Deformations of a holomorphic map and its degeneracy locus (Q1943233)

The author studies deformations of holomorphic maps. Given a deformation \(\Phi : \mathcal{X} \to Y \times M\) of a holomorphic map \(\Phi_0 : \mathcal{X}_0 \to Y\) between holomorphic manifolds, one can consider two Kodaira-Spencer maps. The first one has its image in the infinitesimal deformation space of \(\Phi_0 : \mathcal{X}_0 \to Y\) (here we allow both \(\mathcal{X}_0\) and the map to deform, however \(Y\) is kept fixed). The second one has its image in the infinitesimal deformation space of the degeneracy locus \(\Delta\). The author shows a compatibility between these two Kodaira-Spencer maps with the additional assumptions that \(f\) is surjective, \(\Delta\) is a smooth divisor and the ramification locus \(R\) maps isomorphically onto \(\Delta\). The method of the proof is by using the explicit descriptions of the above Kodaira-Spencer maps given by [\textit{E. Horikawa}, Math. Ann. 222, 275--282 (1976; Zbl 0334.32021)] and [\textit{K. Kodaira}, Ann. Math. (2) 75, 146--162 (1962; Zbl 0112.38404)]. The main application of the above deformation theoretic result is the rigidity of ordinary conic bundles \(f : X \to \mathbb{P}^m\) associated to vector bundles \(\mathcal{E}\) that are direct sums of line bundles. Here rigidity means that there are no deformations of \(f\) that fixes \(Y\) and the degeneracy locus \(\Delta\). This extends earlier results of the author on deformations of conic bundles.