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A Bochner principle and its applications to Fujiki class $mathcal C$ manifolds with vanishing first Chern class | | Indranil Biswas
; Sorin Dumitrescu
; Henri Guenancia
; | | Date: |
9 Jan 2019 | | Abstract: | We prove a Bochner type vanishing theorem for compact complex manifolds $Y$
in Fujiki class $mathcal C$, with vanishing first Chern class, that admit a
cohomology class $[alpha] in H^{1,1}(Y,mathbb R)$ which is numerically
effective (nef) and has positive self-intersection (meaning $int_Y alpha^n
,>, 0$, where $n,=,dim_{mathbb C} Y$). Using it, we prove that all
holomorphic geometric structures of affine type on such a manifold $Y$ are
locally homogeneous on a non-empty Zariski open subset. Consequently, if the
geometric structure is rigid in the sense of Gromov, then the fundamental group
of $Y$ must be infinite. In the particular case where the geometric structure
is a holomorphic Riemannian metric, we show that the manifold $Y$ admits a
finite unramified cover by a complex torus with the property that the pulled
back holomorphic Riemannian metric on the torus is translation invariant. | | Source: | arXiv, 1901.2656 | | Services: | Forum | Review | PDF | Favorites | |
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