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Deformations of complex hypersurfaces on $$G/P$$ and an infinitesimal Torelli theorem. (English) Zbl 1023.14024
From the summary: This paper addresses the problem whether the family of complex hypersurfaces of a given multidegree on a generalized flag manifold $$(G/P)$$ is a local complete deformation or not. An explicit solution to this problem is given for the classical groups, the special groups are considered independently. The question was previously considered by J. Wehler [Math. Z. 198, 21-38 (1988; Zbl 0662.14029)] for hypersurfaces in a manifold of complete flags (that is $$G= {\mathbf S}L_n$$ and $$P=B$$ a Borel subgroup of $$G)$$. We will actually follow the lines of Wehler’s paper and this reduces the question to the vanishing of the group $$H^2(G/P,\tau (-d))$$. By the same token one can give conditions under which the infinitesimal Torelli theorem holds for every smooth hypersurface of given multidegree of a generalized flag manifold. Here we use a criterion developed by H. Flenner [Math. Z. 193, 307-322 (1986; Zbl 0613.14010)] to reduce the question to the vanishing of certain cohomology groups and the surjectivity of certain maps.
This article is divided in two parts, part 1 (especially sections 2-4) considers only the case of classical groups, while the special groups (except for $$G_2$$, where our method only work for the case of the Borel subgroup) are treated in part two (sections 5 and 6).
##### MSC:
 14M17 Homogeneous spaces and generalizations 14J70 Hypersurfaces and algebraic geometry 14D15 Formal methods and deformations in algebraic geometry 14C25 Algebraic cycles