# A remark on the decomposition theorem for direct images of canonical sheaves tensorized with semipositive vector bundles.

The decomposition theorem for direct images of canonical sheaves was proved by J. Kollar [1, Theorem 3.1]. Inspired by the work of S. Matsumura [2], here we note that the decomposition theorem also holds for direct images of canonical sheaves tensorized with Nakano semipositive vector bundles. Although Theorem 1 below is a direct consequence of Takegoshi's results in [3], it was not stated explicitly there. Therefore we give the precise statement of the decomposition theorem and prove it explicitly here. We remark that Theorem 1 below immediately implies the weaker form of the decomposition theorem [3, I Decomposition Theorem] (cf. Corollary 2).Theorem 1. Let X be a Kahler manifold of pure dimension, Y a complex analytic space and f: X [right arrow] Y a proper surjective morphism such that all the connected components of X are mapped surjectively to Y. For a Nakano semipositive vector bundle (E, h) on X, we have an isomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in the derived category of [O.sub.Y]-modules.

Proof. The sheaf of E-valued [C.sup.[infinity]] (p, q)-forms on X is denoted by [A.sup.p,q.sub.X](E). Then we have the Dolbeault quasi-isomorphism

[[omega].sub.X] [cross product] E [right arrow] ([A.sup.n,*.sub.X] (E), [bar.[partial derivative]]),

which is an [f.sub.*]-acyclic resolution of [[omega].sub.X] [cross product] E. There

fore we have an isomorphism

R[f.sub.*]([[omega].sub.X] [cross product] E) [equivalent] ([f.sub.*] [A.sup.n,*.sub.X] (E), [bar.[partial derivative]])

in the derived category of [O.sub.Y]-modules.

In the proof of Theorem 6.4 in [3], Takegoshi defined an [O.sub.Y]-subsheaf [R.sup.0][f.sub.*][H.sup.n,q] (E) of Ker([bar.[partial derivative]]: [f.sub.*][A.sup.n,q.sub.X](E) [right arrow] [f.sub.*], [A.sup.n,q+1.sub.X](E)) such that the canonical inclusion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

induces an isomorphism of [O.sub.Y]-modules

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for every q. The composite of the inclusions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is denoted by [[phi].sup.q]. Then [[phi].sup.q] defines a morphism of complexes

[R.sup.0][f.sub.*][H.sup.n,q](E)[-q] [right arrow] [f.sub.*][A.sup.n,*.sub.X](E)

for every q. Since we have the isomorphism (1.1) for every q, we obtain a quasi-isomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by taking the direct sum for all q. Combining with the isomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we obtain an isomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in the derived category as desired.

As a corollary of the theorem above, we have the following:

Corollary 2. In addition to the situation in Theorem 1, let g: Y [right arrow] Z be any morphism of complex analytic spaces. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for every n. In particular, we have

(2.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for every n.

Remark 3. For the case of X being compact, the decomposition (2.1) of the cohomology groups is proved by S. Matsumura [2, Corollary 1.2].

doi: 10.3792/pjaa.92.84

By Taro FUJISAWA

Department of Mathematics, School of Engineering, Tokyo Denki University, 5 Senju-Asahi-cho, Adachi-ku, Tokyo 120-8551, Japan

(Communicated by Masaki KASHIWARA, M.J.A., June 13, 2016)

Acknowledgments. The author would like to thank Professor Osamu Fujino and Professor Shin-ichi Matsumura for their helpful discussion.

References

[1] J. Kollar, Higher direct images of dualizing sheaves. II, Ann. of Math. (2) 124 (1986), no. 1, 171-202.

[2] S. Matsumura, A vanishing theorem of Kollar-Ohsawa type, Math. Ann. (2016), DOI: 10.1007/s00208-016-1371-8.

[3] K. Takegoshi, Higher direct images of canonical sheaves tensorized with semi-positive vector bundles by proper Kahler morphisms, Math. Ann. 303 (1995), no. 3, 389-416.

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Author: | Fujisawa, Taro |
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Publication: | Japan Academy Proceedings Series A: Mathematical Sciences |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jul 1, 2016 |

Words: | 653 |

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