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- Title
On effective existence of symmetric differentials of complex hyperbolic space forms.
- Authors
Wong, Kwok-Kin
- Abstract
For a noncompact complex hyperbolic space form of finite volume X=Bn/Γ<inline-graphic></inline-graphic>, we consider the problem of producing symmetric differentials vanishing at infinity on the Mumford compactification X¯<inline-graphic></inline-graphic> of X similar to the case of producing cusp forms on hyperbolic Riemann surfaces. We introduce a natural geometric measurement which measures the size of the infinity X¯-X<inline-graphic></inline-graphic> called canonical radius of a cusp of Γ<inline-graphic></inline-graphic>. The main result in the article is that there is a constant r∗=r∗(n)<inline-graphic></inline-graphic> depending only on the dimension, so that if the canonical radii of all cusps of Γ<inline-graphic></inline-graphic> are larger than r∗<inline-graphic></inline-graphic>, then there exist symmetric differentials of X¯<inline-graphic></inline-graphic> vanishing at infinity. As a corollary, we show that the cotangent bundle TX¯<inline-graphic></inline-graphic> is ample modulo the infinity if moreover the injectivity radius in the interior of X¯<inline-graphic></inline-graphic> is larger than some constant d∗=d∗(n)<inline-graphic></inline-graphic> which depends only on the dimension.
- Publication
Mathematische Zeitschrift, 2018, Vol 290, Issue 3/4, p711
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-017-2038-1