On the plurisubharmonicity of the leafwise Poincaré metric on projective manifolds
A \textit{skew cylinder} (in Yu. S. Ilyashenko's terminology, see [\textit{Yu. S. Il'yashenko} and \textit{A. A. Shcherbakov}, Tr. Mat. Inst. Steklova. 213, 112--122 (1997; Zbl 0911.32033)]) is a complex manifold holomorphically fibered over another one by simply connected holomorphic curves that admits a holomorphic section. The paper under review studies an important class of skew cylinders that are related to foliations with singularities (of complex codimension at least 2) by analytic curves on complex projective manifolds. To each (topologically trivial) cross-section \(S\) of a foliation one associates the \textit{universal covering manifold} \(U_S\) [see \textit{Y. S. Ilyashenko}, Topol. Methods Nonlinear Anal. 11, No.~2, 361--373 (1998); remark ibid. 23, No.~2, 377--381 (2004; Zbl 0927.32020), Math. USSR, Sb. 17(1972), 551--569 (1973); translation from Mat. Sb., N. Ser. 88(130), 558--577 (1972; Zbl 0243.57011)]. This is a union of a family of simply connected Riemann surfaces (with marked points) that is parametrized by points of \(S\): for each \(s\in S\) the corresponding surface is the universal covering over the leaf passing through \(s\) with the marked point \(s\). It is known (see [\textit{M. Brunella}, On entire curves tangent to a foliation, preprint (2004)]) that in almost all the cases the space \(U_S\) admits a natural structure of complex manifold that is a skew cylinder fibered over \(S\). A similar result in the case, when the foliated manifold is Stein was proved by Yu. S. Ilyashenko (see his two latter loc. cit.), who had also shown that in this case the universal covering manifold is Stein. There is an important open question: is it true that the universal covering manifold \(U_S\) is always \textit{Stein} (whenever it is a manifold and the foliated manifold is compact projective)? The paper under review provides a proof of a metric version of this statement. It is known [\textit{H. Yamaguchi}, Jap. J. Math. 7, 319--377 (1981; Zbl 0485.32010)] that if a skew cylinder is Stein, then the Poincaré metrics on the fibers have plurisubharmonic variation, i.e., the Laplacian of the \(\log\) of the metric coefficient (in a holomorphic chart) taken along any holomorphic curve is nonnegative. A very recent result [\textit{F. Maitani} and \textit{H. Yamaguchi}, Math. Ann. 330, No.~3, 477--489 (2004; Zbl 1077.32006)] provides a similar statement in the case, when the fibers are not simply connected. It is formulated for the \textit{Bergman metric} of the fibers, which generalizes the Poincaré metric. In his previous papers [\textit{M. Brunella}, Invent. Math. 152, No.~1, 119--148 (2003; Zbl 1029.32014), Int. J. Math. 14, No.~2, 139--151 (2003; Zbl 1052.32027), On entire curves tangent to a foliation, preprint (2004)] the author of the paper under review proved that the fiberwise Poincaré metric of \(U_S\) always has a plurisubharmonic variation. In the present paper he gives a short and nice proof of the same result and some its extensions. (By definition, the Poincaré metric of a non-hyperbolic Riemann surface is zero.) To do this, he throws away from \(U_S\) an appropriate hypersurface and considers its complement \(U''_S\). The Bergman metric on the fibers of \(U_S''\) coincides with the restriction of the fiberwise Poincaré metric on \(U_S\) (the \(L_2\)-singularity erasing principle for holomorphic functions). The key argument says that the (smaller) manifold \(U_S''\) is ``almost'' Stein. This appears to be sufficient to conclude (by the previously mentioned Maitani-Yamaguchi's result, see the corresponding loc. cit. above) that the Bergman metric of the fibers of \(U''_S\) has a plurisubharmonic variation. This implies the similar result for the fiberwise Poincaré metric of \(U_S\). It seems that the arguments of this new proof could have important further applications.
scientific article; zbMATH DE number 5014811
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