On polarized manifolds of Δ genus two; part I
1984; Mathematical Society of Japan; Volume: 36; Issue: 4 Linguagem: Inglês
10.2969/jmsj/03640709
ISSN1881-1167
Autores Tópico(s)Advanced Algebra and Geometry
ResumoBy a polarized manifold we mean a pair $(M, L)$ of a projective manifold $M$ and an ample line bundle $L$ on $M$ .Set $n=\dim M,$ $d(M, L)=L^{n}$ and $\Delta(M,$ $L\rangle$ $=n+d(M, L)-h^{0}(M, L)$ .Then $\Delta(M, L)\geqq 0$ for any polarized manifold $(M, L)$(see [F2]).We have classified polarized manifolds with $\Delta=0$ in [F2] and those with $\Delta=1$ in [F5] (as for positive characteristic cases, see [F6]).In this series of papers we will study polarized manifolds with $\Delta=2$ .However, because of various technical reasons, we assume here that things are deflned over the complex number field $C$ , although some arguments work in positive characteristic cases too.This series is an improved version of [F1], which contains most results here, but, unfortunately, is hardly readable.We remark that Ionescu [I] obtained independently the classification of $(M, L)$ with $\Delta=2$ such that $L$ is very ample.\S 0. Outline of the classification.In this section we give a brief account of the classification of polarized manifolds with $\Delta=2$ .We freely use the notation in [F2], [F5], [F6], etc.The following result is used to reduce various problems to lower dimensional cases.(0.1) THEOREM.Let $(M, L)$ be a polarized manifold with dimM $=n\geqq 3$ , $d(M, L)=d\geqq 2$ and $\Delta(M, L)=2$ .Then any general member $D$ of $|L|$ is non- srngular.Moreover, the restriction homomorphism $r:H^{0}(M, L)arrow H^{0}(D, L_{D})$ is surjective and $\Delta(D, L_{D})=2$ .PROOF.[F7; (4.1)] shows that $D$ is smooth.If $r$ is not surjective, we have $H^{1}(M, O_{M})>0$ and $\Delta(D, L_{D})<2$ .The latter implies $H^{1}(D, L_{D})=0$ by [F2] and [F5].This is absurd because we have an exact sequence $H^{1}(M, -L)arrow$ $H^{1}(M, O_{M})arrow H^{1}(D, O_{D})$ and $H^{1}(M, -L)=0$ by Kodaira's vanishing theorem.Thus $r$ is surjective and hence $\Delta(D, L_{D})=2$ .(0.2) THEOREM.Let $(M, L)$ be a polarized manifold with dimM $=n\geqq 2$ , $\Delta(M, L)=2$ and $g(M, L)\leqq 1$ , where $g(M, L)$ is the sectional genus.Then $M\cong P(E\rangle$ T. $p_{UJITA}$ for an ample vector bundle $E$ of rank two over an elliptic curve $C$ and $L$ is the tautological line bundle on it.PROOF.We consider first the case $d(M, L)=d=1$ .Then $h^{0}(M, L)=n+d-\Delta$ $=n-1$, while dimBsl $L|\leqq 1$ by [F2; Theorem 1.9].Therefore, if $D_{1}$ , , $D_{n-1}$ are general members of $|L|$ and if $C=D_{1}\cap\cdots\cap D_{n-1}$ , then Bsl $L|=Supp(C)$ is a curve.Moreover $LC=L^{n}=1$ .Hence the scheme theoretic intersection $C$ is an irreducible reduced curve.By [F2; Proposition 1.3] we have $h^{1}(C, O_{C})$ $=g(M, L)\leqq 1$ .Assume that $H^{1}(M, O_{M})=0$ .Then we claim $H^{i}(V_{j}, (1-i)L)=0$ for each $j=1,$ $\cdots$ , $n$ and $i=1,$ $\cdots$ , $j-1$ , where $V_{j}=D_{j}\cap D_{j+1}\cap\cdots\cap D_{n-1}$ (set $V_{n}=M$ and $V_{1}=C)$ .Indeed, this is true when $j=n$ by the assumption and Kodaira's vanish- ing theorem.In case $j<n$ , we use the exact sequence $H^{i}(V_{j+1}, (1-i)L)arrow$ $H^{i}(V_{j}, (1-i)L)arrow H^{i+1}(V_{j+1}, -iL)$ and the descending induction on $j$ from above to prove the claim.Thus we have $H^{1}(V_{j}, O)=0$ for each $j\geqq 2$ , which implies $\Delta(M, L)=\Delta(V_{n}, L)=\ldots=\Delta(V_{1}, L)=\Delta(C, L)$ .However $\Delta(C, L)\leqq 1$ because $h^{1}(C_{y}O_{C})\leqq 1$ .This contradiction shows that $H^{1}(M, O_{M})\neq 0$ .On the other hand, by a similar argument as above, we get $H^{i}(V_{j}, -tL)=0$ for any $i 0$ by the descending induction on $j$ and hence $H^{1}(V_{j+1}, O)arrow H^{1}(V_{j}, O)$ is injective for each $j\geqq 1$ .Therefore $h^{1}(M, O_{M})\leqq h^{1}(C, O_{C})\leqq 1$ .So we conclude that $H^{1}(M, O_{M})arrow H^{1}(C, O_{C})$ is bijective and $g(M, L)=h^{1}(C, O_{C})=1$ .Since $h^{1}(M, O_{M})=1$ , the Albanese variety $A$ of $M$ is an elliptic curve.Let $\alpha;Marrow A$ be the Albanese morphism.Then $\alpha(C)=A$ because $H^{1}(A, O_{A})arrow$ $H^{1}(M, O_{M})arrow H^{1}(C, O_{C})$ is bijective.In view of $h^{1}(C, O_{C})=1$ , we infer that $C$ is a non-singular elliptic curve.Now, when $n=2$ , we apply [F5; (1.11)] to prove the theorem.So we will consider the case $n\geqq 3$ by induction on $n$ .Let $\pi;M'arrow M$ be the blowing-up with center $C$ , let $E=\pi^{-1}(C)$ be the exceptional divisor, and let $D_{j}'$ and $V_{j}'$ be the proper transforms of $D_{j}$ and $V_{j}$ respectively.Since $C$ is the ideal theoretical intersection of $D_{j}' s$ , we have $D_{1}'\cap\cdots\cap D_{n-1}'=\emptyset$ .So Bs $|\pi^{*}L-E|=\emptyset$ because $D_{j}'\in|\pi^{*}L-E|$ .This linear system gives a morphism $\rho$ : $M'arrow P^{n-2}$ , whose restriction to each fiber of $Earrow C$ is an isomorphism.From this we infer $E\cong$ $C\cross P^{n-2},$ $D_{j}'\cap E\cong C\cross P^{n-3}$ and $V_{j}'\cap E\cong C\cross P^{j-2}$ .This implies that $V_{j}$ is smooth along $C$ and $V_{j}'$ is the blowing-up of $V_{j}$ with center $C$ .Thus, by Bertini's theorem, $V_{j}$ is a submanifold of $M$ .So, to prove the theorem, it suffices to derive a contradiction assuming $n=3$ .When $n=3$ , any general member $D$ of $|L|$ is a $P^{1}$ -bundle over $A=Alb(M)$ $\cong Alb(D)$ by [F5; (1.11)].Hence $\alpha;Marrow A$ is a $P^{2}$ -bundle by [F4; (4.9)].Moreover $M\cong P_{A}(\mathcal{E})$ for some ample vector bundle $\mathcal{E}$ of rank 3 on $A$ and $L$ is the tautological line bundle on it.Then, as is well-known (cf., e.g., [I; Prop- osition 3.11]), we have $h^{0}(M, L)=h^{0}(A, \mathcal{E})=\deg(\det \mathcal{E})=L^{3}=d$ , contradicting
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