\documentclass{eudml-infty} \usepackage{latexsym} \usepackage{amsmath} \usepackage{amssymb} \usepackage{bm} \pagestyle{empty} \begin{document} {\it Compositio Mathematica} 89: 217-240, 1993. \copyright 1993 {\it Kluwer Academic Publishers. Printed in the Netherlands}. Kahler manifolds with numerically effective Ricci class JEAN-PIERRE DEMAILLY {\it Universite}' {\it de Grenoble} $I$, {\it Institut Fourier}, $BP74$ {\it U.R.A. 188} $du$ {\it C.N.R.S}., {\it 38402 Saint-Martin} $d' H\grave{e}res$, {\it France} THOMAS PETERNELL and MICHAEL SCHNEIDER {\it Universität Bayreuth, Mathematisches Institut, D-95440 Bayreuth, Deutschland} Received 5 March 1992; accepted in final form 29 October 1992 Introduction Compact Kähler manifolds with semipositive Ricci curvature have been in- vestigated by various authors. S. Kobayashi [Ko61] first proved the simple connectedness of Fano manifolds, namely manifolds with positive Ricci cur- vature or equivalently, with ample anticanonical line $\mathrm{bundle}-K_{X}$. Later on, generalizing results of Y. Matsushima [Ma69], A. Lichnerowicz [Li71, 72] proved the following interesting fibration theorem: if $X$ is a compact Kähler manifold with semipositive Ricci class, then $X$ is a smooth fibration over its Albanese torus and there is a group of analytic automorphisms of $X$ lying over the group of toms translations (see also Section 2 for another proof of these facts based on the solution of Calabi's conjecture and on Bochner's technique). Finally, there were extensive works in the last decades to study the structure and classification of Ricci flat Kähler manifolds, see e.g. [Ca57], $[\mathrm{Bo}74\mathrm{a},\mathrm{b}]$, [Be83] and [Kr86]; of special interest for physicists is the subclass of so-called Calabi-Yau manifolds, i.e. Ricci flat compact Kähler manifolds with finite fundamental group, which appear as a natural generalization of K3 surfaces. To make things precise, one says that $X$ has semipositive Ricci class $c_{1}(X)$ if $c_{1}(X)$ contains a smooth semipositive closed (1,1)-form, or equivalently if $-K_{X}$ carries a smooth hermitian metric with semipositive curvature. By the $\mathrm{Aubin}-\mathrm{Calabi}-\mathrm{Yau}$ theorem, this is equivalent to $X$ having a Kähler metric with semipositive Ricci curvature. On the other hand, recent developments of algebraic geometry (especially those related to Mori's minimal model program) have shown the importance of the notion of numerical effectivity, 218 {\it J}.-{\it P. Demailly et al}. which generalizes hermitian semipositivity but is much more flexible. It would thus be important to extend the above mentioned results to the case when $-K_{X}$ is numerically effective. The purpose of this paper is to contribute to the following two conjectures. CONJECTURE 1. {\it Let} $X$ {\it be a compact Kdhler manifold with numerically} {\it effective anticanonical bundle} $-K_{X}$. {\it Then the fundamental group} $\pi_{1}(X)$ {\it has} {\it polynomial growth}. CONJECTURE 2. {\it Let} $X$ {\it be a compact Kdhler manifold with} $-K_{X}$ {\it numeri}- {\it cally effective. Then the Albanese map} $\alpha:X\rightarrow$ Alb({\it X}) {\it is surjective}. Before we state the results, let us recall the definition of a numerically effective line bundle $L$ on a compact complex manifold (see [DPS91] for more details). The abbreviation ``nef'' will be used for ''numerically {\it effective} . DEFINITION. {\it Let} $X$ {\it be a compact complex manifold with a fixed hermitian} {\it metric} $\omega$. {\it A holomorphic line bundle} $L$ {\it over} $X$ {\it is} nef {\it if for every} $\epsilon>0$ {\it there} {\it exists a smooth hermitian metric} $h_{\epsilon}$ {\it on} $L$ {\it such that the curvature satisfies} $\Theta_{h_{\epsilon}}\geq-\epsilon\omega$. Of course this notion does not depend on the choice of $\omega$. If $X$ is projective, $L$ is nef precisely if $L\cdot C\geq 0$ for all curves $C\subset X$. Our main contribution to Conjecture 1 is THEOREM 1. {\it Let} $X$ {\it be a compact Kdhler manifold with} $-K_{X}$ nef. {\it Then} $\pi_{1}(X)$ {\it is a group of subexponential growth}. The main tool to prove this result is the solution of the Calabi conjecture by Aubin [Au76] and Yau [Y77], combined with volume bounds for geodesic balls due to Bishop [Bi63] and Gage [Ga80] (see Section 1 for details). In fact, the volume of a geodesic ball of radius $R$ in the universal covering of $X$ essentially counts the number of words of $\pi_{1}(X)$ of length $\leq R$. The difficulty is that we have to deal with a sequence of metrics with Ricci curvature closer and closer to being semipositive, but nevertheless slightly negative in some points, and moreover the diameter of $X$ need not remain uniformly bounded; this difficulty is solved by observing that a large fraction of the volume of $X$ remains at bounded distance without being disconnected (Lemma 1.3). A by-product of our proof is that Conjecture 1 holds in the semipositive case. This was in fact already known since a long time in the context of riemannian manifolds (cf. e.g. [HK78]); our method is then no- thing more than the usual riemannian geometry proof combined with the $\mathrm{Aubin}-\mathrm{Calabi}-\mathrm{Yau}$ theorem. In this way we get: {\it Kähler manifolds with numerically effective Ricci} $case$ 219 THEOREM 2. {\it Let} $X$ {\it be a compact Kiihler manifold with} $-K_{X}$ {\it hermitian} {\it semipositive. Then} $\pi_{1}(X)$ {\it has polynomial growth of degree} $\leq 2\dim X$, {\it in} {\it particular} $h^{1}(X, \mathcal{O}_{X})\leq\dim X$. Note that there are simple examples of compact {\it Kähler} manifolds $X$ with $-K_{X}$ nef but not hermitian semipositive, e.g. some ruled surfaces over elliptic curves (see examples 1.7 and 3.5 in [DPS91]). Also, to give a more precise idea of what Conjecture 1 means, let us recall Gromov's well-known result [Gr81]: a finitely generated group has polynomial growth if and only if it contains a nilpotent subgroup of finite index. Much more might be perhaps expected in the present situation: QUESTION. {\it Let} $X$ {\it be a compact Kähler manifold} $with-K_{X}$ nef. {\it Does there} {\it exist a finite e}'{\it tale covering} $\tilde{X}$ {\it of} $X$ {\it such that Albanese map} $\tilde{X}\rightarrow$ Alb $(\tilde{X})$ {\it induces an isomorphism of fundamental groups}? If this would be the case, $\pi_{1}(X)$ would always be an extension of a finite