\documentclass{eudml-infty} \usepackage{latexsym} \usepackage{amsmath} \usepackage{amssymb} \usepackage{bm} \pagestyle{empty} \begin{document} {\it Compositio Mathematica} 101: 217-224, 1996. \copyright 1996 {\it KluwerAcademic Publishers. Printed in the Netherlands}. 217 Compact Kähler manifolds with hermitian semipositive anticanonical bundle JEAN-PIERRE DEMAILL$\mathrm{Y}^{}$ , THOMAS $\mathrm{PETERNELL}^{2}$ and MICHAEL SCHNEIDER${}^{\text{2}}$ $\iota_{Universit\acute{e}de}$ {\it Grenoble} $I$, {\it Institut Fourier, U.R.A. 188} $du$ {\it C.N.R.S}., $BP74$, {\it Saint-Martin} $d' H\grave{e}res$, {\it France. e-mail}: {\it demailly}@{\it fourier.grenet}.$fr$ 2 {\it Universität Bayreuth, Mathematisches Institut, D-95440 Bayreuth, Deutschland}. {\it e-mail}: {\it thomas.peternell}@{\it uni-bayreuth.de, michael.schneider}@{\it uni-bayreuth.de} Received21 October 1994; accepted in final form 9 March 1995 Abstract. This note states a structure theorem for compact Kahler manifolds with semipositive Ricci curvature: any such manifold has a fnite étale covering possessing a De Rham decomposition as a product of irreducible compact Kahler manifolds, each one being either Ricci flat (torus, symplectic or Calabi-Yau manifold), or Ricci semipositive without non trivial holomorphic forms. Related questions and conjectures conceming the latter case are discussed. Key words: Compact Kahler manifold, semipositive Ricci curvature, complex toms, symplectic manifold, Calabi-Yau manifold, Albanese map, fundamental group, Bochner formula, De Rham decomposition, Cheeger-Gromoll theorem, nef line bundle, Kodaira-Iitaka dimension, rationally connected manifold 1. Main results This short note is a continuation of our previous work [DPS93] on compact Kähler manifolds $X$ with semipositive Ricci curvature. Our purpose is to state a splitting theorem describing the stmcture of such manifolds, and to raise some related questions. The foundational background will be found in papers by Lichnerowicz [Li67], [Li71], and Cheeger-Gromoll [CG71], [CG72]. Recall that a {\it Calabi-Yau} {\it manifold} $X$ is a compact Kahler manifold with $c_{1}(X)=0$ and finite fundamental group $\pi_{1}(X)$, such that the universal covering $\tilde{X}$ satisfes $H^{0}(\displaystyle \overline{X}, \Omega\frac{p}{X})=0$ for all $1\leq p\leq\dim X-1$. A {\it symplectic manifold} $X$ is a compact Kähler manifold admitting a holomorphic symplectic 2-form $\omega$ (of maximal rank everywhere); in particular $K_{X}=\mathcal{O}_{X}$. We denote here as usual $\Omega_{X}=\Omega_{X}^{1}=T_{X}^{\star},\ \Omega_{X}^{p}=\Lambda^{p}T_{X}^{\star},\ \mathrm{A}_{X}^{r}=\det(T_{X}^{\star})$. The following structure theorem generalizes the stmcture theorem for Ricci-flat manifolds (due to Bogomolov $[\mathrm{Bo}74\mathrm{a}],\ [\mathrm{Bo}74\mathrm{b}]$, Kobayashi [Ko81] and Beauville [Be83] $)$ to the Ricci semipositive case. 218 J.