Some theorems of algebraicity for complex spaces - Project Euclid

J. Math. Soc. Japan

Vol. 27, No. 2, 1975

Some theorems of algebraicity for complex spaces By J. MORROW and H. ROSSI (Received Oct. 2, 1972)

Introduction. This paper consists of a review of a certain collection of known theorems, which prove that, under suitable hypotheses, a given complex space is projective–or nearly projective. Many of the original proofs depended on use of the Riemann-Roch theorem, and thus were confined to low dimensions. Our technique is to find positive line bundles by patching together plurisubharmonic functions. Thus, a fundamental criterion for algebraicity is Kodaira’s theorem that a Hodge manifold is projective [10], as generalized by Grauert in [4]. THEOREM. A compact complex space which admits a positive line bundle is a projective variety. In the first section we collect the definitions and fundamental theorems in the positivity-pseudoconvexity-plurisubharmonicity cycle of ideas. Lemma 1.3 is fundamental, and we are indebted to J. E. Fornass for showing us the crucial trick. Section 2 is devoted to a new proof of Grauert’s theorem on the extension of positivity of line bundles ([4], \S 3 Satz 4), and in Section 3 this theorem is applied to give a new proof of Van de Ven’s theorem that a compactificais algebraic [16]. In Section 4 we show that projective space is tion of characterized by its hyperplane section (in dimensions greater than 2). We turn next to Kodaira’s theorem that a surface with one meromorphic function is elliptic [11]. This result was generalized to dimension 3 by Kawai [9], and in his review of Kawai’s Paper, Hironaka [6] indicated a proof of the appropriate generalization to arbitrary dimension. We prove the following generalization of Hironaka’s argument: Let be a holomorphic mapping of a compact complex space $X$ of dimension onto an algebraic variety . Suppose is a coherent sheaf whose restriction to the generic fiber is the sheaf of sections of a positive line bundle. Then $X$ is Moisheson: $X$ algebraically independent meromorphic functions. This result (as Hirohas naka has shown) easily gives the generalization: if $X$ is of dimension and $n-1$ , then the transcendence degree of its field of meromorphic functions is $X$ bimeromorphically equivalent to an elliptic fibration over a projective is variety. If, in the preceding result it is assumed that the sheaf is a sheaf $C^{2}$

$\pi:X\rightarrow\Delta$

$n$

$F\rightarrow X$

$\Delta$

$n$

$n$

$F$

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J. MORROW and H. ROSSI

of sections of a line bundle which is positive on every fiber, then the conclusion is that $X$ is algebraic. Finally, in section 6 we give new proofs of the analogous theorems for surfaces, where the results are sharper than those obtainable in higher dimensions. We add, using the techniques of negativity, a proof that the Hopf process preserves algebraicity in both directions. $\sigma-$

\S 1. Preliminaries. The basic techniques of this paper are those developed by Grauert in his fundamental paper on exceptional sets [4]. We shall need some basic definitions for complex spaces (see Narasimhan [13]). 1.1. DEFINITIONS. (a) Let $X$ be a complex space, $U$ an open subset of $X$ . such that $U$ can be embedded as a closed subvariety of a domain $D$ in is a well-dePned Define $C^{\infty}(U)=\{f|_{U} ; f\in C_{R}^{\infty}(D)\}$ . The association presheaf; we call the associated sheaf of real-valued functions on $X$. (b) For $x\in X$ , let $T_{x}(X)$ be the (Zariski) tangent space of $X$ at . The set $T(X)=UT_{x}(X)x$ has, in a natural way the structure of a linear space (com$C^{n}$

$U\rightarrow C^{\infty}(U)$

$C^{\infty}$

$C_{X}^{\infty}$

$x$

on the

This is the tangent bundle to X. will denote the sheaf of holomorphic sections of $T(X)$ . (c) A Hermitian form on $X$ is a complex-valued function on $T(X)$ $\oplus T(X)$ which is hermitian symmetric on the fibers. is an hermitian metric if it is positive dePnite. (d) For denoted $H_{x}(f)$ , as , we define the complex hessian of at the hermitian form on $T_{x}(X)$ :

plex space with linear structure

fibers).

