Mathematical Research Letters

Volume 22 (2015)

Number 6

Collapsing of negative Kähler–Einstein metrics

Pages: 1843 – 1869

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n6.a16

Author

Yuguang Zhang (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Abstract

In this paper, we study the collapsing behaviour of negative Kähler–Einstein metrics along degenerations of canonical polarized manifolds. We prove that for a toroidal degeneration of canonical polarized manifolds with the total space $\mathbb{Q}$-factorial, the Kähler–Einstein metrics on fibers collapse to a lower dimensional complete Riemannian manifold in the pointed Gromov–Hausdorff sense by suitably choosing the base points. Furthermore, the most collapsed limit is a real affine Kähler manifold.

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