On the pluricanonical map and the canonical volume of projective 4-folds of general type

Abstract

For nonsingular projective 4-folds V of general type with plurigenus $P_{m_0}(V)\ge 2$ for some positive integer m₀, we show that ${\phi }_m$ is birational onto its image for all integers $m\ge 76m_0+77$ and the canonical volume $\text{ Vol}(V)$ has the lower bound $\frac{1}{1680m_0{(m_0+1)}^3}$. This improves earlier results of Meng Chen.

KEYWORDS:

• 4-folds of general type
• canonical volume
• pluricanonical maps

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 14J35
• 14E05
• 14C20

Acknowledgments

The author would like to express great gratitude to Professor Meng Chen for his suggestion of this problem and his guidance over this paper. The author is also obliged to Yong Hu for the enlightening discussions of the problem and his help in making the paper much more readable. The author would also like to thank the referee for valuable suggestions and comments.

Additional information

Funding

The work is supported by the Fundamental Research Funds for the Central Universities (Grant No. N2205012).