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Hodge metrics and positivity of direct images. (English) Zbl 1128.14030
The authors consider positivity properties of direct images of adjoint line bundles. Their main results are as follows: Let $$X$$ and $$Y$$ be complex manifolds and $$\phi:Y\rightarrow X$$ a submersion. Let $$L$$ be a semi-ample line bundle on Y and $$K_{Y/X}=K_Y\otimes(\phi^*K_X)^{-1}$$ the relative canonical line bundle on $$Y$$. If $$\phi_*(K_{Y/X}\otimes L)$$ is a holomorphic vector bundle (e. g. if $$\phi$$ is projective) then it admits a continuous metric with Griffiths semi-positive curvature. If $$X$$ and $$Y$$ are projective and $$L$$ is ample, then $$\phi_*(K_{Y/X}\otimes L)$$ admits a smooth metric with Griffiths positive curvature. A particular application of these results is related to the old problem whether ample vector bundles are Griffiths positive: If $$E$$ is an ample vector bundle on a complex projective manifold, then the vector bundles $$S^kE\otimes$$ det$$E$$ are Griffiths positive for all integers $$k\geq 1$$.
An analogous statement is true for a semi-ample bundle $$E$$ on a complex manifold and continuous metrics on $$S^k\oplus$$ det$$E$$ with Griffiths semi-positive curvature. For the proof of the theorems the authors compute Hodge metrics on the direct image of the relative canonical sheaf under proper Kähler submersions between complex manifolds. The main problem to be solved is to determine the singularities of those metrics and to remove them. As a main technical tool they use and generalize a method of T. Fujita, introduced in [J. Math. Soc. Japan 30, 779–794 (1978; Zbl 0393.14006)].

##### MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 32Q10 Positive curvature complex manifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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