$\Theta_{X}$

$C^{\infty}$

$\omega$

$\omega$

$f$

$f\in C_{X}^{\infty}$

$x$

$H_{x}(f)(X, Y)=\partial\partial f(X,\overline{Y})$

, and (the differentiation is taken in the ambient $f$ ). $f$ is (strictly) plurisubharmonic (spsh) at sion dePnite (dePnite) on $T_{x}(X)$ . $C^{n}$

is independent of the extenif $H_{x}(f)$ is positive semi-

$x$

A domain $D\Subset X$ is strongly pseudoconvex (spsc) if there is a neighand a spsh function $f\in C^{\infty}(U)$ such that borhood $U$ of (e)

$\partial D$

(i) (ii)

$D\cap U=\{x\in U;f(x)0$

$\partial D$

$ p(x)\geqq-\epsilon$

$\epsilon_{0}>0$

$\epsilon0$

$\psi(t)=t$

.

, for

, for

$(-2\epsilon_{0}-\eta)/3-2\epsilon_{0}/3$ , the tive ( second line is positive. The lemma now follows by aPplying the following in some fiber metric on lemma to the bundle of unit tangent vectors to the tangent bundle and letting $q=g/A$ . continuous functions 1.4. LEMMA. Let $K$ be a compact metric space, $K$ and $g(x)>0$ $K$ $f(x)=0$ is an $A>0$ such There that on such . on if that $g+Af>0$ on $K$. be a line bundle over the complex space $X$ . Then there Now let a natural $c*$ action which is fiber-preserving. is defined on the space defined on function A fiber metric on is a non-negative real-valued $\rho(l)=0$ only of . If on the zero section and such that $\varphi=1$

$\overline{D\cup U}$

$g,$

$f$

$f\geqq 0$

$L\rightarrow\pi X$

$L$

$L$

$\rho(t\cdot l)=|t|^{2}\rho(l)$

$l\rightarrow t\cdot l$

$L$

$C^{\infty}$

$\rho$

$Z_{L}$

$L$

$U_{\alpha}$

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J. MORROW and H. ROSSI

is a coordinate neighborhood of

$L,$

$\rho(x, \xi_{\alpha})=\rho_{\alpha}(x)|\xi_{\alpha}|^{2}(\rho_{\alpha}(x)=\rho(x, 1))$

.

, we obtain plex Hessians of on $X$, denoted

is given by and the transition functions for

$L|U_{\alpha}=U_{\alpha}\times C$

If

$\{f_{a}^{\beta}\}$

, then

$\rho|U_{\alpha}$

on the overlap. Thus the comagree on so define a global hermitian form , and called the curvature form of the metric . Note that . We shall need this fact, see Grauert [4], p. 341. a line bundle. The follow1.5. LEMMA. Let $X$ be a complex sPace, ing are equivalent. (a) has a fiber metric whose curvature is Positive definite. , a spsh function (b) There are an open set in , an defined in $L$

$\rho_{\alpha}|f_{a}^{\beta}|^{2}=\rho_{\beta}$

$ln\rho_{a},$

so

,

$\partial\overline{\partial}ln\rho_{\alpha}=\partial\partial ln\rho_{\beta}$

$U_{\alpha}\cap U_{\beta}$

$ln\rho_{\beta}$

$\Theta_{\rho}$

$\rho$

$\pi^{*}\Theta_{\rho}=\partial\partial ln\rho$

$L\rightarrow X$

$L$

$\rho$

$L$

$U$

$U$

$\epsilon>0$

$p$

such that (i)

$1+\epsilon)$

for each

$x$

in

$X,$

$p$

maPs

$U\cap\pi^{-1}(x)$

Properly onto the interval

$(1-\epsilon$

,

,

if $p(l)=1$ , then infinitesimal generator of the (ii)

$\partial p/\partial|\xi|(l)>0$

$c*$

and

$e^{i\theta}l\in U$

for all

$\theta$

(where

$\partial/\partial\xi$

is the

action).

and PROOF. (a) trivially implies (b) by taking $U=\{l;0C$

$R_{j}>0$

.

This

.

is not positive, where

$\epsilon>0$

In the latter case, we are finished. In

.

$\sum_{i=1}^{p}R_{i}a_{ij}=\sum_{i1}^{p-1}R_{i}a_{ij}+a_{pj}(\sum_{i=1}^{p-1}R_{i}a_{ip})-\epsilon a_{pj}$

Because of (3.7) we may choose $j0$ . For $j=p$ we get

if

so that the sum is positive for those

$(a_{pp}=-1)$

$\sum_{i=1}^{\rho}R_{i}a_{tp}=\sum_{i=1}^{p-1}R_{i}a_{tp}+\epsilon-\sum_{i=1}^{p-1}R_{i}a_{ip}=\epsilon>0$

.