group by a free abelian group of even rank. Concerning Conjecture 2, the following result will be proved in Section 2: THEOREM 3. {\it Let} $X$ {\it be a n-dimensional compact Kähler manifold such that} $-K_{X}$ {\it is} nef. {\it Then} (i), {\it The Albanese map} $\alpha:X\rightarrow$ Alb({\it X}) {\it is surjective as soon as the Albanese} {\it dimension} $p=\dim\alpha(X)$ {\it is} $0,1$ {\it or} $n$, {\it and also for} $p=n-1$ {\it if} $X$ {\it is} {\it projective}. (ii) {\it If} $X$ {\it is projective and if the generic fiber} $F$ {\it of} $\alpha$ {\it has} $-K_{F}$ {\it big, then} $\alpha$ {\it is} {\it surjective}. The {\it case} $p=1$ in (i) is a straightforward consequence of Theorem 1, as pointed out to us by F. Campana, if one observes that the growth of the fundamental group of a curve of genus $\geq 2$ is of exponential type. The other interesting case $p=n-1$ is obtained as a consequence of point (ii), which is itself a rather simple consequence of the Kawamata-Viehweg vanishing theorem. Theorem 3 settles Conjecture 2 for projective 3-folds. In that case, we can also obtain a direct algebraic proof of the Albanese surjectivity in most cases by an examination of the stmcture of Mori contractions of $X$. When the contraction is not a modification, we give the description of the fibration structure of $X$. This is done in Section 3. To conclude let us mention that the first theorem was used in the classific- ation of compact {\it Kähler} manifolds with nef tangent bundles [DPS91] in a crucial way. 220 {\it J}.-{\it P. Demailly et al}. Acknowledgement Our collaboration has been made possible by PROCOPE and the DFG Schwerpunktprogramm ''Komplexe Mannigfaltigkeiten''. W{\it e} would like to thank Prof. F.A. Bogomolov, F. Campana and P. Gauduchon for v{\it e} ry useful discussions. 1. Subexponential growth of the fundamental group If $G$ is a finitely generated group with generators $g_{1},\ \ldots$ , $g_{p}$, we denote by $N(k)$ the numb {\it e}r of elements $\gamma\in G$ which can b{\it e} written as words $\gamma=g_{i_{1}}^{\epsilon_{1}}\ldots g_{i_{k}^{k}}^{\epsilon},\ \epsilon_{j}=0,1$ or-l of length $\leq k$ in terms of the generators. The group $G$ is said to have {\it subexponential growth} if for every $\epsilon>0$ there is a constant $C(\epsilon)$ such that $N(k)\leq C(\epsilon)e^{\epsilon k}$ for $k\geq 0$. This notion is independent of the choice of generators. In the free group with two generators, w{\it e} have $N(k)=1+4(1+3+3^{2}+\cdots+3^{k-1})=2\cdot 3^{k}-1$. It follows immediately that a group with subexponential growth cannot con- tain a non abelian fre $\mathrm{e}$ subgroup. The main goal of this section is to prove THEOREM 1.1. {\it Let} $X$ {\it be a compact Kähler manifold such that} $K_{X}^{-1}$ {\it is} nef. {\it Then} $\pi_{1}(X)$ {\it has subexponential growth}. {\it Proof}. The first step consists in th $e$ construction of suitable {\it Kähler} metrics by means of the Aubin-Calabi-Yau theorem. L{\it e} $\mathrm{t}\omega$ be a fixed {\it Kähler} metric on $X$. Since $K_{X}^{-1}$ is nef, for every $\epsilon>0$ there exists a smooth hermitian metric $h_{\epsilon}$ on $K_{X}^{-1}$ such that $ u_{\epsilon}=\Theta_{h_{\epsilon}}(K_{X}^{-1})\geq-\epsilon\omega$. By [Au76] and [Y77, 78] there exists a unique {\it Kähler} metric $\omega_{\epsilon}$ in the cohomology class $\{\omega\}$ such that Ricci $(\omega_{\epsilon})=-\epsilon\omega_{\epsilon}+\epsilon\omega+u_{\epsilon}$. $(+)$ {\it Kdhler manifolds with numerically effective Ricci class} 221 In fact $u_{\epsilon}$ belongs to the Ricci class $c_{1}(K_{X}^{-1})=c_{1}(X)$, {\it hence} so does the right hand side $-\epsilon\omega_{\epsilon}+\epsilon\omega+u_{\epsilon}$. In particular there exists a function $f_{\epsilon}$ such that $u_{\epsilon}=\mathrm{Ri}cc\mathrm{i}(\omega)+i\partial\overline{\partial}f_{\epsilon}$. If we set $\omega_{\epsilon}=\omega +i\partial\overline{\partial}\varphi$ (where $\varphi$ depends on e), $e$ quation $(+)$ is equivalent to the Monge-Ampère equation $(\omega+i\partial\overline{\partial}\varphi)^{n}$ $\overline{\omega^{n}}=e^{\epsilon\varphi-f_{\epsilon}} (++)$ because $i\partial\overline{\partial}\log(\omega+i\partial\overline{\partial}\varphi)^{n}/\omega^{n}=$ Ricci $(\omega)-$ Ri {\it cci} $(\omega_{\epsilon})$ $=\epsilon(\omega_{\epsilon}-\omega)+$ Ricci $(\omega) -u_{\epsilon}$ $$ =i\partial\overline{\partial}(\epsilon\varphi-f_{\epsilon}). $$ It follows from the results of [Au76] that $(++)$ has a unique solution $\varphi$, thanks to the fact the right hand side of $(++)$ is increasing in $\varphi$. Since $ u_{\epsilon}\geq-\epsilon\omega$, equation $(+)$ implies in particular that Ricci $(\omega_{\epsilon})\geq-\epsilon\omega_{\epsilon}$. W{\it e} now recall a w{\it e} ll-known differential geometric t{\it e} chnique used to get bounds for $N(k)$ (this technique has b{\it een} explained to us in a v{\it e} ry efficient way by S. Gallot). Let $(M, g)$ be a compact Riemannian {\it m}-fold and let $E\subset\tilde{M}$ b{\it e} a fundamental domain for the action of $\pi_{1}(M)$ on the universal {\it c}ove ring $\tilde{M}$. Fix $a\in E$ and set $\beta=$ diam $E$. Since $\pi_{1}(M)$ acts isometrically on $\tilde{M}$ with r{\it e} spect to the pull-back metric $\tilde{g}$, w{\it e} infer that $E_{k}=\displaystyle \bigcup_{1\gamma\in\pi(M),1\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}(\gamma)\leq k}\gamma(E)$ has volume equal to $N(k)\mathrm{Vol}(M)$ and is contained in the geodesic ball $B(a, \alpha k+\beta)$, where $\alpha$ is the maximum of the length of loops repre senting the generators $g_{j}$. Therefore $N(k)\displaystyle \leq\frac{\mathrm{V}\mathrm{o}\mathrm{l}(B(a,\alpha k+\beta))}{\mathrm{V}\mathrm{o}1(M)} (\star)$ and it is enough to bound the volume of geodesic balls in $\tilde{M}$. For this w{\it e} use the following fundamental inequality due to R. Bishop [Bi63], Heintze- Karcher [HK78] and M. Gage [Ga80]. 222 {\it J}.