标题： 纤维双全纯映射的特征及双全纯类型的特殊化 I：非负 Kodaira 维数情形 显示英文标题

标题： Characterization of fiberwise bimeromorphism and specialization of bimeromorphic types I: the non-negative Kodaira dimension case

摘要： 受M. Kontsevich–Y. Tschinkel以及J. Nicaise–J. C. Ottem关于平滑族（在概型范畴内）的双有理类型特殊化的近期工作，以及J. Kollár关于纤维双比密态的工作的启发，我们专注于刻画纤维双比密态，并利用这一刻画来研究复解析情形下非平滑族的双有理类型特殊化问题。我们给出了一些关于两个具有相同底空间的族之间的双比密映射为纤维双比密映射的判别准则。 通过结合这些判别准则与D. Mumford–U. Persson、T. de Fernex–D. Fusi的想法，以及K. Timmerscheidt借助相对Barlet循环空间理论的方法，我们建立了局部Moishezon族的双有理类型特殊化结果，其中纤维具有仅可接受奇点且非负Kodaira维数。这些特殊化结果可以轻松导出局部强双比密平凡性的判别准则。 本文始终揭示了双有理几何中的四个经典主题之间的联系：多重亏格（甚至$1$-亏格）的变形行为、纤维双比密态、双有理类型的特殊化以及变形刚性的双有理版本。 显示英文摘要

摘要： 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{\'a}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 some criteria for a bimeromorphic map between two families over the same base to be fiberwise bimeromorphic. By combining these criteria with ideas by D. Mumford--U. Persson and T. de Fernex--D. Fusi, as well as K. Timmerscheidt's approach via the relative Barlet cycle space theory, 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.