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APPLICATIONS OF HODGE MODULES : KOLLAR CONJECTURE AND KODAIRA VANISHING(Algebraic Geometry and Hodge Theory) SAITO, MASA-HIKO

数理解析研究所講究録 (1992), 803: 107-124

1992-08

http://hdl.handle.net/2433/82895

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Departmental Bulletin Paper

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Kyoto University

数理解析研究所講究録 第 803 巻 1992 年 107-124

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107

APPLICATIONS OF HODGE MODULES KOLL\’AR CONJECTURE AND KODAIRA VANISHING

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BY

MASA-HIKO SAITO Department of Mathematics, Faculty of Science, Kyoto University

\S 0 Hodge Modules. (0.1). In this note, we will give an exposition of some applications of Morihiko Saito’s theory of Hodge modules. All of these applications are due to Moriniko Saito himself. Though there is a good exposition by Shimizu[Sh], in \S 0, we will recall quickly the definition of the category $MH(X, \mathbb{Q}, n)$ of Hodge Modules of weight , mainly for preparing the notations. In \S 1, we will give the statements of the stability of polarized Hodge modules by projective direct images and the decol aposition theorem for the intersection complexes of Beilinson-Bernstein-Deligne-Gakber type, and we will explain how these results imply existence of the natural pure Hodge structures on the intersection cohomology groups. In \S 2, we will discuss about Saito’s proof of Koll\’ar conjecture on the direct images of the edge components of “generic variation of Hodge structures“. \S 3 is devoted to a generalization of vanishing theorem of Kodaira-type, which follcws naturally from the theory of Hodge modules. $n$

(0.2). Let $X$ be a complex manifold. In this note, we will use the filtered right be the category of filtered Modules. Let -Modules $(M, F)$ such that $M$ is regular holonomic and $Gr^{F}(M)$ is coherent over . $13y$ Kashiwara, we have a faithful and exact functor $DR$ : $MF_{h}(D_{X})arrow Perv(\mathbb{C}_{X})$ (Riemann-Hilbert correspondence), and we define to be a fiber product of over Perv . That is, the objects are $((M, F),$ $K$ ) $\in MF_{h}(D_{X})\cross$ and Perv with an isomorphism : , and the morphisms are the pairs of the morphisms compatible with . $\mathcal{D}_{X^{-}}$

$j\psi F_{h}(\mathcal{D}_{X})$

$\mathcal{D}_{X}$

$Gr^{F}\mathcal{D}_{X}$

$MF_{h}(\mathcal{D}_{X})$

$MF_{h}(\mathcal{D}_{X}, \mathbb{Q}_{X})$

$(\mathbb{Q}_{X})$

$Perv(\mathbb{Q}_{X})$

$(\mathbb{C}_{X})$

$\alpha$

$DR(M)arrow K\otimes_{\mathbb{Q}_{X}}\mathbb{C}_{X}$

$\alpha$

(0.3). Let $i:Xarrow+Y$ be a closed embedding locally defined by $X=\{x_{1}=\cdots=x_{k}=$ with $(x_{1}, \cdots x_{m})$ local coordinates of Y. Then for a filtered holonomic -modules $(M, F)$ , the direct image $(\tilde{M}, F)=i_{*}(A/I, F)$ is defined by (see [Sh]), and locally we have

$0\}$

$\mathcal{D}_{X}$

$(M, F)\otimes_{\mathcal{D}_{X}}(\mathcal{D}_{Xarrow Y}, F)$

$\tilde{M}=M\otimes_{\mathbb{C}}\mathbb{C}[\partial_{1}, \cdots\partial_{k}]$

,

108

$F_{p}\tilde{M}=\oplus_{\nu\in N^{k}}F_{p-|\nu|}M\otimes\partial^{\nu}$

where we get the functor

$\partial^{\nu}=\prod_{1