-P DEMAILLY ET AL. STRUCTURE THEOREM. {\it Let} $X$ {\it be a compact Kdhler manifold with} $-K_{X}$ {\it hermitian semipositive. Then} (i) {\it The universal covering} $\overline{X}$ {\it admits a holomorphic and isometric splitting} $$ \overline{X}\simeq \mathbb{C}^{q}\times\prod X_{i} $$ {\it with} $X_{i}$ {\it being either a Calabi-Yau manifold or a symplectic manifold or a} {\it manifold} $with-K_{X_{t}}$ {\it semipositive and} $H^{0}(X_{i}, \Omega_{X_{t}}^{\otimes m})=0$ {\it for all} $m>0$. (ii) {\it There exists ajfinite étale Galois covering} $\hat{X}\rightarrow X$ {\it such that the Albanese} {\it variety} Alb({\it X}) {\it is a q-dimensional torus and the Albanese map} $\alpha$ : $\hat{X}\rightarrow \mathrm{Alb}(\hat{X})$ {\it is a locally trivial holomorphic fibre bundle whose fibres} {\it are products} $\Pi X_{i}$ {\it of the type described in} (i), {\it all} $X_{i}$ {\it being simply connected}. (iii) {\it We have} $\pi_{1}(\hat{X})\simeq \mathrm{Z}^{2q}$ {\it and} $n_{1}(X)$ {\it is an extension of a finite group} $\Gamma$ {\it by the} {\it normal subgroup} $\pi_{1}(\hat{X})$. {\it In particular there is an exact sequence} $$ 0\rightarrow \mathrm{Z}^{2q}\rightarrow 7|.1(X)\rightarrow\Gamma\rightarrow 0, $$ {\it and the fundamental group} $\pi_{1}(X)$ {\it is almost abelian}. Recall that a line bundle $L$ is said to be hermitian semipositive if it can be equipped with a smooth hermitian metric of semipositive curvature form. A suffi- cient condition for hermitian semipositivity is that some multiple of $L$ is spanned by global sections; on the other hand, the hermitian semipositivity condition im- plies that $L$ is numerically effective $(\mathrm{nef})$ in the sense of [DPS94], which, for $X$ projective algebraic, is equivalent to saying that $L\cdot C\geq 0$ for every curve $C$ in $X$. Examples contained in [DPS94] show that all three conditions are different (even for $X$ projective algebraic). By $\mathrm{Yau}$'s solution of the $\mathrm{Ca}1$ abi conjecture (see [Au76], [Yau78] $)$, a compact {\it Kahler} manifold $X$ has a hermitian semipositive anticanonical $\mathrm{bundle}-K_{X}$ if and only if $X$ admits a Kähler metric $g$ with Ricci $(g) \geq 0$. The isometric decomposition described in the theorem refers to such Kähler metrics. In view of `standard conjectu res' in minimal model theory it is expected that projective manifolds $X$ with no nonzero global sections in $H^{0}(X, \Omega_{X}^{\otimes m}),\ m>0$, are rationally connected. We hope that most of the above results will continue to hold under the weaker assumption $\mathrm{t}h\mathrm{at}-K_{X}$ is nef instead of hermitian semiposi- tive. However, the technical tools needed to treat this case are still missing. We would like to thank the DFG-Schwerpunktprogramm Komplexe Mannig- faltigkeiten' and the Institut Universitaire de France and for making our work possible. 2. Bochner formula and holomorphic differential forms Our starting point is the following well-known consequence of the Bochner for- mula. $\mathrm{K}\ddot{\mathrm{A}}$ HLER MANIFOLDS WITH SEMIPOSITIVE ANTICANONICAL BUNDLE 219 LEMMA. {\it Let} $X$ {\it be a compact n-dimensional Kdhler manifold} $with-K_{X}$ {\it hermi}- {\it tian semipositive. Then every section of} $\Omega_{X}^{\otimes m},\ m\geq 1$ {\it is parallel with respect to the} {\it given Kdhler metric}. {\it Proof}. The Lemma is an easy consequence