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Algebraic reverse Khovanskii–Teissier inequality via Okounkov bodies

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Abstract

Let X be a projective variety of dimension n over an algebraically closed field of arbitrary characteristic and let A, B, C be nef divisors on X. We show that for any integer \(1\le k\le n-1\),

$$\begin{aligned} (B^k\cdot A^{n-k})\cdot (A^k\cdot C^{n-k})\ge \frac{k!(n-k)!}{n!}(A^n)\cdot (B^k\cdot C^{n-k}). \end{aligned}$$

The same inequality in the analytic setting was obtained by Lehmann and Xiao for compact Kähler manifolds using the Calabi–Yau theorem, while our approach is purely algebraic using (multipoint) Okounkov bodies. We also discuss applications of this inequality to Bézout-type inequalities and inequalities on degrees of dominant rational self-maps.

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Acknowledgements

We are grateful to Jian Xiao for drawing our attention to this topic and many helpful comments. We would like to thank Mingchen Xia for discussions on Okounkov bodies. We would like to thank the referee for useful suggestions. The authors were supported by NSFC for Innovative Research Groups (Grant No. 12121001) and partially supported by National Key Research and Development Program of China (Grant No. 2020YFA0713200). The second author was also supported by Shanghai Pilot Program for Basic Research (No. 21TQ00). The authors are members of LMNS, Fudan University.

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  1. Shanghai Center for Mathematical Sciences, Fudan University, Jiangwan Campus, Shanghai, 200438, China

    Chen Jiang & Zhiyuan Li

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Jiang, C., Li, Z. Algebraic reverse Khovanskii–Teissier inequality via Okounkov bodies. Math. Z. 305, 26 (2023). https://doi.org/10.1007/s00209-023-03349-9

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