-{\it P. Demailly et al}. LEMMA 1.2. {\it Let} $\Phi:T_{M^{-},a}\rightarrow\tilde{M},\ \Phi(\zeta)=e\mathrm{xp}_{a}(\zeta)$ {\it be the} ({\it geodesic}) {\it exponential map. Denote by} $\Phi^{\star}\mathrm{d}V_{g}=a(t, \zeta)$ d{\it t} $\mathrm{d}\sigma(\zeta)$ {\it the expression of the volume element in spherical coordinates with} $t\in \mathbb{R}_{+}and$ $\zeta\in S_{a}(1)=unit$ {\it sphere in} $T_{M^{-},a}$. {\it Suppose that} $a(t, \zeta)$ {\it does not vanish for} $t\in]0,\ \tau(\zeta)[$, {\it with} $\tau(\zeta)=+\infty$ {\it or} $a(\tau(\zeta), \zeta)=0$. {\it Then} $b(t, \zeta)=a(t, \zeta)^{1/(m-1)}$ {\it satisfies on} ] $0,\ \tau(\zeta)[$ {\it the inequality} $\displaystyle \frac{\partial^{2}}{\partial t^{2}}b(t, \zeta)+\frac{1}{m-1}$ Ri $cci_{g}(v(t, \zeta), v(t, \zeta))b(t, \zeta)\leq 0$ {\it where} $v(t, \displaystyle \zeta)=\frac{\mathrm{d}}{\mathrm{d}t}e\mathrm{xp}_{a}(t\zeta)\in S_{\Phi(t\zeta)}(1)\subset T_{M^{-},\Phi(t\zeta)}.\ \square $ If Ri $cci_{g} \geq-$ Eg, we infer in particular $\displaystyle \frac{\partial^{2}b}{\partial t^{2}}-\frac{\epsilon}{m-1}b\leq 0$ and therefore $b(t, \zeta)\leq\alpha^{-1}\sinh(\alpha t)$ with $\alpha=\sqrt{\epsilon/(m-1)}$ (to {\it check} this, ob- serve that $b(t, \zeta)=t+o(t)$ at $0$ and that $\sinh(\alpha t)\partial b/\partial t-\alpha\cosh(\alpha t)b$ has a negative derivative). Now, every point $x\in B(a, r)$ can b{\it e} joined to $a$ by a minimal geodesic arc of length $0$, {\it there are closed subsets} $U_{1,\epsilon,\delta}\subset U_{1}$ {\it and} $U_{2,\epsilon,\delta}\subset U_{2}$ {\it with} Vo $ 1_{\omega}(U_{j}\backslash U_{j,\epsilon,\delta})<\delta$, {\it such that any two points} $x_{1}\in U_{1,\epsilon,\delta},\ x_{2}\in U_{2,\epsilon,\delta}$ {\it can be joined by a path of} {\it length} $\leq C\delta^{-1/2}$ {\it with respect to} $\omega_{\epsilon}$, {\it where} $C$ {\it is a constant independent of} $\epsilon$ {\it and} $\delta$. {\it Proof} The basic observation is that $\displaystyle \int_{X}\omega_{\epsilon}\wedge\omega^{n-1}=\int_{X}\omega^{n}$ does not depend on $\epsilon$, therefore $\Vert\omega_{\epsilon}\Vert_{L^{1}(X)}$ is uniformly bounded. First suppose that $U_{1}=U_{2}=K$ where $K$ is a compact convex set in some coordinate open set $\Omega$ of $\tilde{X}$. We simply join $x_{1}\in K,\ x_{2}\in K$ by the segment $[x_{1}, x_{2}]\subset K$ and compute the average $\omega_{\epsilon}\mathrm{w},1\mathrm{ength}$ of this segment when $x_{1},\ x_{2}$ vary (the average being computed in $L^{2}$ norm with respect to the L{\it e} besgue m{\it e} are of $\Omega$). By Fubini and the $\mathrm{Cauchy}-\mathrm{S}ch\mathrm{warz}$ ine quality we get $\displaystyle \int_{K\times K}(\int_{0}^{1}\sqrt{\omega_{\epsilon}((1-t)x_{1}+tx_{2})(x_{2}-x_{1})}\mathrm{d}t)^{2}$ d{\it x} 1 $\mathrm{d}x_{2}$ $\displaystyle \leq|x_{2}-x_{1}|\int_{0}^{1}$ d{\it t} $\displaystyle \int_{K\times K}\omega_{\epsilon}((1-t)x_{1}+tx_{2})$ d{\it x} 1 $\mathrm{d}x_{2}$ $\leq 2^{2n}$ diam $K\cdot \mathrm{Vol}(K)\cdot\Vert\omega_{\epsilon}\Vert_{L^{1}(K)}\leq C_{1}$ where $C_{1}$ is independent of $\epsilon$; the last inequality is obtained by integrating first with r{\it e} spe ct to $y=(1-t)x_{1}$ wh {\it e}n $t\displaystyle \leq\frac{1}{2}$, resp. $y=tx_{2}$ when $t\displaystyle \geq\frac{1}{2}$ (observe that $\mathrm{u}_{j}\leq 2^{2n}$ d{\it y} in both cases). It follows that the set $S$ of pairs $(x_{1}, x_{2})\in K\times K$ such that $1e\mathrm{ngth}_{\omega_{\epsilon}}$ $([x_{1}, x_{2}])$ {\it exceeds} $(C_{1}/\delta)^{1/2}$ has measure $<\delta$ in $K\times K$. By Fubini, the set $Q$ 224 {\it J}.-{\it P. Demailly et al}. of points $x_{1}\in K$ such that the slice $S(x_{1})=\{x_{2}\in K;(x_{1}, x_{2})\in S\}$ has volume $\displaystyle \mathrm{Vol}(S(x_{1}))\geq\frac{1}{2}\mathrm{Vol}(K)$ itself has volume $\mathrm{Vol}(Q)<2\delta/\mathrm{Vol}(K)$. Now for $x_{1}$, $x_{2}\in K\backslash Q$ w{\it e} have by definition $\displaystyle \mathrm{Vol}(S(x_{j}))<\frac{1}{2}\mathrm{Vol}(K)$, therefore $ K\backslash (S(x_{1})\cup(K\backslash S(x_{2}))\neq\emptyset$. If $y$ is a point in this set, then $(x_{1}, y)\not\in S$ and $(x_{2}, y)\not\in S$, hence $1e\mathrm{ngth}_{\omega_{\epsilon}}([x_{1}, y]\cup[y, x_{2}])\leq 2(C_{1}/\delta)^{1/2}$. By continuity, a similar estimate still holds for any two points $x_{1},\ x_{2}\in$ $\overline{K\backslash Q}$, with some $y\in K$. When $U_{1}=U_{2}=K$, the lemma is thus proved with $U_{j,\epsilon,\delta}=\overline{K\backslash Q}$; note that Vo $ 1_{\omega}(U_{j}\backslash U_{j,\epsilon,\delta})\leq$ Vo $1_{\omega}(Q)\leq C_{2}\mathrm{Vol}(Q)<2C_{2}\delta/\mathrm{Vol}(K)$ and replace $\delta$ by $ c\delta$ with $c=\mathrm{Vol}(K)/(2C_{2})$ to get the desired bound $\delta$ for the volume of $U_{j}\backslash U_{j,\epsilon,\delta}$. If $U_{1},\ U_{2}$ are isomorphic to compact convex subsets in $\mathbb{C}^{n}$, w{\it e} find a chain of such sets $V_{1},\ \ldots$ , $V_{N}$ {\it w}ith $V_{1}=U_{1},\ V_{N}=U_{2}$ and $ V_{\mathring{j}}\cap V_{\mathring{j}+1}\neq\emptyset$. By the first {\it c}ase, there exists for {\it e}ach $j=1,\ \ldots$ , $N a$ subset $V_{j,\epsilon,\delta}\subset V_{j}$ with path of length $\leq C_{3}\delta^{-1/2}$ Vo $ 1_{\omega}(V_{j}\backslash V_{j,\epsilon,\delta})<\delta$ such $\mathrm{thatanypairof}_{\frac{}{2}}\mathrm{ointsin}V_{j,\epsilon,\delta}c\mathrm{anb}e\mathrm{join}e\mathrm{dbyaIfw}e\mathrm{tak}e\delta<^{\mathrm{P}}\mathrm{Vo}1_{\omega}(V_{j}\cap V_{j+1})\mathrm{for}e\mathrm{v}e\mathrm{ry}j,\mathrm{th}e\mathrm{n}$ $(V_{j}\backslash V_{j,\epsilon,\delta})\cup(V_{j+1}\backslash V_{j+1,\epsilon,\delta})$ cannot contain $V_{j}\cap V_{j+1}$ and therefore $ V_{j,\epsilon,\delta}\cap V_{j+1,\epsilon,\delta}\neq\emptyset$. This implies that any $x\in U_{1,\epsilon,\delta}:=V_{1,\epsilon,\delta}$ can be joined to any $y\in U_{2,\epsilon,\delta}:=V_{N,\epsilon,\delta}$ by a piecewise line ar path of length $\leq NC_{3}\delta^{-1/2}$. The {\it c}ase when $U_{1},\ U_{2}$ are arbitrary is obtained by cove ring these sets with finitely many compact convex coordinate patches. $\square $ W{\it e} take $U$ to b{\it e} a compact set containing the fundamental domain $E$, so large that $ U^{\mathrm{o}}\cap g_{j}(U^{\mathrm{o}})\neq\emptyset$ for each generator $g_{j}$. W{\it e} apply Lemma 1.3 with $U_{1}=U_{2}=U$ and $\delta>0$ fixed such that $\displaystyle \delta<\frac{1}{2}$ Vo $1_{\omega}(E),\ \displaystyle \delta<\frac{1}{2}$ Vo $1_{\omega}(U\cap g_{j}(U))$. W{\it e} get $U_{\epsilon,\delta}\subset U$ with Vo $ 1_{\omega}(U\backslash U_{\epsilon,\delta})<\delta$ and diam $\omega_{\epsilon}(U_{\epsilon,\delta})\leq C\delta^{-1/2}$. The in- equalities on volumes imply that $\displaystyle \mathrm{Vo}1_{\omega}(U_{\epsilon,\delta}\cap E)\geq\frac{1}{2}\mathrm{Vo}1_{\omega}(E)$ and $ U_{\epsilon,\delta}\cap g_{j}(U_{\epsilon,\delta})\neq\emptyset$ for {\it eve} ry $j$ (note that all $g_{j}$ preserve volumes). It is then clear that {\it Kähler manifolds with numerically effective Ricci class} 225 $W_{k,\epsilon,\delta}:=\displaystyle \bigcup_{1\gamma\in\pi(X),1\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}(\gamma)\leq k}\gamma(U_{\epsilon,\delta})$ satisfies Vo $1_{\omega}(W_{k,\epsilon,\delta})\geq N(k)$ Vo $1_{\omega}(U_{\epsilon,\delta}\displaystyle \cap E)\geq N(k)\frac{1}{2}$ Vo $1_{\omega}(E)$ and $\mathrm{di}a\mathrm{m}_{\omega_{\epsilon}}(W_{k,\epsilon,\delta})\leq k\mathrm{diam}_{\omega_{\epsilon}} U_{\epsilon,\delta}\leq kC\delta^{-1/2}$ Since $m=\dim_{\mathrm{R}}X=2n$, ine quality $(\star\star)$ implies $\mathrm{Vo}1_{\omega_{\epsilon}}(W_{k,\epsilon,\delta})\leq \mathrm{Vo}1_{\omega_{\epsilon}}(B(a, kC\delta^{-1/2}))\leq C_{4}k^{2n}\mathrm{e}^{C_{5}k}\sqrt{\epsilon}$. Now $X$ is compact, so there is a constant $C(\epsilon)>0$ such that $\omega^{n}\leq C(\epsilon)\omega_{\epsilon}^{n}$. W{\it e} conclude that $N(k)\displaystyle \leq\frac{2\mathrm{V}\mathrm{o}1_{\omega}(W_{k,\epsilon,\delta})}{\mathrm{V}\mathrm{o}1_{\omega}(E)}\leq C_{6}C(\epsilon)k^{2n}e^{c_{5}k}\sqrt{\epsilon}$. Th $e$ proof of Th eorem 1.1 is complete. $\square $ REMARK 1.4. In the non-Kähler case, one might try inst $e$ ad to use hermi- tian metrics $\omega_{\epsilon}$ in the same conformal {\it class} as $\omega$, such that $\displaystyle \int_{x}\omega_{\epsilon}^{n}=\int_{X}\omega^{n}$ and $\Theta_{\omega_{\epsilon}}(K_{X}^{-1})=u_{\epsilon}\geq-\epsilon\omega$. T{\it hen} Lemma 1.3 still holds. The major difficulty is that the first Then form $\Theta_{\omega_{\epsilon}}(K_{X}^{-1})$ differs from the Riemannian {\it Ricci} tensor Ricci $(\omega_{\epsilon})$ and there is no known analogue of Inequality 1.2 in that {\it c}ase. The fact that w{\it e} control $\Theta_{\omega_{\epsilon}}(K_{X}^{-1})$ by $-\epsilon\omega$ instead $\mathrm{of}-\epsilon\omega_{\epsilon}$ could b{\it e} also a source of difficulties. REMARK 1.5. It is well known and easy to check that the $e$ quation $(++)$ implies $C(\displaystyle \epsilon)=\max\frac{\omega^{n}}{\omega_{\epsilon}^{n}}\leq e\mathrm{xp}(\max_{X}f_{\epsilon}-\min_{X}f_{\epsilon})$. In fact, this follows from the observation that $i\partial\overline{\partial}\varphi\geq 0$ at a minimum point, thus $(\omega+i\partial\overline{\partial}\varphi)^{n}/\omega^{n}\geq 1$ and $(++)$ implies $\displaystyle \epsilon\min\varphi\geq\min f_{\epsilon}$. Similarly we have $\displaystyle \epsilon\max\varphi\leq \mathrm{m}a\mathrm{x}f_{\epsilon}$. Since $f_{\epsilon}$ is a potential of $\Theta_{h_{\epsilon}}(K_{X}^{-1})-\mathrm{Ri}c\mathrm{ci}(\omega)$ and con- v{\it e} rges to an almost plurisubharmonic function as $\epsilon$ tends to $0$, it is reasonable to {\it e}xpect that $C(\epsilon)$ has polynomial growth in $\epsilon^{-1}$; this would imply that $\pi_{1}(X)$ has polynomial growth by taking $\epsilon=k^{-2}$. When $K_{X}^{-1}$ has a semiposi- 226 {\it J}.-{\it P. Demailly et al}. tive metric, $we$ can {\it e}ven t{\it ake} $\epsilon=0$ and find a metric $\omega_{0}$ with Ricci $(\omega_{\mathrm{O}})=$ $u_{0}\geq 0$. This gives THEOREM 1.6. {\it If} $X$ {\it is Kdhler and} $K_{X}^{-1}$ {\it is hermitian semipositive} (e.g. {\it if} $K_{X}^{-m}$ {\it is generated by sections for some} $m$) {\it then} $\pi_{1}(X)$ {\it has polynomial growth} {\it of degree} $\leq 2n$. {\it In particular} $q(X)=h^{1}(X, \displaystyle \mathcal{O})=\frac{1}{2}\mathrm{rank}_{\mathrm{Z}}H_{1}(X, \mathrm{Z})\leq n$. REMARK 1.7. If $X$ is a Fano manifold, i.{\it e}. if $K_{X}^{-1}$ is ample, the above techniques can b{\it e} used to show that $X$ i{\it s} simply connected, as observed long ago by S. Kobayashi [Ko61]. In fact the $\mathrm{Aubin}-\mathrm{Calabi}-\mathrm{Yau}$ theorem pro- vides a Kähler metric $\omega$ with Ricci $(\omega)=u>0$, say $ u\geq\epsilon\omega$. Then L{\it e} mma 1.2 implies $\partial^{2}b/\partial t^{2}+\epsilon/(2n-1)b(t, \zeta)\leq 0$, thus $b(t, \zeta)\leq\alpha^{-1}\sin(\alpha t)$ with $\alpha=\sqrt{\epsilon/(2n-1)}$. In particular $\tau(\zeta)\leq\pi/\alpha$, hence the universal cove ring $\tilde{X}$ is compact of diameter $\leq\pi/\alpha$ and $\pi_{1}(X)$ is finite (all this was already settled by S. Myers [My41] for arbitrary Riemannian manifolds). The Hirzebruch- Riemann-{\it R}och formula implies $\chi(\tilde{X}, \mathcal{O}_{X})=k\chi(X, \mathcal{O}_{X})$ with $k=\neq\pi_{1}(X)$. Moreover, the Kodaira vanishing theorem applied to {\it the} ample line bundle $L=K_{X^{-}}^{-1}$ gives $H^{q}(\tilde{X}, \theta_{\overline{\chi}})=H^{q}(\tilde{X}, K_{X^{-}}\otimes L)=0$ for $q\geq 1$, hence $\lambda'(\tilde{X}, \mathcal{Y})=h^{0}(\tilde{X}, \mathcal{O}_{X^{-}})=1$ and $k=1.\ \square $ 2. Surjectivity of the Albanese map Let $X$ b{\it e} a comp act Kähler manifold and let $q(X)=h^{1}(X, \mathcal{O}_{X})$ be its irregu- larity. {\it R}ecall that the Albanese map is $a$ holomorphic map from $X$ to the Albanese torus Alb({\it X}) $:=H^{0}(X, \Omega_{X})^{\star}/H_{1}(X, \mathrm{Z}),\ \dim$ Alb({\it X}) $=q(X)$, defined by $\alpha(x)(h) :=\displaystyle \int_{x}^{X}0h,\ h\in H^{0}(X, \Omega_{X}^{1})$; {\it Kdhler manifolds with numerically effective Ricci} $case$ 227 the path from $x_{\mathrm{O}}$ to $x$ in the integral is taken modulo an arbitrary loop at $x_{\mathrm{O}}$, i.e. modulo $H_{1}(X, \mathrm{Z})$. We first reprove Li chnerowicz' fibration theorem [Li71] by a simpler method based on the Bochner technique (of course Lichnerowicz had somehow to circumvent the Aubin-Calabi-Yau theorem, {\it which} was not available at that time). Our starting point is the following basic formula. FO RMULA 2.1. {\it Let} $\#$ {\it be the conjugate linear} $C^{\mathrm{x}}$ {\it isomorphism} $T_{X}\rightarrow\Omega_{X}^{1}$, $ v\mapsto i\overline{v}\lrcorner\omega$, {\it given by aKdhler metric} $\omega$. {\it Denote also by} $\# :\Lambda^{p}T_{X}\rightarrow\Omega_{X}^{p}$ {\it the} {\it induced} $C^{\infty}$ {\it isomorphism from p-vectors to p-forms. Then for an arbitrary} {\it smooth section} $v$ {\it of} $\Lambda^{p}T_{X}$ {\it we have} $\displaystyle \int_{X}\Vert\overline{\partial}(*v)\Vert^{2}\mathrm{d}V_{\omega}=\int_{X}\Vert\overline{\partial}v\Vert^{2}\mathrm{d}V_{\omega}+\int_{X}\langle \mathcal{R}(v),\ v\rangle \mathrm{d}V_{\omega}$ {\it where} $\mathrm{d}V_{\omega}$ {\it is the Kiihler element of volume and} !