of the Bochner formula $\triangle(\Vert u\Vert^{2})=\Vert$ {\it Vu} $\Vert^{2}+Q(u)$, {\it where} $u\in H^{0}(X, \Omega_{X}^{\otimes m})$ and $Q(u)\geq m\lambda_{0}\Vert u \Vert^{2}$. Here $\lambda_{0}$ is the smallest eigen- $\mathrm{v}$ alue of the Ricci curvature tensor. For details see for instance [Ko83]. $\square $ The following definition of a modified Kodaira dimension $\kappa_{+}(X)$ is taken from Campana [Ca93]. As the usual $\mathrm{Kod}$ {\it a}i {\it r}a dimension $\kappa(X)$, this is a birational invariant of $X$. Other similar inv ariants have also been considered in [BR90] and [Ma93]. DEFINITION. Let $Y$ be a comp act complex manifold. We define (i) $\displaystyle \kappa+(Y)=\max$\{ $\kappa(\det \mathcal{F}):\mathcal{F}$ is a subsheaf of $\Omega_{Y}^{p}$ for some $p>0$\}, (ii) $\displaystyle \kappa_{\dagger+}(Y)=\max$\{ $\kappa(\det \mathcal{F}):\mathcal{F}$ {\it is} a subshe {\it a}f of $\Omega_{Y}^{\otimes m}$ for some $m>0$\}. Here we let as usu {\it a}l det $\mathcal{F}=(\Lambda^{r}\mathcal{F})^{**}$, where $r=$ rank $\mathcal{F}$ and $\kappa$ is the usual Iitaka dimension of a line bundle. Clearly, we have $-\infty\leq\kappa(Y)\leq\kappa_{+}(Y)\leq\kappa_{++}(Y)$ where $\kappa(Y)=\kappa(K_{Y})$ is the usual Kodaira dimension. It would be interesting to know whether there are precise relations between $\kappa_{+}(Y)$ and $\kappa_{++}(Y)$, {\it a}s well as with the weighted Kodaira dimensions defined by Manivel [Ma93]. The above lemma implies: PROPOSITION. {\it Let} $X$ {\it be a compact Kdhler manifold with} $-K_{X}$ {\it hermitian} {\it semipositive. Then} $\kappa_{++}(X)\leq 0$. {\it Proof}. Assume that $\kappa_{++}(X)>0$. Then we can find an integer $m>0$ and a subsheaf $\mathcal{F}\subset\Omega_{X}^{\otimes m}$ with $\kappa(\det \mathcal{F})>0$. Hence there is some $\mu\in \mathrm{N}$ and $s \in H^{0}(X, (\det \mathcal{F})^{\mu})$ with $s\neq 0$. Since $\kappa(\det \mathcal{F})>0,\ s$ must have zeroes. Hence the induced section $\tilde{s}\in H^{0}(X, \Omega_{X}^{\otimes\mu rm})$ has zeroes too, $r$ being the rank of $\mathcal{F}$.This contradicts the previous Lemma. $\square $ COROLLARY. {\it Let} $X$ {\it be a compact Kdhler manifold} $with-K_{X}$ {\it hermitian semi}- {\it positive. Let} $\phi:X\rightarrow Y$ {\it be a surjective holomorphic map to a normal compact} {\it Kdhler space. Then} $\kappa(Y)\leq 0$. ({\it Here} $\kappa(Y)=\kappa(\hat{Y})$, {\it where} $\hat{Y}$ {\it is an arbitrary} {\it desingularization of} $Y.$) {\it Proof}. This follows from the inequ alities $0\geq\kappa_{+}(X)\geq\kappa_{+}(Y)\geq\kappa(Y)$. For the second inequality, which is easily checked by a pulling-b ack argument, see [Ca93]. $\square $ 220 J.-P. DEMAILLY ET AL. 3. Proof of the structure theorem W{\it e} suppose here that $X$ is equipped with $a$ Kähler metric $g$ such that Ricci $(g) \geq 0$, and we set $n =\dim_{\mathbb{C}}X$. (i) Let $(\overline{X},g) \simeq\Pi(X_{i},g_{i})$ be the De Rh {\it a}m decomposition of $(\tilde{X},g)$, induced by a decomposition of the holonomy representation in $\mathrm{i}$ rreducible representations. Since the holonomy is contained in $\mathrm{U}(n)$, all factors $(X_{i},g_{i})$ are Kähler manifolds with irreducible holonomy and holonomy group $H_{i}\subset U(ni),\ ni=\dim X_{i}$. By Cheeger-Gromoll [CG71], there i{\it s} possibly a flat factor $X_{0}=\mathbb{C}^{?