Now we can finish the proof of Theorem 3.4. Let be the divisor are the integers found in Lemma 3.6 for the matrix where the $D$

$\{\Gamma_{p}\}^{n_{p}}$

$\{\Gamma_{1}\}^{n_{1}}$

$n_{j}$

$(\Gamma_{i}\cdot\Gamma_{j}),$

$|D|=A^{\prime}=U\Gamma_{j}n_{j}\nearrow 0$

and for

$i$

such that

$n_{j}>0$

$c([D]|_{\Gamma_{j}})=\sum_{i=1}^{p}n_{\{}c([\Gamma_{i}]|_{\Gamma_{j}})=\sum_{i=1}^{n}n_{i}(\Gamma_{i}\cdot\Gamma_{j})>0$

.

is positive. Then by [12], p. 62, Thus is positive, so $D$ has a positive normal bundle. By Theorem 3.3 every exceptional subvariety of is strongly negatively embedded, so Corollary 2.4 applies: $X$ is algebraic. $[D]|_{\Gamma_{j}}$

$[D]|_{A^{\prime}}$

$X-A^{\prime}$

\S 4. Characterizations of projective space. In dimensions at least 2, Hartogs’ theorem tells us that a normal projective variety is determined by a neighborhood of a hyperplane section. We would like that neighborhood to be an infinitesimal neighborhood. The following

Theorems

of

algebraicity

for

complex

spaces

177

results tell us that for projective space, a Prst order neighborhood will suffice. 4.1. THEOREM. Let $X$ be a connected compact complex manifold of dimension . Let $K$ be a submanifold of codimension 1 such that there is a biholomorphic map $j:P^{p-1}\rightarrow K$ with $j_{*}([K]|K)$ being the hyperplane section bundle of $P^{n-1}$ . Then if $X-K$ has no excepti0nal varieties, $X=P^{n}$ . Before we begin the proof we first prove a lemma. 4.2. LEMMA. If $X$ satisfies the hyp0thesis of Theorem 4.1, then $H^{1}(X, O)=0$ . PROOF. We have assumed dim $X=n\geqq 2$ . We have the following exact $Z)$ denotes cohomology with compact support). sequence (where $n\geqq 2$

$H_{c}^{*}(-,$

$\rightarrow H_{c}^{1}(X-K, Z)\rightarrow H^{1}(X, Z)\rightarrow H^{1}(K, Z)\rightarrow H_{c}^{2}(X-K)\rightarrow\cdots$

.

Now $X-K$ is holomorphically convex since $[K]|K>0$ . Since $X-K$ has

no

compact subvarieties it is Stein, and thus has the homotopy type of a $CW$ complex of (real) dimension . Thus $H_{c}^{1}(X-K, Z)=H_{2n-1}(X-K, Z)=0$ . Since $K=P^{n-1},$ $H^{1}(K, Z)=0$ . Then the exact sequence implies that $H^{1}(X, Z)=0$ , so $b_{1}(X)=b_{1}=0$ , where is the first Betti number of $X$ . Now by Theorem 2.4, $n$

$b_{1}$

$X$

is projective and hence

$=0$

$b_{1}=2h^{01}$

where

$h^{0,1}=\dim H^{1}(X, O)$

. Thus

$H^{1}(X, O)$

.

, PROOF (of Theorem 4.1). By hypothesis $[K]|K$ has sections be the which define a biholomorphic map of $K$ onto . Let canonical section, so that $K=\{x\in X:\sigma_{0}(x)=0\}$ . From the exact sequence $\sigma_{1}^{0},$

$n$

$P^{n- 1}$

$\cdots$

$\sigma_{n}^{0}$

$\sigma_{0}\in H^{0}(X, \underline{[K]})$

$0\rightarrow 0\rightarrow^{\sigma_{0}}\underline{[K]}\rightarrow^{\pi}[K]|K\rightarrow 0$

and Lemma 4.2

we obtain

the exact sequence

$0\rightarrow H^{0}(X, \mathcal{O})\rightarrow H^{0}(X, [K])\rightarrow H^{0}(K\pi_{*}[K]|K)\rightarrow 0$

.

