Total geodesy of proper holomorphic immersions between complex hyperbolic space forms of finite volume
We prove the following Theorem: Let \(X\) and \(Y\) be complex hyperbolic space forms of complex dimensions \(n\) and \(m\) respectively. Assume that both \(X\) and \(Y\) are of finite volume with respect to their Poincaré metrics and \(m \leq 2n-1\). Suppose \(f:X \to Y\) is a proper holomorphic immersion. Then up to a multiplicative constant, \(f\) is an isometric totally geodesic immersion with respect to their Poincaré metrics. The above theorem generalizes a result of \textit{H.-D. Cao} and \textit{N. Mok} [Invent. Math. 100, No. 1, 49-62 (1990; Zbl 0698.53035)], who proved the above theorem in the case when \(X\) is compact. The main difficulty in proving the above theorem in the case of quotients of finite volume is to justify the vanishing of certain integral over the domain manifold whose integrand is an exact \((n,n)\)-form related to the second fundamental form associated to the holomorphic immersion. We overcome this difficulty by first extending the proper holomorphic immersion to the compactifications of \(X\) and \(Y\). It is known that \(X\) (resp. \(Y)\) admits smooth toroidal compactifications \(\overline X\) (resp. \(\overline Y)\) such that \(\overline X-X\) (resp. \(\overline Y-Y)\) consists of disjoint union of abelian varieties of codimension one in \(\overline X\) (resp. \(\overline Y)\). The important observation is that \(f\) extends to a holomorphic map \(\tilde f:\overline X \to \overline Y\) without `blowing down' the compactifying divisors. The jacobian of \(\tilde f\) induces a `bundle immersion' between certain vector bundle extensions \(\overline{TX}\) of \(TX\) to \(\overline X\) and \(\overline{TY}\) of \(TY\) to \(\overline Y\). Then we show that the pull-back of the Poincaré metric under \(f\) is a Hermitian metric `good' on \(\overline X\) with respect to \(\tilde f^* \overline{TY}\) in the sense of \textit{D. Mumford} [Invent. Math. 42, 239- 272 (1977; Zbl 0365.14012)], and its restriction on \(TX\) is good on \(\overline X\) with respect to \(\overline{TX}\). A study of the Chern classes of the quotient bundle \(\tilde f^* \overline{TY}/ \overline{TX}\) over \(\overline X\) (instead of \(f^* TY/TX\) over \(X\) as in the compact case) allows us to justify the necessary integral formula over \(X\).
scientific article; zbMATH DE number 537199
scientific article