}$ and the other factors $X_{i},\ i\geq 1$, are compact. Also, the product stmcture shows th $a\mathrm{t}-K_{X_{t}}$ is hermitian semipositive. It suffices to prove that $\kappa_{++}(X_{i})=0$ implies that $X_{i}$ i{\it s} a Calabi-Yau manifold o{\it r} a symplectic manifold. In view of Section 2, the condition $\kappa_{++}(X_{i})=0$ means that there is a nonzero section $u\in H^{0}(X_{i}, \Omega_{X_{t}}^{\otimes m})$ for some $m>0$. Since $u$ i{\it s} parallel by the lemma, it is invari ant under the holonomy action, and therefore the holonomy group $H_{i}$ is not the full unitary $\mathrm{g}$ roup $\mathrm{U}(n_{i})$ (indeed, the trivial rep {\it r}e sentation does not occur in the decomposition of $(\mathbb{C}^{n_{t}})^{\otimes m}$ in $\mathrm{i}$ rreducible $\mathrm{U}(n_{i})$-rep resentations, all weights being of length $m$). By Berger's classification of holonomy groups [Bg55] there are only two remaining possibilities, namely $H_{i}=$ SU $(n_{i})$ or $H_{i}=\mathrm{Sp}(n_{i}/2)$. The case $H_{i}=$ SU $(n_{i})$ leads to $X_{i}$ being a Calabi-Yau manifold. The remaining case $H_{i}=\mathrm{Sp}(n_{i}/2)$ implies that $X_{i}$ i{\it s} symplectic (see e.g. [Be83]). (ii) Set $X'=\Pi_{i\geq 1}X_{i}$. The group of covering transformations acts on the product $\overline{X}=\mathbb{C}^{j}\times X'$ by holomorphic isometries of the form $ x=(z, x')\mapsto$ $(u(z), v(x'))$. {\it At} this point, the argument is slightly mo {\it r}e involved than in Beauville's paper [Be83], because the group $G'$ of holomorphic isometries of $X'$ need not be finite ( $X'\mathrm{m}$ {\it a}y be for instance a projective space); instead, we imitate the proof of ([CG72], Theo rem 9.2) and use the fact that $X'$ and $G'=\mathrm{Isom}(\mathrm{X}')$ are compact. Let $E_{q}=\mathbb{C}^{?}\ltimes U(q)$ b{\it e} the group of unitary motions of $\mathbb{C}^{?}$. Then $\pi_{1}(X)$ can be seen as $a$ discrete subgroup of $E_{q}\times G'$. As $G'$ is compact, the kemel of the projection map $\pi_{1}(X)\rightarrow E_{q}$ is finite and the image of $\pi_{1}(X)$ in $E_{q}$ is still discrete with compact quotient. This shows th {\it a}t there i{\it s a} subgroup $\Gamma$ of finite index in $\pi_{1}(X)$ which is isomorphic to a crystallographic subgroup of $\mathbb{C}^{?}$. By Bieberbach's theorem, the subgroup $\Gamma_{0}\subset\Gamma$ of elements which are tran slations is a subgroup of finite index. Taking the intersection of all conjugates of $\Gamma_{0}$ in $\pi_{1}(X)$, we find a normal subgroup $\Gamma_{1}\subset\pi_{1}(X)$ of finite index, acting by translations on $\mathbb{C}^{1}$. Then $\hat{X}=\tilde{X}/\Gamma_{1}$ is a fibre bundle over the toms $\mathbb{C}^{1}/\Gamma_{1}$ with $X'$ {\it a}s fibre and $\pi_{1}(X')=1$. Therefore $\hat{X}$ is the desi red finite étale covering of $X$. (iii) is an immediate consequence of (ii), using the homotopy exact sequence of a fibration. $\square $ COROLLARY 1. {\it Let} $X$ {\it be a compact