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When is the Albanese morphism an algebraic fiber space in positive characteristic?

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Abstract

In this paper, we study the Albanese morphisms in positive characteristic. We prove that the Albanese morphism of a variety with nef anti-canonical divisor is an algebraic fiber space, under the assumption that the general fiber is F-pure. Furthermore, we consider a notion of F-splitting for morphisms, and investigate it in the case of Albanese morphisms. We show that an F-split variety has F-split Albanese morphism, and that the F-split Albanese morphism is an algebraic fiber space. As an application, we provide a new characterization of abelian varieties.

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Acknowledgments

The author wishes to express his gratitude to his supervisor Professor Shunsuke Takagi for suggesting problems, valuable comments and helpful advice. He is deeply grateful to Professors Zsolt Patakfalvi and Yoshinori Gongyo for fruitful discussions and valuable comments. He would like to thank Professors Osamu Fujino, Nobuo Hara and Doctor Yuan Wang for stimulating discussions, questions and comments. He also would like to thank the reviewer for a careful reading and helpful suggestions. Part of this work was carried out during his visit to Princeton University with support from The University of Tokyo/Princeton University Strategic Partnership Teaching and Research Collaboration Grant, and from the Program for Leading Graduate Schools, MEXT, Japan. He was also supported by JSPS KAKENHI Grant Number 15J09117.

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Ejiri, S. When is the Albanese morphism an algebraic fiber space in positive characteristic?. manuscripta math. 160, 239–264 (2019). https://doi.org/10.1007/s00229-018-1056-6

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  • DOI: https://doi.org/10.1007/s00229-018-1056-6

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