\% {\it is the hermitian operator} $v=\displaystyle \sum_{|I|=p}v_{I}\frac{\partial}{\partial z_{I}}\mapsto\Re(v)=\sum_{|I|=p}(\sum_{k\in I}\rho_{k})v_{I}\frac{\partial}{\partial z_{I}}$ {\it associated to the Ricci curvature form}: $\rho_{k}$ {\it denotes the eigenvalues of Ricci}({\it co}) {\it in an co-orthonormal frame} $(\partial/\partial z_{k})$. {\it Proof} W{\it e} first make a pointwise calculation of $\overline{\partial}^{\star}\overline{\partial}v$ and $\overline{\partial}^{\star}\overline{\partial}(\# v)$ in a normal coordinate system for the {\it Kähler} metric $\omega$. In such coordinates $we$ can write $\omega =j\displaystyle \sum_{\leq m\leq n}\mathrm{d}z_{m}\wedge \mathrm{d}\overline{z}_{m^{-;\sum_{<}}}c_{jklm}z_{j}\overline{z}_{k}\mathrm{d}z_{l}\wedge \mathrm{d}\overline{z}_{m}+O(|z|^{3})11\sim j,k,l,m\leq n$ where $(c_{jklm})$ is the curvature tensor of $T_{X}$ {\it with} respect to $\omega$. The {\it Kähler} property shows that w{\it e} have the symmetry relations $c_{jklm}=c_{lkjm}=c_{jmlk}$, and the Ricci tensor $R=\Sigma R_{lm}\mathrm{d}z_{l}\wedge \mathrm{d}\overline{z}_{m}$ is obtained {\it a}s the {\it trace}: $R_{lm}=\Sigma_{j}c_{jjlm}$. Since $\omega$ is tangent of order 2 to a flat metric {\it a}t the center $x_{0}$ of the chart, {\it we} easily {\it see} that the first order operator $\overline{\partial}^{\star}$ has the same formal expression {\it a}t $x_{\mathrm{O}}$ {\it a}s in the {\it case} of the flat metric on $\mathbb{C}^{n}$: if $w$ is a smooth $(0, q)$ for with values in $a$ holomorp {\it hic} vector bundle $E$ trivialized locally by a holomorphic frame $(e_{\lambda})$ such t{\it hat} $(e_{\lambda}(x_{\mathrm{O}}))$ i{\it s} orthonormal and $De_{\lambda}(x_{\mathrm{O}})=0$, w{\it e} have at $\chi_{0}$ {\it the} formula $w=\displaystyle \sum_{\lambda,|J|=q}w_{\lambda,J}e_{\lambda}\otimes \mathrm{d}\overline{z}_{J},\ \displaystyle \overline{\partial}^{\star}w=-\sum_{\lambda,|J|=q,k}\frac{\partial w_{\lambda,J}}{\partial z_{k}}e_{\lambda}\otimes(\frac{\partial}{\partial\overline{z}_{k}}\lrcorner \mathrm{d}\overline{z}_{J})$. 228 {\it J}.-{\it P. Demailly et al}. This applies of course to the {\it c}ase of sections of $\Lambda^{p}T_{X}$ or $\Omega_{X}^{p}$ expressed in terms of the frames $\partial/\partial z_{I}$ and $\mathrm{d}z_{I},\ |I|=p$. From this, $we$ immediately find that for any smooth sections $v=\Sigma v_{I}\partial/\partial z_{I}$ and $w=\Sigma w_{I}\mathrm{d}z_{I}$ w{\it e} have $\displaystyle \overline{\partial}^{\star}\overline{\partial}v=-\sum_{I,k}\frac{\partial^{2}v_{I}}{\partial z_{k}\partial\overline{z}_{k}}\frac{\partial}{\partial z_{I}},\ \displaystyle \overline{\partial}^{\star}\overline{\partial}w=-\sum_{I,k}\frac{\partial^{2}w_{I}}{\partial z_{k}\partial\overline{z}_{k}}\mathrm{d}z_{I}$ at the point $x_{0}$. Now, w{\it e} get $\# \displaystyle \frac{\partial}{\partial z_{m}}=i\frac{\partial}{\partial\overline{z}_{m}}\lrcorner\omega =\displaystyle \mathrm{d}_{Z_{m}}-\sum_{j,kl},c_{jklm}z_{j}\overline{z}_{k}\mathrm{d}_{Z\iota}+O(|z|^{3})$, $\neq v=\displaystyle \sum_{I}v_{I}\mathrm{d}z_{I^{-\sum_{I,j,k,l,m}}}v_{1}c_{jklm}z_{j}\overline{z}_{k}\mathrm{d}z_{l}\wedge(\frac{\partial}{\partial z_{m}}\lrcorner \mathrm{d}z_{I})+O(|z|^{3})$. Computing $\overline{\partial}^{\star}\overline{\partial}(\neq v)$ at $x_{\mathrm{O}}$ {\it we} obtain $\displaystyle \overline{\partial}^{\star}\overline{\partial}(\# v)=-\sum_{I,k}\frac{\partial^{2}v_{I}}{\partial z_{k}\partial\overline{z}_{k}}\mathrm{d}z_{I}+\sum_{I,k,lm},v_{1}c_{kklm}\mathrm{d}z_{l}\wedge(\frac{\partial}{\partial z_{m}}\lrcorner \mathrm{d}z_{I})$ $$ =*(\overline{\partial}^{\star}\overline{\partial}v)+\sum_{I,l,m}v_{I}R_{lm}\mathrm{d}z_{l}\wedge(\frac{\partial}{\partial z_{m}}\lrcorner \mathrm{d}_{Z_{I}})=*(\overline{\partial}^{\star}\overline{\partial}v)+\#\mathcal{R}(v). $$ Formula 2.1 then follows from this identity by writing $\displaystyle \int_{X}\Vert\overline{\partial}(\# v)\Vert^{2}\mathrm{d}V_{\omega}=\int_{X}\langle\overline{\partial}^{\star}\overline{\partial}(*v), \neq v\rangle \displaystyle \mathrm{d}V_{\omega}=\int_{X}\langle\overline{\partial}^{\star}\overline{\partial}v+\mathcal{R}(v),\ v\rangle \mathrm{d}V_{\omega}.\ \square $ W{\it e} easily d{\it educe} from this the fibration theorem of Lichnerowicz [Li71, 72], as well as analogous results of [Li67] in the case $\mathrm{Ric}c\mathrm{i}(\omega)\leq 0$. THEOREM 2.2. {\it Let} $(X, \omega)$ {\it be a compact Kiihler manifold. Consider the} {\it natural contraction pairing} $\Psi:H^{0}(X, \Lambda^{p}T_{X})\times H^{\mathrm{o}}(X, \Omega_{X}^{p})\rightarrow \mathbb{C},\ 0\leq p\leq n=\dim X$. (i) {\it If} Ricci $(\omega)\geq 0$, {\it then} $\Psi$ {\it has zero kernel in} $H^{\mathrm{o}}(X, \Omega_{X}^{p})$. {\it In that case, the} {\it Albanese map} $\alpha;X\rightarrow$ Alb({\it X}) {\it is a submersion and every holomorphic} {\it vector field of} Alb({\it X}) {\it admits a lifting to X. Therefore, there is a group} {\it of analytic automorphisms of} $X$ {\it lying over the group of translations of} Alb({\it X}). {\it Kähler manifolds with numerically effective Ricci} $case$ 229 (ii) {\it If} Ri $cc\mathrm{i}(\omega) \leq 0$, {\it then} $\Psi$ {\it has zero kernel in} $H^{\mathrm{o}}(X, \Lambda^{p}T_{X})$. {\it In that case the} {\it identity component} Aut $(X)^{\mathrm{o}}$ {\it of} Aut({\it X}) {\it is abelian and leaves invariant all} {\it global holomorphic p-forms or p-vector fields}. {\it Proof}. Let $v$ b{\it e} a smooth section of $\Lambda^{p}T_{X}$ and let $w=\# v$ b{\it e} the associated smooth $(p, 0)$-form. By definition of $\#$ w{\it e} have $v\lrcorner w=\Vert v\Vert^{2}$. Now, when Ricci $(\omega)\geq 0$, Formula 2.1 shows that $\displaystyle \int_{X}\Vert\overline{\partial}w\Vert^{2}\mathrm{d}V_{\omega}\geq\int_{X}\Vert\overline{\partial}v\Vert^{2}\mathrm{d}V_{\omega}$, thus $v$ is holomorphic as soon as $w$ is. Therefore {\it we} get an {\it injective} conjugate line ar map $\#^{-1}:H^{0}(X, \Omega_{X}^{p})\rightarrow H^{0}(X, \Lambda^{p}T_{X})$ with the property that $(\#^{-1}w)\lrcorner w$ is a non zero constant for $w\neq 0$. This shows that the kemel of $\Psi$ in $H^{\mathrm{o}}(X, \Omega_{X}^{p})$ is zero. On the other hand, when Ricci $(\omega)\leq 0$, the inequality is reversed and w{\it e} get an injection $\neq:H^{\mathrm{o}}(X, \Lambda^{p}T_{X})\rightarrow H^{0}(X, \Omega_{X}^{p})$. Hence in that case the kemel of $\Psi$ in $H^{0}(X, \Lambda^{p}T_{X})$ is zero. (i) By the above with $p=1$, every holomorphic 1-form $h$ which is not identically zero does not vanish at all, because there is a vector field $v$ such that $v\lrcorner h=1$. Let $(h_{1}, \ldots , h_{q})$ b{\it e} $a$ basis of $I\theta(X, \Omega_{X}^{1})$. Then for $e$ ach point $x\in X$ the 1-forms $h_{1}(x),\ \ldots$ , $h_{q}(x)$ must be independent in $T_{X,x}^{\star}$. In the basis of $T_{\mathrm{A}\mathrm{l}\mathrm{b}(X)}$ provided by {\it the} $h_{j}' \mathrm{s}$, w{\it e} have $\mathrm{d}\alpha(x)=(h_{1}(x), \ldots , h_{q}(x))$ and so $\mathrm{d}\alpha(x)$ is surjective. Now, there are holomorphic vector fields $v_{1},\ \ldots,\ v_{q}$ on $X$ su {\it c}h that $v_{i}\lrcorner h_{j}=\delta_ij$. These v{\it e} ctor fields clearly generate a subgroup $G$ of Aut $(X)^{\mathrm{o}}$ which l{\it ies} over the group of translations of Alb({\it X}). (ii) Let $v_{1},\ \ldots$ , $v_{q}$ b{\it e} a basis of the Li $e$ algebra of Aut $(X)^{\mathrm{o}}$. Th {\it e}n all Lie brackets $[v_{j}, v_{j}]$ vanish, because w{\it e} have $[v_{i}, v_{j}]\lrcorner h=v_{i}\cdot(v_{j}\lrcorner h)-v_{j}\cdot(v_{i}\lrcorner h)=0$ for every holomorphic 1-form $h$ (just observe that $v_{i}\lrcorner h$ and $v_{j}\lrcorner h$ are constant functions). Thus Aut $(X)^{\mathrm{o}}$ is abelian. Moreover, for any holomorphic {\it p}-form $w$, the Lie derivative $\mathcal{D}_{v_{\iota}}(w)$ vanishes: $\mathcal{D}_{v_{l}}(w)=\mathrm{d}(v_{i^{\lrcorner}}w)+v_{j}\lrcorner(\mathrm{d}w)=0$, because all holomorphic forms on a compact Kähler manifold are d-closed. Hence $w$ is invariant under Aut $(X)^{\mathrm{o}}$. By duality, w{\it e} $e$ asily conclude that th $e$ holomorphic {\it p}-vectors are also kept invari ant. $\square $ We now discuss Conjecture 2 for compact {\it Kähler} manifolds $X$ with $-K_{X}$ 230 {\it J}.-{\it P. Demailly et al}. being only nef. Th $e$ proof of the following theorem has b{\it een} communicated to us by F. Camp {\it ana}. THEOREM 2.3. {\it Let} $X$ {\it be a compact Kdhler manifold with} $-K_{X}$ nef. {\it Let} $\alpha:X\rightarrow \mathrm{Alb}(\mathrm{X})$ {\it be the Albanese map. If} $\dim\alpha(X)=1$, {\it then} $\alpha$ {\it is surjective}. {\it Proof}. $\alpha(X)$ is a smooth curve $C$. Assume that $C$ has genus $g\geq 2$. Then $\pi_{1}(C)$ has exponential growth; in fact it contains $a$ f{\it ree} group with $2g-1$ $$ \alpha_{\star} $$ generators. Because of the exact sequence $\pi_{1}(X)\rightarrow\pi_{1}(C)\rightarrow\pi_{\mathrm{O}}(F)$, the image of $\alpha_{\star}$ has finite index in $\pi_{1}(C)$, hence $\pi_{1}(X)$ is of exponential growth, contradicting Theorem 1.1. $\square $ First suppose $\dim\alpha(X)=\dim X$. If $\alpha(X)\neq$ Alb({\it X}), there would b{\it e} at least two independe nt sections of $K_{X}$ coming from $H^{0}(\Omega_{\mathrm{A}\mathrm{l}\mathrm{b}(X)}^{n});s\mathrm{in}ce-K_{X}$ is nef, these sections would not vanish and so $K_{X}=\mathcal{O}_{X}$, contradiction. The next interesting case is $\dim\alpha(X)=\dim X-1$, which w{\it e} tre at next. W{\it e} first prove a more general statement. THEOREM 2.4. {\it Let} $X$ {\it be a compact Kiihler manifold with} $-K_{X}$ nef. {\it Then} {\it there is no holomorphic surjective map} $\varphi;X\rightarrow \mathrm{Y}$ {\it to a projective variety} $\mathrm{Y}$ {\it with} $\kappa(\mathrm{Y})>0$ {\it such} $that-K_{F}$ {\it is big for the general fiber} $F$ {\it of} $\varphi$. By definition the Kodaira dimension $\kappa(\mathrm{Y})$ is the Kodaira dimension of $a$ desingularisation. {\it Proof}. Assume there is a map $\varphi$ as above. W{\it e} may assume that $\mathrm{Y}$ is normal by passing to the normalization, and moreover that the fibers are connected by taking the Stein factorization. Choose a very ample divisor $H$ on Y. L{\it e} tting $m=\dim \mathrm{Y}$, w{\it e} pick a curve $C=H_{1}\cap\cdots\cap H_{m-1}$ with $H_{i}\in|H|$ in general position. Then $C$ is smooth as well {\it a}s $X_{C}=\varphi^{-1}(C)$ by Bertini's lemma. Moreover $c_{1}(\omega_{\mathrm{Y}})\cdot C>0$, (1) since $ C\cap$ Sing $(\mathrm{Y})=\emptyset$ and since $\omega_{Y}=\pi_{\star}(\omega_{\mathrm{Y}^{-}})$ on $\mathrm{Y}\backslash $ Sing(Y) for eve ry d{\it e}- singularisation $\pi:\tilde{\mathrm{Y}}\rightarrow$ Y. Let $f=\varphi_{|X_{C}}$. W{\it e} claim: $\omega_{X_{C}/C}^{-1}$ is big and nef. (2) In fa{\it c}t, {\it Kdhler manifold with numerically effective Ricci last} 231 $\omega_{X_{C}/C}^{-1}=\omega_{X/\mathrm{Y}_{|X_{C}}}^{-1}=\omega_{X|X_{C}}^{-1}\otimes f^{\star}(\omega_{Y|C})$ which is nef {\it because} of (1). Letting $p=\dim X_{c}$ and taking {\it p-th} p{\it owers}, we o{\it bse} rve that $c_{1}(f^{\star}\omega_{\mathrm{Y}_{|C}})\geq c_{1}(\mathcal{O}(F))$ by (1), $F$ being a g{\it ene} $\mathrm{ric}$ fibre, thus $c_{1}(\omega_{X_{C}/C}^{-1})^{p}\geq c_{1}(\omega_{X|X_{c}}^{-1})^{p-1}\cdot c_{1}(f^{\star}\omega_{Y|C})\geq c_{1}(\omega_{X|X_{C}}^{-1})^{p-1}\cdot F=c_{1}(\omega_{F}^{-1})^{p-1}$, which is positive by our assumption that $-K_{F}$ is big. Now w{\it e} can apply $\mathrm{Ka}w\mathrm{amata}-\mathrm{Vie}h\mathrm{w}e\mathrm{g}$'s vanishing theorem [Ka82, Vi82] to obtain $H^{1}(X_{c}, \omega_{X_{C}}\otimes\omega_{\overline{x}_{C}^{1}/c})=0$ But $\omega_{X_{C}}\otimes\omega_{X_{C}/C}^{-1}=f^{\star}(\omega_{C})$, {\it s}o via the Leray spectral sequence {\it we} conclude $H^{1}(C, \omega_{C})=0$, which i{\it s} absurd. $\square $ COROLLARY 2.5. {\it Let} $X$ {\it be a n-dimensional projective} ({\it or Moishezon}) {\it manifold} $with-K_{X}$ nef. {\it Assume that the Albanese map} $\alpha$ {\it has} $(n-1)$-{\it dimen}- {\it sional image. Then} $\alpha$ {\it is surjective}. {\it Proof}. If $\alpha$ i{\it s} not surjective, the image $\mathrm{Y}=\alpha(X)$ automatically has $\kappa(\mathrm{Y})>0$ since {\it w}e get at l{\it e} ast two independent holomorphic forms of maxi- mum degree from the Albanese toms. We may thus assume $\kappa(\mathrm{Y})>0$. Let $F$ b{\it e} the general fiber of $\alpha$, which i{\it s a} curve. Since $-K_{F}=-K_{X|F}$ i{\it s} nef, $F$ i{\it s} rational or elliptic. In the first case, $\alpha$ is surjective by the previous theorem. If $F$ is elliptic, then $\kappa(X)=0$, {\it s}o $h^{0}(X, mK_{X})\neq 0$ for {\it s}ome $m$ and conse- quently $mK_{X}=G_{X}$. Therefore $\alpha$ i{\it s} onto by Theorem 2.2 (i). $\square $ The {\it last} part of the proof shows more generally that conjecture 2 holds if $\kappa(X)=0$. A different proof of Corollary 2.5 has b{\it een} given by F. Campana. 