Finiteness of projective pluricanonical representation for automorphisms of complex manifolds
Konstantin Loginov, Constantin Shramov
TL;DR
This paper establishes finiteness of the projective pluricanonical representation $\overline{\rho}(\mathrm{Bim}(X))$ for a compact complex manifold $X$ in the case $\kappa(X)=\dim X-1$, extending known Moishezon results. The authors develop a canonical bundle formula in relative dimension $1$ and define invariant discriminant and moduli divisors to control the action of bimeromorphic automorphisms on the pluricanonical image. By constructing $\mathrm{Bim}(X)$-equivariant bimeromorphic modifications and analyzing elliptic fibrations, they reduce the problem to finiteness of automorphism groups of pairs with big and nef canonical data. The work also clarifies the analytic Prokhorov–Shokurov conjecture in low relative dimension and demonstrates its failure in dimension two, contributing both new methods and deeper insight into the structure of automorphism groups in complex geometry.
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 $2$.