On Automorphisms of A Kählerian Structure

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Abstract
Is every isometry, or more generally, every affine transformation of a Kählerian manifold a complex analytic transformation? The answer is certainly negative in the case of a complex Euclidean space. This question has been recently studied by Lichnerowicz [8] and Schouten-Yano [11] from the infinitesimal point of view; they have found some conditions in order that every infinitesimal motion of a Kählerian manifold preserve the complex structure. (As a matter of fact, [11] has dealt with the case of a pseudo-Kählerian manifold, which does not differ essentially from a Kählerian manifold as far as the question at hand is concerned.)In the present paper, we generalize their results by a different approach. In order to explain our main idea, we shall first give a few definitions (1 and 2) and state our main results (3). The proofs are given in the subsequent sections.

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