CRM: Centro De Giorgi
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Dynamical systems and systems of equations

Equivariant Kähler model for Fujiki's class

speaker: Sheng Meng (Korea Institute for Advanced Study)

abstract: Let $X$ be a compact complex manifold in Fujiki's class $\mathcal{C}$,i.e., admitting a big $(1,1)$-class $\alpha$. Consider $Aut(X)$ the group of biholomorphic automorphisms and $Aut{\alpha}(X)$ the subgroup of automorphisms preserving the class $\alpha$ via pullback. We show that $X$ admits an $Aut{\alpha}(X)$-equivariant K\"{a}hler model: there is a bimeromorphic holomorphic map $\sigma \colon \widetilde{X}\to X$ from a K\"{a}hler manifold $\widetilde{X}$ such that $Aut{\alpha}(X)$ lifts holomorphically via $\sigma$. There are several applications. We show that $Aut{\alpha}(X)$ is a Lie group with only finitely many components. This generalizes an early result of Fujiki and Lieberman on the K\"{a}hler case.We also show that every torsion subgroup of $Aut(X)$ is almost abelian, and $Aut(X)$ is finite if it is a torsion group. This is a joint work with Jia Jia.

Thu 23 Jun, 10:20 - 11:20, Aula Dini
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