A remark on the decomposition theorem for direct images of canonical sheaves tensorized with semipositive vector bundles. (English) Zbl 1368.14017

Fujisawa, Taro

Proc. Japan Acad., Ser. A 92, No. 7, 84-85 (2016).

J. Kollár’s decomposition theorem [Ann. Math. (2) 124, 171–202 (1986; Zbl 0605.14014)] is an important tool for studying the higher direct images of the canonical bundle of a projective manifold mapping onto a projective variety. In this short note the author gives a decomposition theorem for the twisted canonical bundle in a non-compact setting. More precisely, let \(X\) be a connected Kähler manifold, and let \(f: X \rightarrow Y\) be a proper surjective morphism onto a complex analytic space \(Y\). Let \(E\) be a Nakano semipositive vector bundle on \(X\), then we have an isomorphism \[ \bigoplus_q R^q f_* (\omega_X \otimes E) [-q] \simeq R f_* (\omega_X \otimes E) \] in the derived category of \(\mathcal O_Y\)-modules. As a corollary one obtains \[ \bigoplus_{p+q=n} H^p(Y, R^q f_* (\omega_X \otimes E)) \simeq H^n(X, \omega_X \otimes E). \]

Reviewer: Andreas Höring (Nice)

MSC:

14D06	Fibrations, degenerations in algebraic geometry
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
32J25	Transcendental methods of algebraic geometry (complex-analytic aspects)

Keywords:

decomposition theorem; higher direct images; semipositive vector bundles

Citations:

Zbl 0605.14014