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arXiv · 2504.06654

Finiteness of projective pluricanonical representation for automorphisms of complex manifolds

Konstantin Loginov·Constantin Shramov
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Abstract

We study the action of the group of bimeromorphic automorphisms $\mathrm{Bim}(X)$ of a compact complex manifold $X$ on the image of the pluricanonical map, which we call the projective pluricanonical representation of this group. If $X$ is a Moishezon variety, then the image of $\mathrm{Bim}(X)$ via such a representation is a finite group by a classical result due to Deligne and Ueno. We prove that this image is a finite group under the assumption that for the Kodaira dimension $κ(X)$ of $X$ we have $κ(X)=\dim X-1$. To this aim, we prove a version of the canonical bundle formula in relative dimension $1$ which works for a proper morphism from a complex variety to a projective variety. In particular, this establishes the analytic version of Prokhorov--Shokurov conjecture in relative dimension $1$. Also, we observe that the analytic version of this conjecture does not hold in relative dimension $2$.

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Konstantin Loginov, Constantin Shramov. 2025-04-09. Finiteness of projective pluricanonical representation for automorphisms of complex manifolds. https://arxiv.org/abs/2504.06654

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