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On the bicanonical map of primitive varieties with \(q(X) = \mathrm{dim }X\): the degree and the Euler number

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Abstract

Let \(X\) be a variety of maximal Albanese dimension and of general type. Assume that \(q(X) = \mathrm{dim }X\), the Albanese variety \(\mathrm {Alb} (X)\) is a simple abelian variety, and the bicanonical map is not birational. We prove that the Euler number \(\chi (X, \omega _X)\) is equal to 1, and \(|2K_X|\) separates two distinct points over the same general point on \(\mathrm {Alb} (X)\) via \(\mathrm {alb}_X\) (Theorem 1.1).

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Acknowledgments

Part of this note appears in the author’s doctoral thesis submitted to Peking University (2011). The author expresses appreciations to Prof. Jinxing Cai and Dr. Wenfei Liu for many useful discussions. He thanks Prof. Meng Chen, Dr. Fan Peng and Ze Xu for their help on the inequality appearing in the appendix, and thanks Olivier Debarre and Yifei Chen for their suggestions in improving the English. He also thanks Sofia Tirabassi for some suggestions, and thanks the authors of [1] for their stimulating ideas. Finally, the author owes too much to an anonymous referee, who shares his or her ideas on improving the result of Theorem 3.1 and simplifying the proof of Corollary 2.5 and Theorem 3.1. The author is supported by NSFC (No. 11226075).

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  1. College of Mathematics and Information Sciences, Shaanxi Normal University, Xi’an, 710062, People’s Republic of China

    Lei Zhang

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Zhang, L. On the bicanonical map of primitive varieties with \(q(X) = \mathrm{dim }X\): the degree and the Euler number. Math. Z. 277, 575–590 (2014). https://doi.org/10.1007/s00209-013-1266-2

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