Kähler manifold with} $-Kx$ {\it hermitian} {\it semipositive. If} $\tilde{X}$ {\it is indecomposable and} $K+(X)=0$, {\it then} $X$ {\it is Ricci-flat}. $\mathrm{K}\ddot{\mathrm{A}}$ HLER MANIFOLDS WITH SEMIPOSITIVE ANTICANONICAL BUNDLE 221 COROLLARY 2. {\it Let} $X$ {\it be a compact Kdhler manifold with} $-1K_{X}$ {\it hermitian} {\it semipositive. Then}, $\iota f\hat{X}\rightarrow X$ {\it is an arbitrary finite etale covering} $\kappa_{+}(X)=-\infty \Leftrightarrow\kappa_{++}(X)=-\infty$ $$ \Leftrightarrow\forall\hat{X}\rightarrow X,\ \forall p\geq 1,\ H^{0}(\hat{X}, \Omega_{\hat{X}}^{p})=0. $$ {\it If} $\kappa_{+}(X)=-\infty$, {\it then} $\chi(X, \mathcal{O}_{X})=1$ {\it and} $X$ {\it is simply connected}. {\it Proof}. Th $e$ equivalence of all three properties is $a$ direct consequence ofthe stmc- ture theorem. Now, any étale covering $\hat{X}\rightarrow X$ sati sfies $\kappa_{+}(\hat{X})=\kappa_{+}(X)=-\infty$, hence $\chi(\hat{X}, \mathcal{O}_{\hat{X}})=\chi(X, O_{X})=1$ (by Hodge symmetry we $\mathrm{h}$ ave $h^{p}(X, O_{X})=0$ for $p\geq 1$, whilst $h^{0}(X, \mathcal{O}_{X})=1)$. However, if $d$ i{\it s} the covering degree, the Riemann-Roch formula implies $\chi(\hat{X}, \mathcal{O}_{\hat{X}})=d\chi(X, \mathcal{O}_{X})$, hence $d=1$ and $X$ mu {\it s}t be simply connected. $$ \square $$ 4. Rel ated questions for the $\mathrm{c}as\mathrm{e}-K_{X}$ nef In order to make the stmcture theorem more explicit, it would be necess {\it ary} to characterize more precisely the manifolds for which $\kappa_{+}(X)=-\infty$. W{\it e} expect these manifolds to be rationally connected, even $\mathrm{when}-I\iota_{X}^{r}$ i{\it s} just supposed to be nef. CONJECTURE. {\it Let} $X$ {\it be a compact Kähler manifold such} $that-K_{X}$ {\it is} $nef$ {\it and} $\kappa_{+}(X)=-\infty$. {\it Then} $X$ {\it is rationally connected}, $i.e$. {\it any two points of} $X$ {\it can be} {\it joined by a chain of rational curves}. $\mathrm{C}$ amp {\it a}n $a$ even conjectures this to be tme without $a\mathrm{ssuming}-K_{X}$ to be nef. Another hope we have i{\it s} that $a$ similar stmcture theorem might also hold in the $\mathrm{c}as\mathrm{e}-I\iota_{X}' \mathrm{nef}$. A sm all $\mathrm{p}$ {\it art} of it would be to unders tand better the stmcture of the Alb(X ese map. We proved in [DPS93] that the Albanese map i{\it s} surjective when $\dim X\leq 3$, and if $\dim X\leq 2$ it is well-known th {\it a}t the Albanese map i{\it s} a locally trivi {\it a}l fb ration. It is thus natural to state the following PROBLEM. {\it Let} $X$ {\it be a compact Kdhler manifold} $with-I\zeta_{xnef}$. {\it Is the Albanese} {\it map} $\alpha;X\rightarrow$ Alb({\it X}) {\it a smooth locally trivial fibration} ? The following simple example shows, even in the case of a locally trivial ibration, that the stmcture group of transition automorphisms need not be a group of isometries, in contrast with the $\mathrm{case}-K_{X}$ hermitian semipositive. EXAMPLE 1 (see [DPS94], Example 1.7). Let $C=\mathbb{C}/(\mathrm{Z}+\mathrm{Z}\tau)$ be {\it a}n elliptic 222 J.