Communications in Analysis and Geometry

Volume 20 (2012)

Number 4

On an efficient induction step with Nklt(X,D) — notes to Todorov

Pages: 765 – 779

DOI: http://dx.doi.org/10.4310/CAG.2012.v20.n4.a3

Author

Meng Chen (Institute of Mathematics & LMNS, Fudan University, Shanghai, China)

Abstract

Applying the effective induction on $\text{Nklt}(X,D)$ developed by Hacon–McKernan and Takayama, Todorov provedthat $\varphi_5$ is birational for projective 3-fold $V$ with $\vol(V)\gg 0$, which was recently improved byBiagio in loosing the volume constraint. The observation is that the least efficient induction step can be studiedin an alternative way, which allows us to assert Todorov’s statement for $\vol(V)>12^3$. The four-dimensional analogis also given in this note. The idea works well for all dimensions.

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