν ′∗ : PEff( Y ) → PEff( X ′ ).Now, let σ : X ′ → X be a birational holomorphic map such that thecomposition ν ′ := ν ◦ σ : X ′ → Y is a holomorphic map. Due to the aboveremarks, ν ′∗ L is a pseudo-effective line bundle on X ′ . Furthermore, thepushforward σ ∗ of divisors induces a linear map σ ∗ : N ( X ′ ) R → N ( X ) R .Since the pushforward of an effective divisor is still effective (or zero), weget an induced map σ ∗ : PEff( X ′ ) → PEff( X ). Since ν ∗ = σ ∗ ◦ ν ′∗ , ourclaim is proven. (cid:3) In [HW12, Section 2], a linear algebra argument was given for the factthat any pseudo-effective line bundle P on M satisfies Z M c ( P ) ∧ ω n − ≥ . Alternatively, the non-negativity of this integral can be justified by arguingthat it holds if the numerical class of P is a positive scalar multiple ofthe numerical class of an effective line bundle. The non-negativity is thenpreserved when taking limit.Since det E = f ∗ K ∗ Z , we have c ( E ) = − c ( f ∗ K Z ). Also, by Proposition 3.1, the pseudo-effectivityof K Z implies the pseudo-effectivity of f ∗ K Z , and we obtain the desired in-equality as follows. Z M c ( E ) ∧ ω n − = Z M c ( f ∗ K ∗ Z ) ∧ ω n − = − Z M c ( f ∗ K Z ) ∧ ω n − ≤ . In the rest of this section, we shall infer that the value of the above integralis positive under the assumptions of Theorems 1.1 and 1.5, respectively,resulting in contradictions. Our argument is based on the well-known factthat, as a quotient bundle of
T M over M \ W , the vector bundle E | M \ W N PROJECTIVE K ¨AHLER MANIFOLDS OF PARTIALLY POSITIVE CURVATURE 11 carries an induced hermitian metric h whose curvature is equal to or morepositive than that of g on T M over M \ W .To be precise, let us recall the following standard setup. Locally on M \ W ,we choose a unitary frame e , . . . , e n for T M such that e , . . . , e n − m is aframe for S := T ( M \ V/f ( M \ V )) . The connection matrix for the metric con-nection on T M is θ T M = (cid:18) θ S ¯ A T A θ E (cid:19) , where θ S , θ E are the respective connection matrices for S and E , and A ∈ A , (Hom( S, E )) is the second fundamental form of S in T M . Now, thecurvature matrix in terms of the unitary frame e , . . . , e n isΘ T M = (cid:18) dθ S − θ S ∧ θ S − ¯ A T ∧ A ∗∗ dθ E − θ E ∧ θ E − A ∧ ¯ A T (cid:19) , which implies that Θ E = Θ T M | E + A ∧ ¯ A T and, in particular,(1) Θ E ≥ Θ T M | E . We let the (1 , η be the trace of the matrix Θ E over M \ W , i.e., η = m X α =1 (Θ E ) αα . Let ˜ h be an arbitrary smooth hermitian metric on det E over the entire M with curvature form ˜ η . If τ is a local nowhere zero holomorphic section ofdet E , the ratio q = (det h )( τ, τ )˜ h ( τ, τ )is independent of the choice of τ and constitutes a smooth positive functionon M \ W . Over that same set, we have the following relationship:˜ η = η + ∂ ¯ ∂ log q. By standard techniques from the theory of resolution of singularities andthe removal of indeterminacy, there is a compact complex manifold M ∗ anda surjective holomorphic map ρ : M ∗ → M such that ρ | M ∗ \ ρ − ( W ) : M ∗ \ ρ − ( W ) → M \ W is biholomorphic, the total transform ρ ∗ ( W ) is a divisor with simple normalcrossing support in M ∗ , and f ◦ ρ is holomorphic. With positive integers a i , b j , write ρ ∗ ( W ) = X i ∈ I a i D (1) i + X j ∈ J b j D (2) j , where the D (1) i are the irreducible components of ρ ∗ ( W ) such that ρ ( D (1) i )has codimension one and the D (2) j are the irreducible components of ρ ∗ ( W )such that ρ ( D (2) j ) has codimension at least two. Furthermore, Z M c ( E ) ∧ ω n − = Z M √− π ˜ η ∧ ω n − = Z M \ W √− π ˜ η ∧ ω n − (2) = Z M \ W √− π η ∧ ω n − + Z M \ W √− π ∂ ¯ ∂ log q ∧ ω n − . (3)The equality (2) holds because ˜ η is a smooth (1 , Z M \ W √− π ∂ ¯ ∂ log q ∧ ω n − = Z M ∗ \ ρ − ( W ) √− π ∂ ¯ ∂ log ρ ∗ q ∧ ρ ∗ ω n − , is a non-negative number. To be more precise, we will see that this integralis always non-negative and, additionally, positive if and only if there existsa divisor along which f is not smooth. The reason is the following chain ofequalities. Z M ∗ \ ρ − ( W ) √− π ∂ ¯ ∂ log ρ ∗ q ∧ ρ ∗ ω n − = X i ∈ I a i Z D (1) i ρ ∗ ω n − + X j ∈ J b j Z D (2) j ρ ∗ ω n − (4) = X i ∈ I a i Z ρ ( D (1) i ) ω n − + X j ∈ J b j Z ρ ( D (2) j ) ω n − = X i ∈ I a i Z ρ ( D (1) i ) ω n − + 0(5)Note that equality (4) is due to the Poincar´e-Lelong equation and equality(5) is due to the fact that dim ρ ( D (2) j ) ≤ n −
2. Moreover, if I = ∅ , then thevalue of (5) is zero. If I = ∅ , then the value of (5) is positive.We now observe that on M \ W : √− η ∧ ω n − = √− m X α =1 (Θ E ) αα ∧ ω n − ≥ √− m X α =1 (Θ T M ) ( n − m + α ) ( n − m + α ) ∧ ω n − (6) = √− m X α =1 (Θ T M ) e ( n − m + α ) ¯ e ( n − m + α ) ∧ ω n −
1N PROJECTIVE K ¨AHLER MANIFOLDS OF PARTIALLY POSITIVE CURVATURE 13 = √− m X α =1 n X i,j =1 R ( ∂∂z i , ∂∂ ¯ z j , e ( n − m + α ) , ¯ e ( n − m + α ) ) dz i ∧ d ¯ z j ∧ ω n − = √− m X α =1 n X i,j =1 R ( e ( n − m + α ) , ¯ e ( n − m + α ) , ∂∂z i , ∂∂ ¯ z j ) dz i ∧ d ¯ z j ∧ ω n − (7) = 2 n m X α =1 Ric ( e ( n − m + α ) , ¯ e ( n − m + α ) ) ! ω n (8)The inequality in (6) is due to the curvature increasing property (1). Theequality (7) is due to the K¨ahler symmetry of the curvature 4-tensor, andequality (8) is due to the trace formula.To obtain the desired contradiction, let us assume that a general fiber ofthe MRC fibration is of dimension no greater than n − κ . This condition isclearly equivalent to m ≥ κ . The technical reason behind our argument is thetheory of minimax formulae and extremal partial traces for the eigenvaluesof hermitian matrices. For a nice account of this material we refer to [Tao12,Section 1.3.2]. In a nutshell, the key point is the following.Given an n × n matrix T and an m -dimensional subspace S ⊂ C n , onecan define the partial trace of T with respect to S and a fixed hermitianinner product to be the expressiontr S T := m X i =1 v ∗ i T v i , where v , . . . , v m is any unitary basis of S . We simply write tr T for tr C n T .The displayed expression is easily seen to be independent of the choice ofthe unitary basis and thus well-defined. If we assume that T is hermitianand let λ ≥ . . . ≥ λ n be the eigenvalues of T , then for any 1 ≤ m ≤ n , onehas λ + . . . + λ m = sup dim( S )= m tr S T and(9) λ n − m +1 + . . . + λ n = inf dim( S )= m tr S T. Now, the sum P mα =1 Ric ( e ( n − m + α ) , ¯ e ( n − m + α ) ) is the partial trace of thehermitian n × n matrix R i ¯ j with respect to S = span { e ( n − m +1) , . . . , e n } .Thus, according to (9), the expression (8) is bounded below by2 n m X α =1 τ n − m + α ! ω n , (10)where τ n − m +1 , . . . , τ n denote the m smallest eigenvalues of the hermitian n × n matrix R i ¯ j . Due to m ≥ κ and the assumption of κ -positivity, P mα =1 τ n − m + α is positive everywhere on M , and the expression (10) is there-fore bounded below by εω n for some positive number ε . To conclude theproof, we observe that altogether Z M c ( E ) ∧ ω n − ≥ Z M \ W √− π η ∧ ω n − (11) ≥ π Z M \ W εω n > . Note that the inequality (11) is due to the non-negativity of the secondsummand in (3). We have obtained the desired contradiction to completethe proof of Theorem 1.1.3.2.
Proof of Theorem 1.5.
