Communications in Analysis and Geometry

Volume 29 (2021)

Number 7

On an open problem of characterizing the birationality of $4K$

Pages: 1545 – 1557



Meng Chen (School of Mathematical Sciences & Shanghai Centre for Mathematical Sciences, Fudan University, Shanghai, China)

Yong Hu (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China)


We answer an open problem raised by Chen–Zhang in 2008 and prove that, for any minimal projective $3$-fold $X$ of general type with the geometric genus $p_g (X) \geq 5$, the $4$-canonical map $\varphi_{4,X}$ is non-birational if and only if $X$ is birationally equivalent to a fibration, onto a curve, of which the general fiber is a minimal surface of general type with $(c^2_1, p_g) = (1, 2)$. The statement does not hold for those with the geometric genus $p_g (X) \leq 4$ according to our examples.

The first author was supported by National Natural Science Foundation of China (#12071078, #11571076, #11731004, #11421061) and Program of Shanghai Subject Chief Scientist (#16XD1400400).

The second author is supported by National Researcher Program of National Research Foundation of Korea (Grant No. 2010- 0020413) and the Shanghai Pujiang Program Grant No. 21PJ1405200.

Received 17 June 2018

Accepted 29 April 2019

Published 17 May 2022