, and let be Let $\sigma_{i}\in H^{0}(X, [K])$ be such that $F(p)=[\sigma_{0}(p), \cdots , \sigma_{n}(p)]$ , where we use homogeneous the meromorphic map and if $P\in K$ coordinates in is actually holomorphic: if . for some $i>0$ . $V=$ is compact and Let { $x\in X$ ; rank $dF(x)0,$ $x\in K$ and is independent of the gives a biholomorphic map onto . But has codimension 1 in $X$ (it is defined locally by the equation det $dF(x)=0$ , and det $dF$ is not identically zero). ; and is an unbranched Since $X-K$ has no compact subvarieties, covering. is biholomorphic. is simply connected so $F:X\rightarrow P^{n}$

$\pi_{*}(\sigma_{i})=\sigma_{i}^{0},1\leqq i\leqq n$

$P^{n}$

$F$

$P\not\in K,$

$\sigma_{0}(p)\neq 0$

$\sigma_{i}(p)\neq 0$

$V$

$d\sigma_{0}(x)$

$[\sigma_{1}(x), \cdots , \sigma_{n}(x)]$

$d\sigma_{i}(x)$

$P^{n-1}$

$V$

$ V=\emptyset$

$P^{n}$

$ V\cap K=\emptyset$

$F:X\rightarrow P^{n}$

$F$

\S 5. Fiberings over projective varieties.

its be a compact complex space, reduced and irreducible, and field of meromorphic functions. By tr (X) we mean the transcendence degree

Let

$X$

$\mathcal{M}(X)$

178

J. MORROW and H. Rossi

of over $C$. If tr $(X)=\dim X,$ $X$ is said to be a Moisheson space. It is clear that $X$ is Moisheson if and only if there is a set of meromorphic functions, all holomorphic at some regular point of $X$ , with spanning differentials there. be a coherent analytic sheaf. As in [15], we see that there Now let is a Zariski open set $U$ in $X$ on which $F$ is locally free. If $F|U$ is locally free of rank , we shall say that $F$ is of generic rank , writing grk $(F)=k$ . The construction in [15] demonstrates that any $m$ sections of a coherent sheaf of generic rank 1 defines a meromorphic map into $P^{m-1}$ . Our next result generalizes (as we shall see) Kodaira’s theorem on surfaces of transcendence degree 1 ([11], p. 131 and p. 134); the proof is not too different from, but more general than that of Hironaka [6]. a Projective variety 5.1. THEOREM. Let $X$ be a comPact comPlex sPace, and a surjective holomorphic map. Supp0se there exists a coherent over a such that sheaf of generic rank 1, and there is a $X$ $U$ $L|\tau^{-1}(p)>0$ neighborhood . Then is Moisheson. of , and [3], By PROOF. Grauert’s theorem the 0th direct image of $F$ is a coherent sheaf on . By Corollary 1.10, $\{q\in U;L|\tau^{-1}(q)>0\}$ is open. Thus we may select our so that is free, is regular in , and there is a regular $X$ point for on . In this case is isomorphic to the sheaf of germs of holomorphic maps into for every . Choose large enough so that the sections of on . Now let $D$ into embed be a holomorphic divisor on is with $[D]>0$ and $p\not\in|D|$ (such exist since projective). There is a $\mu>0$ such that the sections of generate as , -module (see Grauert [4], p. 344). Thus there are sections , span such that , . $X$ ] , are thus sections of on whose restrictions to , the meromorphic functions define a projective embedding. If, say spanning . Since is have differentials , , regular at algebraic, there are meromorphic functions on , . Thus are also meromorphic whose differentials span $X$ , , so the differentials , and constant on on $k+1\leqq j\leqq n$ span $T_{q}(X)$ . Thus $X$ is Moisheson. 5.2. COROLLARY. Let $X$ be a $comPact$ comPlex space of dimension , with tr $(X)=n-1$ . There is a Proper modification of $X$ for which there , inducing an isomorphism of function fields is a holomorphic map generically a curve of genus 1. a Projective variety and with , then PROOF. Let be a projective variety with the function field . Let be the normalidim $\Delta=n-1$ and there is a meromorphic map the two projeczation of the closure of the graph of $\mathcal{M}(X)$

