Arithmetic fake projective spaces and arithmetic fake grassmannians

Arithmetic of a fake projective plane and related elliptic surfaces

The purpose of the present paper is to explain the fake projective plane constructed by J.H. Keum from the point of view of arithmetic ball quotients. Beside the ball quotient associated with the fake projective plane, we also analize two further naturally related ball quotients whose minimal desingularizations lead to two elliptic surfaces, one already considered by J.H. Keum as well as the one constructed by M.N. Ishida in terms of p-adic uniformization.

高度对称的Calabi-Yau Grassmannian超曲面的形变

Deformations of highly symmetric Calabi-Yau Grassmannian hypersurfaces

本文运用算术与Hodge理论技术研究Grassmannians及其商空间中高度对称超曲面实现的Calabi-Yau簇的铅笔，证明其几何性质不同于经典镜像。

Arithmetic structure of Mumford's fake projective plane

This is a revised version of a part of the author's preprint "On p-adic uniformization of fake projective planes" (preprint, Max-Planck-Institut fuer Mathematik, 1998 (121)). In this paper we construct explicitly a Shimura surface of PEL-type, associated to a certain unitary group, whose connected components are isomorphic to Mumford's fake projective plane. We also give the canonical model, Shimura field and number of connected components of this Shimura surface. As a consequence of the construction we prove that the Mumford's fake projective plane is an arithmetic complex unit-ball quotient,…

Fake projective planes

A fake projective plane is a smooth complex surface which is not the complex projective plane but has the same Betti numbers as the complex projective plane. The first example of such a surface was constructed by David Mumford in 1979 using p-adic uniformization. Two more examples were found by Ishida and Kato by related method. Keum has recently given an example which is possibly different from the three known earlier. It is an interesting problem in complex algebraic geometry to determine all fake projective planes. Using the arithmeticity of the fundamental group of fake projective planes, …

Commensurability Classes of Fake Quadrics

A fake quadric is a smooth projective surface that has the same rational cohomology as a smooth quadric surface but is not biholomorphic to one. We provide an explicit classification of all irreducible fake quadrics according to the commensurability class of their fundamental group. To accomplish this task, we develop a number of new techniques that explicitly bound the arithmetic invariants of a fake quadric and more generally of an arithmetic manifold of bounded volume arising from a form of SL_2 over a number field.

Explicit equations of a fake projective plane

Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the two-dimensional ball by explicitly written arithmetic subgroups. In this paper we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order seven automorphism.

Moduli of double EPW-sextics

We study the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of \bigwedge^3{\mathbb C}^6 by the natural action of SL_6, call it M. This is a compactification of the moduli space of smooth double EPW-sextics. We determine the stable points, the irreducible components of the GIT boundary and their dimensions. There are strong analogies with the moduli space of cubic 4-folds: we prove a result which is analogous to a theorem of Laza stating that cubic 4-folds with simple singularities are stable. Our final goal (not achieved in the present paper) is to understand th…