Let us assume that a general fiber of theMRC fibration is of dimension no greater than r + M −
1. This condition isclearly equivalent to m ≥ n − r + M + 1. We enter the proof of Theorem 1.1 atthe point where it was determined that it remained to prove the positivityof Z M \ W √− π η ∧ ω n − (12)in order to obtain a contradiction.Now, we recall that the set W is the union of the indeterminacy locus V of the MRC fibration f and the locus where the map does not have full rank.Since f is almost holomorphic and due to the standard generic smoothnessproperty of holomorphic maps, the set W does not intersect a generic fiberof f . Thus, there is a dense Zariski-open subset Z ′ ⊂ Z such that f | f − ( Z ′ ) : f − ( Z ′ ) → Z ′ is a holomorphic submersion. Moreover, since the integrand in (12) is asmooth form, the domain of integration can be replaced with the densesubset f − ( Z ′ ). Thus, it suffices to take an arbitrary but fixed small openset V ⊂ Z ′ and prove that Z f − ( V ) √− π η ∧ ω n − (13)is non-negative in general and positive for a certain choice of V .Let w , . . . , w m be local coordinates on a given V . For each q ∈ V , wewrite M q := f − ( q ) for the fiber over q . For each p ∈ f − ( V ), we can choose N PROJECTIVE K ¨AHLER MANIFOLDS OF PARTIALLY POSITIVE CURVATURE 15 tangent vectors ( g ∂∂w , . . . , g ∂∂w m ) forming a smooth frame for (( T p M f ( p ) ) ⊥ ) p ∈ f − ( V ) and having the propertythat df p g ∂∂w i ! = ∂∂w i ( i = 1 , . . . , m ) . In a small neighborhood U of an arbitrary but fixed p ∈ M q , we can chooseholomorphic coordinate functions z , . . . , z n − m such that for all p ′ ∈ U : T p ′ M f ( p ′ ) = span (cid:26) ∂∂z , . . . , ∂∂z n − m (cid:27) . We write φ = ∂∂z ∗ , . . . , φ n − m = ∂∂z n − m ∗ , φ n − m +1 = g ∂∂w ∗ , . . . , φ n = g ∂∂w m ∗ for the dual fields of one-forms.Let h continue to denote the induced hermitian metric on E | M \ W . Locally,we write η = n X i,j =1 η i ¯ j φ i ∧ ¯ φ j . If we denote by s E the trace of η with respect to g , we can rewrite (13) bythe trace formula as Z f − ( V ) √− π η ∧ ω n − = Z f − ( V ) nπ s E ω n . Returning to the above-described general linear algebra setting for a mo-ment, observe that for any subspace S and its orthogonal complement S ⊥ ,the trace tr T satisfies tr T = tr S ⊥ T + tr S T. If we apply this with S = T p ′ M f ( p ′ ) , then we obtain s E = K + K , where K = n − m X i,j =1 g i ¯ j η i ¯ j = − n − m X i,j =1 g i ¯ j ∂ log det h∂z i ∂ ¯ z j , and K is the partial trace of η with respect to S ⊥ and g . Due to the curva-ture increasing property, K is bounded from below by the scalar curvatureof g with respect to the plane S ⊥ , which we refer to as ˆ K . Due to [Ber66,Lemme 7.4], ˆ K can be expressed as a positive constant times the integral of H over the unit sphere of vectors in S ⊥ . Due to the overall semi-positivity of H , we can conclude ˆ K ≥
0. Furthermore, at a point p with n − n ( p ) = r + M ,due to dim S ⊥ = m ≥ n − r + M + 1 = n ( p ) + 1 , we actually have ˆ K ( p ) >
0. Note that due to the openness of the set { p ∈ M | n − n ( p ) = r + M } , there exists a V with { p ∈ M | n − n ( p ) = r + M } ∩ f − ( V ) = ∅ . We can thus conclude Z f − ( V ) K ω n ≥ Z f − ( V ) ˆ K ω n > , and it remains to show that the integral R f − ( V ) K ω n is non-negative. Adirect computation yields (we write ~ for det h ) K = − n − m X i,j =1 g i ¯ j ∂ log ~ ∂z i ∂ ¯ z j = − n − m X i,j =1 g i ¯ j ~ ∂ ~ ∂z i ∂ ¯ z j + n − m X i,j =1 g i ¯ j ~ ∂ ~ ∂z i ∂ ~ ∂ ¯ z j = ∆ ′ ~~ + n − m X i,j =1 g i ¯ j ~ ∂ ~ ∂z i ∂ ~ ∂ ¯ z j , (14)where ∆ ′ is the Laplacian operator on the fibers with respect to the restric-tion of the metric g to the fibers. Since the second summand in (14) isalways non-negative, we are done if we can prove that Z f − ( V ) ∆ ′ ~~ ω n = 0 . In order to do this, note that the normal bundle of a general fiber isthe trivial bundle of rank m . Therefore, we can regard ~ as global smoothfunction of any given general fiber. We can rewrite ω n ~ = ω ′ ∧ φ n − m +1 ∧ ¯ φ n − m +1 ∧ . . . ∧ φ n ∧ ¯ φ n , where ω ′ is the volume form of the restriction of the metric g on the fibers.Applying the Fubini Theorem of iterated integrals, we have Z f − ( V ) ∆ ′ ~~ ω n = Z V Z f − ( w ) (∆ ′ ~ ) ω ′ ! dw ∧ d ¯ w ∧ . . . ∧ dw m ∧ d ¯ w m . It follows from the Divergence Theorem that on the compact manifold f − ( w ): Z f − ( w ) (∆ ′ ~ ) ω ′ = 0for all w ∈ V . This finishes the proof. N PROJECTIVE K ¨AHLER MANIFOLDS OF PARTIALLY POSITIVE CURVATURE 17
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