$F\rightarrow X$

$k$

$k$

$\Delta$

$\tau;X\rightarrow\Delta$

$F\rightarrow X$

$ p\in\Delta$

$F=\underline{L}$

$p$

$F_{0}$

$\Delta$

$p$

$q$

$\Delta$

$p$

$F_{0,p}$

$\tau^{-1}(p)$

$F_{0.p}^{\nu}$

$H^{0}(\tau^{-1}(p), L^{\nu}|\tau^{-1}(p))$ $L$ “

$\nu$

$\nu$

$\tau^{-1}(p)$

$P^{k}$

$\tau^{-1}(p)$

$\Delta$

$\Delta$

$F_{0.p}^{\nu}$

$[\underline{D}]^{\mu}\otimes F_{0}^{\nu}$

$\mathcal{O}_{\Delta,p}$

$\sigma_{0},$

$H^{0}(\Delta, [D]^{\mu}\otimes F_{0}^{\nu})$

$\sigma_{0}(p),$

$\tau^{*}(\sigma_{k})$

$\cdots$

$\underline{\tau^{*}([D}$

$\sigma_{k}(p)$

$H^{0}(\tau^{-1}(p), L^{\nu}|\tau^{-1}(p))$

$\cdots$

$\sigma_{k}\in$

$\tau^{*}(\sigma_{0})$

$\tau^{-1}(p)$

$)_{-}^{\mu}\otimes F^{\nu}$

$\tau^{*}(\sigma_{0})(q)\neq 0$

$f_{i}=\tau^{*}(\sigma_{i})/\tau^{*}(\sigma_{0}),$

$1\leqq i\leqq k$

$\Delta$

$T_{q}(\tau^{-1}(p))$

$\Delta,$

$f_{k+1}\circ\tau,$

$T_{p}(\Delta)$

$\cdots$

$f_{k+1},$

$p$

$f_{n}$

$\cdots$

$ f_{n}\circ\tau$

$\tau^{-1}(p)$

$df_{i}(q),$

$1\leqq i\leqq k,$

$d(f_{j}\circ\tau)(q)$

$n$

$\pi;x/\rightarrow X$

$\tau;X^{\prime}\rightarrow\Delta^{n-1}$

$\tau^{-1}(p)$

$\Delta$

$\Delta$

$\mathcal{M}(X)$

$ F:X\rightarrow\Delta$

$F,$

$\pi;X^{\prime}\rightarrow X,$

$X^{\prime}$

$\tau;X^{\prime}\rightarrow\Delta$

Theorems of algebraicity

for

spaces

complex

179

$F_{1}=$ be the sheaf of germs of holomorphic vector fields on $V$ $\{v\in\Theta;\tau_{*}(v)=0\},$ $F_{2}=Hom(F_{1}, \mathcal{O})$ . ; since Let be the singular locus of $V\leqq n-2$ $\tau(V)$ normal, , so is dim is a proper subvariety of . Let $S=$ $\{x\in X^{\prime}-\tau^{-1}\tau(V);rank_{x}\tau_{x}0$

$x$

$\Theta_{\rho}(v,\overline{v})=\pi^{*}\Theta_{\rho}(w,\overline{w})=\partial\partial ln\rho(w,\overline{w})=c(x)\partial\partial\rho(w,\overline{w})>0$

$\tau_{*}(v)=0$

.

.

be a negative line bundle on , and a spsh metric on $H$. Then $H=\tau^{*}H_{0}$ , is positive semiis a metric on . The curvature of . Once again we use Lemma definite and if 1.4 to conclude that there is an integer $N$ such that for any $v\in T_{x}(X),$ $v\neq 0$ ,

Let

$\Delta$

$H_{0}$

$\sigma_{0}$

$\tau^{*}\sigma=\tau^{*}\sigma_{0}$

$\Theta_{\sigma}$

$\sigma,$

$\tau_{*}v\neq 0,$

$\Theta_{\sigma}(v,\overline{v})=\Theta_{\sigma_{0}}(\tau_{*}v, \overline{\tau_{*}v})>0$

180

J. MORROW and H. ROSSI $\Theta_{\rho}(v,\overline{v})+N\Theta_{\sigma}(v,\overline{v})>0$

.