-P. DEMAILLY ET AL. curve, and let $E\rightarrow C$ be the fat rank 2 bundle as sociated to the repre sentation $\pi_{1}(C)\rightarrow \mathrm{GL}_{2}(\mathbb{C})$ defined by the monodromy matrices $\left(\begin{array}{ll} 1 & 0\\ 0 & 1 \end{array}\right),\ \left(\begin{array}{ll} 1 & 1\\ 0 & 1 \end{array}\right)$. Then the projectivized bundle $X=\mathbb{P}(E)$ i{\it s} $a$ mled surface over $C$ with $-K_{X}$ nef and not hermitian semipositive (cf. [DPS94]). In this case, the Albanese map $X\rightarrow C$ is $a$ locally trivial $\mathbb{P}_{1}$-bundle, but the monodromy group is not relatively compact in $\mathrm{GL}_{2}(\mathbb{C})$, hence there is no inv ariant Kähler metric on the fibre. EXAMPLE 2. The following example shows that the pictu {\it r}e i{\it s} unclear even in the case of surfaces with $\kappa_{+}(X)=-\infty$. Let $\mathrm{p}=(p_{1}\ldots,p_{9})$ be $a$ configuration of 9 points in $\mathbb{P}_{2}$ and let $\pi:X_{\mathrm{p}}\rightarrow \mathbb{P}_{2}$ be the blow-up of $\mathbb{P}_{2}$ with center $\mathrm{p}$. Here some of the points $p_{i}$ may be infinitely near: $as$ usual, this mean $s$ that the blowing-up process is made inductively, each $p_{i}$ being an arbitrary point in the blow-up of $\mathbb{P}_{2}$ {\it a}t $(p_{1}\ldots,p_{i-1})$. There is {\it a}lways $a$ cubic curve $C$ containing all9 points $(C$ i{\it s} even unique if $\mathrm{p}$ is general enough). The only assumption we $\mathrm{m}$ {\it ake} i{\it s} that $C$ is nonsingular, and we let $C=\{Q(z_{0}, z_{1}, z_{2})=0\}\subset \mathbb{P}_{2},\ \deg Q=3$. Then $C$ is {\it a}n elliptic curve $a\mathrm{nd}-K_{X_{\mathrm{P}}}= \pi\star${\it 0}({\it 3})--$\Sigma E_{i}$ where $E_{i}=\pi^{-1}(p_{i})$ are the excep- tion {\it a}l divisors. Clearly $Q$ defines a section $\mathrm{of}-K_{X_{\mathrm{P}}}$, of divisor equal to the strict transform $C'$ of $C,\ \mathrm{henc}e-K_{X_{\mathrm{P}}}\simeq \mathcal{O}(C')$, and $(-K_{X\mathrm{p}})^{2}=(C')^{2}=C^{2}-9=0$. The $re\mathrm{fo}r\mathrm{e}-K_{X_{\mathrm{P}}}$ i{\it s} alw ays nef. It i{\it s} easy to see $\mathrm{that}-mK_{X_{\mathrm{P}}}$ may be generated or not by sections according to the choice of the 9 points $p_{i}$. In fact, if $p_{i}'$ is the point of $C'$ lying over $p_{i}$, w{\it e} {\it have} $-K_{X_{\mathrm{P}}|C'}=\displaystyle \pi^{\star}(\mathcal{O}(3))_{|C'}\otimes \mathcal{O}(-\sum p_{j}')=\pi^{\star}(\mathcal{O}(3)_{|C}\otimes \mathcal{O}(-\sum p_{j}))$. Since $C'\simeq C$ i{\it s} an elliptic curve $\mathrm{and}-K_{X_{\mathrm{P}}|C'}$ has degree $0$, there are nonzero sections in $H^{0}(C', -mK_{X_{\mathrm{P}}|C'})$ if and only if $L_{\mathrm{p}}=\displaystyle \mathcal{O}(3)_{|C}\otimes \mathcal{O}(-\sum p_{j}))$ is a torsion point in $\mathrm{Pic}^{0}(C)$ of order dividing $m$. Such sections always extend to $X_{\mathrm{p}}$. Indeed, w{\it e} may $as$ sume that $m$ i{\it s} exactly the order. Then $\mathcal{O}(-C')\otimes \mathcal{O}(-mK_{X_{\mathrm{P}}})=$ $\mathcal{O}((m-1)C')$ admits $a$ filtration by its subsheaves $\mathcal{O}(kC'),\ 0\leq k\leq m-1$, and the $H^{1}$ groups of the graded pieces are $H^{1}(X_{\mathrm{p}}, \mathcal{O}_{X_{\mathrm{P}}})=0$ for $k=0$ and $H^{1}(C', \mathcal{O}(kC')_{|C'})=H^{0}(C', \mathcal{O}(-kC'))=0$ for $0