Let f: X → Y be a surjective holomorphic map of compact complex manifolds and Δ the degeneracy locus of f. In this paper we shall discuss relationship between infinitesimal deformations of f and the corresponding infinitesimal displacements of Δ in Y. We shall prove that two kinds of Kodaira-Spencer maps are compatible under certain assumptions. As an application of our main theorem, deformations of quadric bundles shall be discussed.
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