, where . Thus this bundle is is a metric on But negative on $X$ , so $X$ is algebraic. is prOper and light and 5.4. COROLLARY (see Wavrik [17]). If $X$ is algebraic, then so is . PROOF. Take to be the trivial bundle, and apply the proof of the above theorem. 5.5. COROLLARY. Let $X$ be a compact complex space of dimension , and be a holomorPhic map, a projective variety of dimension $n-1$ . Let and $D$ a holomorphic divisor on $X$ . (a) If there is a such that dim $\tau^{-1}(p)=1,$ $\dim|D|\cap\tau^{-1}(p)=0$ and nonempty, then $X$ is Moisheson. (b) If for all , diml $D|\cap\tau^{-1}(p)=0$ , then $X$ is algebraic. PROOF. In case (a) we could replace $|D|$ by a subvariety $K$ of codimension 1. For we could choose satisfying (a) so that the ideal sheaf of $K$ is near . Clearly $L|\tau^{-1}(p)>0$ so theothe sheaf of sections of a line bundle , so rem (5.1) applies: $X$ is Moisheson. In case (b), $[D]|\tau^{-1}(p)>0$ for all $X$ is algebraic. theorem (5.3) applies: $\Theta_{\rho}+N\Theta_{\sigma}=\Theta_{\rho\sigma}N$

$L\otimes H^{N}$

$\rho\sigma^{N}$

$\tau;X\rightarrow\Delta$

$\Delta$

$L$

$n$

$\tau;X\rightarrow\Delta$

$\Delta$

$ p\in\Delta$

$ p\in\Delta$

$p$

$L$

$p$

$ p\in\Delta$

\S 6. More on surfaces. In the case of compact complex spaces of dimension 2 the preceeding theorems can be made sharper. Although the results in the section are well known we include them for purposes of comparison and completeness and because they are easy to prove. be a 2-dimensional compact complex space with 6.1. THEOREM. Let is algebraic. such that tr $(S)=2$ . Then there is a modification PROOF. Let be a (non-singular) projective variety with function field . Then there is a meromorphic map : . with ; map and : the Let be the graph of the meromorphic projection maps. Again, by resolution of singularities, we may assume that ; is a modification. By Stein factorization we is a manifold. The map as . The map is a modification and can write is a . By 5.4 is algebraic and branched covering. Now , it easily follows that . Thus is a modification and since [7] by Hopf is given by a composition of -processes. Since is algebraic so is . 6.2. COROLLARY (Kodaira-Chow [2]). A Moisheson surface is algebraic. PROOF. Let ; be as above. Then (since are manifolds) is given by a composition of -processes. By Kodaira [10], is algebraic $S$

$\pi;S^{\prime}\rightarrow S$

$S^{\prime}$

$\Delta$

$\Phi$

$\mathcal{M}(S)$

$ S\rightarrow\Delta$

$\Phi^{*}:$

$\Phi$

$S^{f}$

$\pi_{1}$

$S^{\prime}\rightarrow S,$

$\mathcal{M}(\Delta)\cong \mathcal{M}(S)$

$ S^{\prime}\rightarrow\Delta$

$\pi_{2}$

$S^{\prime}$

$S^{\prime}\rightarrow S$

$\pi_{1}$

$S‘\rightarrow\hat{\Delta}$

$ S^{\prime}\rightarrow\hat{\Delta}\rightarrow\Delta$

$\hat{\Delta}\rightarrow\Delta$

$\pi_{2}$

$\hat{\Delta}$

$\mathcal{M}(S^{\prime})=\mathcal{M}(\hat{\Delta})=\mathcal{M}(\Delta)$

$\hat{\Delta}=\Delta$

$\mathcal{M}(\hat{\Delta})=\mathcal{M}(\Delta)$

$\pi_{2}$

$\Delta$

$\pi_{2}^{-1}$

$\sigma$

$S^{\prime}$

$S$

$S^{\prime}\rightarrow S$

$S,$

$\pi_{1}$

$\sigma$

$S^{\prime}$

$\pi_{1}^{-1}$

$S$

Theorems

of

algebraicity

for

spaces

complex

181

since is algebraic. 6.3. COROLLARY. If is an algebraic surface and if is a surface which has a holomorPhic surjective map , then is algebraic. 6.4. THEOREM. Let be a 2-dimensional normal compact complex space, , whose and a holomorPhic map tr $(S)=1$ . Then there is a curve generic fiber is an elliptjc curve. PROOF. We know there is a modification and a map is well-defined on . with the required properties. We shall show that $p\in S$ , so that is not a point. Then is a subvariety of . Let , and be a branch of . By Corollary such that Let $\tau^{-1}(q)\cap E=1$ (a), , dim . must so is connected, 5.5 Since have we . Thus that implies that has a continuous extension to is already defined except at an isolated set of points. Since is since normal, is holomorphic on . As a final application of these techniques we prove a generalization of the theorem in ([10], Appendix). 6.5. THEOREM. Let $X$ be a compact complex manifold of dimension and an embedding whose normal bundle is the Hopf bundle . Then (a) the space with identified to a pOint is a manifold, and the $S^{\prime}$

