The Kodaira dimension of complex hyperbolic manifolds with cusps

We prove a bound relating the volume of a curve near a cusp in a complex ball quotient$X=\mathbb{B}/\unicode[STIX]{x1D6E4}$to its multiplicity at the cusp. There are a number of consequences: we show that for an$n$-dimensional toroidal compactification$\overline{X}$with boundary$D$,$K_{\overline{X}}+(1-\unicode[STIX]{x1D706})D$is ample for$\unicode[STIX]{x1D706}\in (0,(n+1)/2\unicode[STIX]{x1D70B})$, and in particular that$K_{\overline{X}}$is ample for$n\geqslant 6$. By an independent algebraic argument, we prove that every ball quotient of dimension$n\geqslant 4$is of general type, and conclude that the phenomenon famously exhibited by Hirzebruch in dimension 2 does not occur in higher dimensions. Finally, we investigate the applications to the problem of bounding the number of cusps and to the Green–Griffiths conjecture.

Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)

This book introduces the theory of complex surfaces through a comprehensive look at finite covers of the projective plane branched along line arrangements. It emphasizes those finite coverings that are free quotients of the complex 2-ball. The book also includes a background on the classical Gauss hypergeometric function of one variable, and a chapter on the Appell two-variable F1 hypergeometric function. The book began as a set of lecture notes, taken by the author, of a course given by Friedrich Hirzebruch at ETH Zürich in 1996. The lecture notes were then considerably expanded over a number of years. In this book, the author has expanded those notes even further, still stressing examples offered by finite covers of line arrangements. The book is largely self-contained and foundational material is introduced and explained as needed, but not treated in full detail. References to omitted material are provided for interested readers. Aimed at graduate students and researchers, this is an accessible account of a highly informative area of complex geometry.

Decomposition of the Laplacian and pluriharmonic Bloch functions

We decompose the invariant Laplacian of the deleted unit complex ball by two directional Laplacians, tangential one and radial one. We give a characterization of pluriharmonic Bloch function in terms of the growth of these Laplacians.