3. Threefolds whose anticanonical bundles are nef In this section {\it w}e want to study the structure of projective 3-folds $X$ with $-K_{X}$ nef in more detail. In particular w{\it e} prove Conjecture 2 in dimension 3 with purely algebraic m{\it e} thods, except in one very {\it spe} cial c{\it ase}. In fact, $we$ prove that the Albanese map is surjective except possibly when all extremal contractions of $X$ are of type (B), defined in Proposition 3.3(2). For the structure of surfaces with $-K_{X}$ nef we refer to [CP91]. Let always $X$ denote a smooth projective 3-fold with $-K_{X}$ nef and let $\alpha:X\rightarrow$ Alb({\it X}) b{\it e} the Albanese map. By the last words of Section 2, the structure of $X$ is clear if $\kappa(X)=0$; {\it s}o w{\it e} will a{\it ssume} $\kappa(X)=-\infty$; note that $K_{X}$ i{\it s} not nef in this {\it c}ase. Th {\it e}n there {\it e}xists {\it a}n extremal ray on $X$ ([Mo82], 232 {\it J}.-{\it P. Demailly et al}. [KMM87] $)$; let $\varphi;X\rightarrow W$ {\it be} the associated contraction. W{\it e} want to analyze the stmcture of $\varphi$. PROPOSITION 3.1. {\it If} $\dim W\leq 2,\ \alpha$ {\it is surjective. More precisely}: (1) {\it If} $W$ {\it is a point}, $X$ {\it is Fano with} $b_{2}=1$, {\it in particular} $X$ {\it is simply connected}. (2) {\it If} $W$ {\it is a} ({\it smooth}) {\it curve, then} $g(W)\leq 1$. {\it In case} $g(W)=1$, {\it we have} $\alpha=$ $\varphi$; {\it if} $g(W)=0$, {\it we have} $q(X)=0$. {\it In all cases} $\varphi$ {\it has the structure of a} $del$ {\it Pezzo fibration}. (3) {\it If} $W$ {\it is a} ({\it smooth}) {\it surface, then either} (a) $\varphi$ {\it is a} $\mathbb{P}_{1}$ {\it bundle} $and-K_{W}$ {\it is} nef (b) $\varphi$ {\it is a proper conic bundle with discriminant locus} $\Delta$ {\it such that} $-(4K_{W}+\Delta)$ {\it is} nef, {\it and} $q(W)\leq 1$. {\it Proof}. (1) If $\dim W=0$, then $q(X)=0$, {\it hence} our claim i{\it s} obvious. (2) L{\it e} $\mathrm{t}\dim W=1$. Since $R^{q}\varphi_{\star}(\mathcal{O}_{X})=0$ {\it for} $q>0$, either $\varphi$ is the Albanese map and then we must {\it s}how that $q(W)=1$ or $q(W)=0$. So assume $q(W)\geq 2$. Then the canonical bundle $K_{W}$ is ample. L{\it e} $\mathrm{t}K_{X/W}$ b{\it e} the relative canonical bundle, {\it s}o $K_{X/W}=K_{X}-\varphi^{\star}(K_{W})$. Since the Picard number $\rho(X)=\rho(W)+1=2$ ({\it see} e.g. [KMM87]), and since $-K_{X}$ i{\it s} nef and $\varphi$-ample ([KMM87]), $-K_{X/W}$ i{\it s} ample. Hence by Kodaira vanishing: $H^{1}(X, \mathcal{O}_{X}(-K_{X/W})\otimes \mathcal{O}_{X}(K_{X}))=0$, so $H^{1}(X, \varphi^{\star}(\mathcal{O}_{W}(K_{W}))=0$ and $H^{1}(W, \mathcal{O}_{W}(K_{W}))=0$, which i{\it s} absurd. (3) Now assume $\dim W=2$. Th {\it e}n $W$ i{\it s} smooth and $\varphi$ is a $\mathbb{P}_{1}$ bundle or $a$ conic bundle ([Mo82]). Since $q(X)=q(W)$, {\it w}e have $a$ diagram $$ \alpha $$ $X \rightarrow \mathrm{Alb}(X)$ $\varphi\downarrow$ $$ \gamma\downarrow $$ $$ \beta $$ $W\rightarrow \mathrm{Alb}(W)$ with $\beta$ being the Albanese map of $W$ and $\gamma$ being finite. (3a) A{\it ssume} $\varphi$ to be $a\mathbb{P}_{1}$-bundle. W{\it e} will prove that $-K_{W}$ i{\it s} nef, hence $\beta$ is onto and {\it s}o i{\it s} $\alpha$. Take any irreducible curve $C\subset W$. Since $\varphi^{-1}(C)=$ $\mathbb{P}(E_{C})$ with a rank 2-bundle $E_{C}$ on $C$, {\it we} $\mathrm{h}$ {\it ave} (after possibly passing to the normalization of $C$): {\it Kdhler manifolds with numerically effective Ricci} $case$ 233 $-K_{X|\mathbb{P}(E_{C})}=-K_{\mathrm{P}(E_{C})}+\varphi^{\star}(N_{C/W})$ {\it b}y the adjunction formula. Since $-K_{\mathrm{P}(E_{C})}=e_{\mathbb{P}(E_{c}\otimes\frac{\det E_{c}^{*}}{2}\otimes\frac{(-K_{c})}{2})^{(2)}}$, w{\it e} have $-K_{X|\mathrm{P}(E_{C})}=\mathcal{O}_{\mathrm{P}(E_{c}\otimes\frac{\det E_{c}^{*}}{2}\otimes\frac{(-K_{W/C})}{2})^{(2)}}$. Since $-K_{X}$ i{\it s} nef, {\it we} conclude that $E_{c}\displaystyle \otimes\frac{\mathrm{d}\mathrm{e}tE_{c}^{\star}}{2}\otimes\frac{(-K_{W|C})}{2}$ i{\it s} nef. Now $ c_{1}(E_{C}\otimes ($det $E_{C}^{\star}/2)=0$, hence $-K_{W|C}$ must b{\it e} nef and $-K_{W}$ itself i{\it s} $\mathrm{nef}$, (3b) Next a{\it ssume} $\varphi$ to $be$ a proper conic bundle. Let $\Delta\subset W$ {\it b}e the discriminant locus, i.e. $\Delta=\{w\in W;X_{w}$ not smooth, From the {\it w}ell-known formula ({\it see} e.g. [Mi81]) $K_{X}^{2}\cdot\varphi^{-1}(C)=-(4K_{W}+\Delta)\cdot C$ for {\it eve} ry curv $eC \subset W$, we deduce from the nefness $\mathrm{of}-K_{X} that-(4K_{W}+\Delta)$ i{\it s} nef. Now {\it we} conclude by m{\it e} ans of the following: LEMMA 3.2. {\it Let} $W$ {\it be a smooth projective surface}, $\Delta\subset W$ {\it be a} ({\it possibly} {\it reducible}) {\it curve. Assume} $that-(4K_{W}+\Delta)$ {\it is} nef. {\it Then} $q(W)\leq 1$. {\it Proof} Obviously $\kappa(W)=-\infty$. We can easily reduce the problem to the {\it case} of $W$ being minimal. If $W\neq \mathbb{P}_{2}$, then $W$ is ruled over a curve $C$. Now it i{\it s} an easy exercise using [Ha77, V.2] {\it t}o prove that $g(C)\leq 1.\ \square $ PROPOSITION 3.3. {\it Assume} $\dim W=3$. {\it Let} $E$ {\it be the exceptional set of} $\varphi$. (1) {\it If} $\dim\varphi(E)=0$, {\it then} $-K_{W}$ {\it is big and} $nef$ {\it and} $q(X)=0$. (2) {\it If} $\dim\varphi(E)=1$, {\it then} $W$ {\it is smooth}, $\varphi$ {\it is the blow-up of the smooth curve} $C_{\mathrm{o}}=\varphi(E)and-K_{W}$ {\it is again} nef {\it except for the following special cases}: $C_{\mathrm{O}}$ {\it is rational and moreover either} 234 {\it J}.-{\it P. Demailly et al}. (A) $N_{C_{0}/W}\simeq \mathcal{O}(-2)\oplus \mathcal{O}(-2)$ {\it or} (B) $N_{C_{0}1W}\simeq \mathcal{O}(-1)\oplus \mathcal{O}(-2)$. {\it Proof}. By [Mo82] $E$ i{\it s} always {\it a}n irre ducible divisor and if $\dim\varphi(E)=1$, $W$ i{\it s} smooth and $\varphi$ is the blow-up of a smooth curve. W{\it e} may always a{\it ssume} $K_{X}^{3}=0$, otherwise $q(X)=0$ by $\mathrm{K}a\mathrm{wamata}-\mathrm{Vi}e\mathrm{h}w\mathrm{eg}$ vanishing. (1) W{\it e} have the following formula of $\mathbb{Q}$-divisors: $K_{X}=\varphi^{\star}(K_{W})+\theta E$ with {\it s}ome $\theta\in \mathbb{Q}_{+}([\mathrm{Mo}82]$, in f{\it act} either $E\simeq \mathbb{P}_{2}$ or $E$ i{\it s} a normal quadric and $\theta=2,1$ or 1/2 $)$. {\it Hence} $-K_{W}$ i{\it s} nef. Furthermore: $K_{X}^{3}=K_{W}^{3}+\theta^{3}E^{3}$ and since $E^{3}>0$ ( $E$ has always negative normal bundle, [Mo82]), w{\it e} con- clude from $K_{X}^{3}=0$ {\it t}hat $K_{W}^{3}<0$, {\it s}o $-K_{W}$ is big and nef (observe that $W$ might b{\it e} singular). Now a ''singular'' version of {\it the} Kawamata-Viehweg vanishing theorem ([KMM87, 1.2.5, 1.2.6] applied to $D=0$) yields $H^{1}(W, \mathcal{O}_{W})=0$. Since $R^{q}\varphi_{\star}(\mathcal{O}_{X})=0$ for $q>0$, w{\it e} get $q(X)=0$. (2) From the formula $K_{X}=\varphi^{\star}(K_{W})+E$, {\it we} immediately {\it see} that $-K_{W}\cdot C\geq 0$ for every curve $C \neq C_{0}$. Let $N_{C_{0}1W}=N$ denote the normal bundle of $C_{0}$. Let $V=N^{\star}\otimes \mathcal{D}$ with $\mathcal{Z}\in$ Pic $(C_{\mathrm{O}})$ be its normalization, i.