$S$

$S^{\prime}$

$\pi;S^{\prime}\rightarrow S^{f}$

$S^{\prime}$

$S$

$\tau;S\rightarrow\Delta$

$\Delta$

$\pi:S^{\prime}\rightarrow S$

$\tau;S^{\prime}\rightarrow\Delta$

$S$

$\tau\circ\pi^{-1}$

$\pi^{-1}(p)$

$\pi^{-1}(p)$

$E$

$\pi^{-1}(p)$

$S^{\prime}$

$\tau^{-1}(q)\cap E\neq\emptyset$

$ q\in\Delta$

$\pi^{-1}(p)$

$E\subset\tau^{-1}(q)$

$\tau\circ\pi^{-1}$

$\pi^{-1}(p)\subset\tau^{-1}(q)$

$p$

$S$

$\tau\circ\pi^{-1}$

$S$

$\tau\circ\pi^{-1}$

$n$

$H^{-1}$

$j:P^{n-1}\rightarrow X$

$P^{n-1}$

$X_{0}$

projection

$X\rightarrow X_{0}$

$p_{0}$

the quadratic transform,

if is algebraic, so is . PROOF. Since has a negative normal bundle, it has a strongly pseudo$U$ convex neighborhood . Let be a line bundle on $X$. Then $L|_{P^{n-1}}=H^{\nu(L)}$ for (b)

$X$

$X_{0}$

$P^{n-1}$

$L$

. some integer 6.6. LEMMA. If $\nu(L)$

$\nu(L)\geqq 0,$

$H^{1}(U, L)=0$

.

PROOF. By [4], p. 355 or [14] there is an integer , is injective, where projection of in $X$. Now, from the exact sheaf sequence $H^{1}(U, \underline{L})\rightarrow H^{1}(U,\underline{L}\otimes O/I^{\nu_{0}})$

such that the is the ideal sheaf

$\nu_{0}\geqq 0$

$I$

$P^{n-1}$

$0\rightarrow\underline{L}\otimes I^{\mu}/I^{\mu+1}\rightarrow\underline{L}\otimes \mathcal{O}/I^{\mu+1}\rightarrow\underline{L}\otimes \mathcal{O}/I^{\mu}\rightarrow 0$

we obtain the exact (6.7)

cohomology sequence

$H^{1}(P^{n- 1}, \underline{L}\otimes I^{\mu}/I^{\mu+1})\rightarrow H^{1}(P^{n-1}, \underline{L}\otimes \mathcal{O}/I^{\mu+1})$

$\rightarrow H^{1}(P^{n-1},\underline{L}\otimes \mathcal{O}/I^{\mu})$

on

so

.

, and $H^{1}(P^{n-1}, H^{\nu(L)+\mu})=0$ if $\nu(L)+\mu\geqq 0$ . Thus, if $\nu(L)\geqq 0$ , the second map of (6.7) is injective for all , , we obtain injectivity of the folso, after composing all these maps, lowing map

$I^{\mu}/I^{\mu+1}=H^{\mu}$

$P^{n-1}$

,

$\underline{L}\otimes I^{\mu}/I^{\mu+1}=H^{\nu(L)+\mu}$

$\mu\geqq 0$

$0\leqq\mu\leqq\nu_{0}$

$H^{1}(U,\underline{L)}\rightarrow H^{1}(P^{n-1},\underline{L}\otimes \mathcal{O}/I)$

$=H^{1}(P^{n- 1}, H^{\nu(L)})=0$

.

