Finiteness of projective pluricanonical representation for automorphisms of complex manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916681077489664 |
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| author | Loginov, Konstantin Shramov, Constantin |
| author_facet | Loginov, Konstantin Shramov, Constantin |
| contents | 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 $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06654 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finiteness of projective pluricanonical representation for automorphisms of complex manifolds Loginov, Konstantin Shramov, Constantin Algebraic Geometry 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 $2$. |
| title | Finiteness of projective pluricanonical representation for automorphisms of complex manifolds |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2504.06654 |