Let $f: X \to Y$ be a surjective holomorphic map of compact complex manifolds and $\Delta$ the degeneracy locus of $f$. In this paper we shall discuss relationship between infinitesimal deformations of $f$ and the corresponding infinitesimal displacements of $\Delta$ 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.
"Deformations of a Holomorphic Map and Its Degeneracy Locus." Tokyo J. Math. 35 (2) 253 - 277, December 2012. https://doi.org/10.3836/tjm/1358951317