182

J. MORROW and H. ROSSI

Thus $H^{1}(U, L)=0$ . We return now to the proof of (a). From the exact sequence $0\rightarrow I^{2}\rightarrow I\rightarrow I/I^{2}\rightarrow 0$

and the just proven fact that $\rightarrow H^{0}(U, I/I^{2})=H^{0}(P^{n-1}, H)$

we have that $H^{0}(U, I)$ is surjective. Thus there are $f_{i}\in H^{0}(U, I),$ $H^{1}(X, I^{2})=0(I^{2}|_{P^{n- 1}}=H^{2})$

,

$0\leqq i\leqq n$

generate the stalk at each point and send to whose projections in $P^{n-1}$ $U$ . Thus $F=$ defines a map of into the quadratic transform $QC^{n}$ , which is one-one on of the origin in and an immersion there. . (a) is proven. Thus $F$ is a biholomorphic map in some neighborhood of $U$ (b). By part (a) we may take to be the ball in with the origin blown $P^{n-1}$ $U$ $H^{1}(U, O)=0$ up. Since , the restriction map and is contractible to is injective. Assuming that $X$ is algebraic, let be $X$ a negative line bundle on , and a spsh metric for . By the above we see that $L\cong[P^{n-1}]^{\nu(L)}$ in $U$ . Now the algebraicity of follows from this [4]. 8, p. generalizing more general fact Grauert’s Satz 364 6.8. LEMMA. Let $X$ be a projective variety, $D$ a holomorphic divisor with be $X$ with $|D|$ blown down to a point a negative normal bundle and (normalized). If there is a negative line bundle such that $L=[D]^{\nu}$ in is also algebraic. some neighborhood of $|D|$ , then PROOF. We may assume that $L=[D]$ in the neighborhood $U$ and that $H^{0}(X, L^{-1})$ projectively embeds $X$ , by replacing , and for some with $D$ (notice that $\nu>0$ necessarily, since by $D$ must be negative). By hypothesis, $L^{-1}|_{(UD|)}-|$ is trivial (since $[D]|_{(X-|D|)}$ is trivial). Let $K$ be the . If $\sigma\in H^{0}(X, L^{-1}),$ to corretrivial line bundle extension of sponds, under the isomorphism $L^{-1}=[D]^{-1}$ , to a function holomorphic in $U$ , (which is $U$ with $|D|$ replaced by ). vanishing on $|D|$ , so drops down to $H^{0}(X, L^{-1})$ $H^{0}(X_{0}, K)$ , , they drop down to , are a basis for Thus if , . Let still denoted . considered as a function on is zero just on the fiber over and otherwise defines a metric on with , since the embed $X,$ $X_{0}-\{p_{0}\}$ into projective space. In is a function with and . By Lemma 1.3 we can find as $ g=ln\rho$ near , with everywhere. Then a $P^{n- 1}$

$I/I^{2}$

$(f_{1}, \cdots , f_{n})$

$P^{n-1}$

$C^{n}$

$P^{n-1}$

$C^{n}$

$L$

$H^{1}(U, O^{*})\rightarrow H^{1}(P^{n-1}, O^{*})$

$L$

$\rho$

$X_{0}$

$X_{0}$

$p_{0}$

$L\rightarrow X$

$X_{0}$

$L$

“

$L^{k}$

$k\geqq 0$

$k$

$L|_{|D|}$

$X_{0}$

$L^{-1}|_{X_{0}-\{p_{0}\}}$

$\sigma|_{U}$

$U_{0}$

$\sigma_{1},$

$\cdots$

$p_{0}$

$\sigma_{N}$

$\sigma_{1},$

$\cdots$

$K^{-1}$

$\rho=\sum|\sigma_{i}|^{2}$

$\sigma_{N}$

$\rho$

$K^{-1}$

$p_{0}$

$\Theta_{\rho}\gg 0$

$U_{0},$

$\sigma_{i}$

$ ln\rho(P)\rightarrow-\infty$

$\partial\partial ln\rho\gg 0$

$g\in C^{\infty}(U_{0}),$

$\partial U_{0}$

$C^{\infty}$

$\rho$

$P\rightarrow p_{0}$

$\partial\partial g\gg O$

on

$X-U_{0}$

$\rho=\left\{\begin{array}{l}\rho\\ e^{g} on\end{array}\right.$

$\overline{U}_{0}$

defines a global metric algebraic.

on

$K^{-1}$

with

$\Theta_{\rho}\gg 0$

, thus

$K^{-1}$

is negative, and

$X_{0}$

is

Theorems

of algebraicity for

complex

spaces

183

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J. MORROW

H. ROSSI

Department of Mathematics University of Washington Seattle, Washington 98195

Department of Mathematics University of Utah Salt Lake City, Utah 84112

U. S. A.

U. S. A.