{\it e}. $H^{0}(V)\neq 0$, but $H^{0}(V\otimes\varphi)=0$ for all line bundles $\wp$ with deg $\wp<0$. Let $\mu=\mathrm{d}e\mathrm{g}\mathcal{D}$. Th {\it e}n $E$ can $be$ written a{\it s} $E=$ $\mathbb{P}(N^{\star})=\mathbb{P}(V)$. The ''tautological'' line bundle $\mathcal{O}_{\mathrm{P}(V)}(1)$ has a ''canonical'' {\it s}e ction $C_{1}$ satisfying $C_{1}^{2}=-e=c_{1}(V)$ ({\it see} [Ha77, V.2]). In this terminology $(K_{X}\cdot C_{1})=(K_{W}\cdot C_{0})+(E\cdot C_{1})$. Let $F$ {\it be} $a$ fiber of $\varphi_{|E}$. Write for nume rical equivalence $-K_{X|E}\equiv aC_{1}+bF$. Since $(K_{X}\cdot F)=-1$, w{\it e} h{\it ave} $a=1$. Moreover {\it Kdhler manifolds with numerically effective Ricci} $case$ 235 $N_{E|X}\equiv-C_{1}+\mu F$ just by definition of $\mu$ and by the fact that $N_{E|X}^{\star}=\mathcal{O}_{\mathrm{P}(N^{\star})}(1)$. {\it Hence} $(K_{W}\cdot C_{0})=(K_{X}\cdot C_{1})-(E\cdot C_{1})=-b-\mu$; {\it s}o $-K_{W}$ is nef if $b+\mu\geq 0.\ (\star)$ Letting $g$ {\it b}e the genus of $C_{0}$, {\it w}e have $K_{E}^{2}=8(1-g)$; on the other hand we compute by adjunction: $K_{E}^{2}=(K_{X}+E)_{|E}^{2}=(-2C_{1}+(\mu-b)F)^{2}$. Thus $ 8(1-g)=4b-4e-4\mu$, and consequently $b+\mu=2b-e+2(g-1).\ (\star\star)$ Since $-K_{X|E}$ i{\it s} nef, w{\it e} have $K_{X}^{2}\cdot E\geq 0$, which i{\it s} equivalent to $b\geq e/2$. Combining this {\it with} $(\star\star)$ shows that $b+\mu\geq 0$ {\it a}t least if $g\geq 1$. Therefore, if $g\geq 1,\ -K_{W}$ i{\it s} nef. Now assume $g=0$. Then $(\star)$ i{\it s} equivalent to $b\displaystyle \geq\frac{e}{2}+1.\ (\star')$ Again by nefness $\mathrm{of}-K_{X|E}$ {\it we} get $0\leq(-K_{X}\cdot C_{1})=(C_{1}+bF\cdot C_{1})=b-e$, {\it so} $b\geq e$. This settles already the {\it case} $e\geq 2$. Since $e\geq 0$, we {\it are} left with $e=0$ and $e=1$ and additionally $e\leq b0$. Th {\it erefore}, putting everything together, $we$ have proved: THEOREM 3.8. {\it Let} $X$ {\it be a smooth projective 3-fold with} $-K_{X}$ nef {\it and} $\kappa(X)=-\infty$. {\it Let} $\varphi;X\rightarrow W$ {\it be a contraction of an extremal ray. Then} $-K_{W}$ $is$ nef {\it expect possibly for the following cases}: (a) $\varphi$ {\it is the blow-up of a smooth rational curve} $C$ {\it such that} $N_{C/W}\simeq\left\{\begin{array}{l} \mathcal{O}(-1)\oplus \mathcal{O}(-2) or\\ \mathcal{O}(-2)\oplus \mathcal{O}(-2). \end{array}\right.$ {\it In the first case} $X$ {\it has irregularity} $0$ {\it and is birational to a Q-Fano variety}. $(b)\varphi$ {\it is a proper conic bundle over a surface} $W$ {\it with} $-(4K_{W}+\Delta) \mathrm{nef},\ \Delta$ {\it being the discriminant locus}. REMARK 3.9. Let $X$ {\it b}e {\it a} smooth projective 3-fold with $-K_{X}$ nef. The {\it above} algebraic considerations again show that the Albanese map $\alpha;X\rightarrow$ Alb({\it X}) i{\it s} surjective, {\it e}xcept possibly if all contractions $X\rightarrow W$ are of type (A) o{\it r} if this situation occurs after finitely many blowing-downs. {\it Proof} W{\it e} may {\it assume} $K_{X}$ not nef. Let $\varphi;X\rightarrow W$ b{\it e} the contraction of an extremal ray. If $\dim W\leq 2,\ \alpha$ is {\it already} surjective by Prop. 3.1. If $\dim W=3$, $\varphi$ must b{\it e} either the blow-up of $a$ point, hence $(-K_{W})^{3}>0$ and $q(X)=$ $q(W)=0$ (except possibly for {\it case} $(\mathrm{A})$), or $\varphi$ i{\it s} the blow-{\it u}p of a smooth curve and $-K_{W}$ is again nef with $W$ smooth. Th {\it e}n w{\it e} proceed by induction on $b_{2}(W).\ \square $ REMARKS 3.10. (1) In 3.8(a) consider the morphism $\psi=\Phi_{|-mK_{X}|}$ with suitable $m$. In {\it case} $N_{C/W}=\mathcal{O}(-1)\oplus \mathcal{O}(-2),\ \psi$ contracts the exceptional {\it curve} of $E\simeq\Sigma_{1}$, in the other case $\varphi$ contracts all curves in $E\simeq \mathbb{P}_{1}\times \mathbb{P}_{1}$ which are ruling {\it lines} not contracted by $\varphi$. It would b{\it e} interesting to know whether $3.8(a)$ can re ally occur. (2) If $\varphi;X\rightarrow W$ is a proper conic bundle with $-K_{X}$ nef, then $(-K_{W}\cdot C)\geq 0$ if $ C\not\subset\Delta$ or $ C\subset\Delta$ but $a$ multiple of $C$ moves. So $-K_{W}$ is ''almost $\mathrm{nef}$. {\it Kdhler manifolds with numerically effective Ricci class} 239 It would b{\it e} interesting to have a rough classification of conic bundles $X$ with $-K_{X} \mathrm{nef}$. (3) For contractions of typ $e(\mathrm{A})$, we have in fact the following additional information: PROPOSITION 3.11. {\it Assume} $that-K_{X}$ {\it is} nef, $K_{X}^{3}=0$ {\it and that} $\varphi;X\rightarrow W$ {\it is of type} (A). {\it Then} $K_{X}^{2}=0$. {\it Proof}. L{\it e} $\mathrm{t}\psi:W\rightarrow W'$ {\it be} the blow-down of $C_{0}=\varphi(E)$. Let $\sigma=\psi\circ\varphi$. Let $ N^{1}(Z)=\mathrm{Pi}c(Z)\otimes_{\mathrm{Z}}\mathbb{R}/\equiv$ for any {\it Z}. $\sigma^{\star}(N^{1}(W'))$ i{\it s} a line ar subspace of codimension 2 in $N^{1}(X)$, in fact $\psi^{\star}(Z^{1}(W'))$ i{\it s} of codimension 1 in $N^{1}(W)$, and $\varphi^{\star}(N^{1}(W))$ is of codimension 1 in $N^{1}(X)$, a{\it s} one {\it checks} immediately from Mori theory: {\it Assume} $K_{X}^{2}\neq 0$. T{\it hen} $(K_{X}^{2})^{\perp}=\{L\in N^{1}(X);L\cdot K_{X}^{2}=0\}$ {\it is} of codimen- {\it s}ion {\it 1 in} $N^{1}(X)$. {\it Hence}: $(K_{X}^{2})^{\perp}=\sigma^{\star}(N^{1}(W'))\oplus \mathbb{R}\cdot K_{X}$ because $K_{X}^{3}=0$. Since $K_{X}^{2}\cdot E=0,\ E$ i{\it s} in $(K_{X}^{2})^{\perp}$, {\it s}o $E=\mu K_{X}+\sigma^{\star}(H)$ with $H\in N^{1}(W')$. Cutting by $a$ fiber of $\varphi$ yields $\mu=1$. 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