Characterization of fiberwise bimeromorphism and specialization of bimeromorphic types I: locally Moishezon case

Inspired by the recent works of M. Kontsevich–Y. Tschinkel and J. Nicaise–J. C. Ottem on specialization of birational types for smooth families (in the scheme category) and J. Koll{á}r’s work on fiberwise bimeromorphism, we focus on characterizing the fiberwise bimeromorphism and utilizing the characterization to investigate the specialization of bimeromorphic types for non-smooth families in the complex analytic setting. We provide several criteria for a bimeromorphic map between two families over the same base to be fiberwise bimeromorphic. By combining these criteria with the relative Barlet cycle space theoretic argument motivated by D. Mumford–U. Persson, K. Timmerscheidt and T. de Fernex–D. Fusi, we establish the specialization of bimeromorphic types for locally Moishezon families with fibers having only canonical singularities and being of non-negative Kodaira dimension. These specialization results can easily lead to criteria for locally strongly bimeromorphic isotriviality. Throughout this paper, we unveil the connections among the four classical topics in bimeromorphic geometry: the deformation behavior of plurigenera (or even $1$-genus), fiberwise bimeromorphism, specialization of bimeromorphic types, and the bimeromorphic version of the deformation rigidity.