Related papers: A Fake Projective Plane with an order 7 automorphi…
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…
A fake projective plane is a complex surface with the same Betti numbers as $\mathbb{C} P^2$ but not biholomorphic to it. We study the fake projective plane $\mathbb{P}_{\operatorname{fake}}^2 = (a = 7, p = 2, \emptyset, D_3 2_7)$ in the…
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…
Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to that of the usual projective plane. Recent explicit constructions of fake projective planes embed them via their bicanonical embedding in…
The addendum updates the results presented in the paper `Fake Projective Plane, Invent Math 168, 321-370 (2007)' and makes some additions and corrections. The fake projective planes are classified into twenty six classes. Together with a…
Recently, Prasad and Yeung classified all possible fundamental groups of fake projective planes. According to their result, many fake projective planes admit a nontrivial group of automorphisms, and in that case it is isomorphic to…
We find explicit equations of the fake projective plane $(a=7,p=2,\emptyset,D_3 X_7)$, which lies in the same class as the fake projective plane $(a=7,p=2,\emptyset,D_3 2_7)$ with $21$ automorphisms whose equations were previously found by…
We study Dolgachev elliptic surfaces with a double and a triple fiber and find explicit equations of two new pairs of fake projective plane with $21$ automorphisms, thus finishing the task of finding explicit equations of fake projective…
We study real rational models of the euclidean plane $\mathbb{R}^2$ up to isomorphisms and up to birational diffeomorphisms. The analogous study in the compact case, that is the classification of real rational models of the real projective…
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…
We show, for several fake projective planes with nontrivial automorphism group, that the bicanonical map is an embedding.
On the projective plane there is a unique cubic root of the canonical bundle and this root is acyclic. On fake projective planes such root exists and is unique if there are no 3-torsion divisors (and usually exists, but not unique,…
This paper is motivated by the real symplectic isotopy problem : does there exists a nonsingular real pseudoholomorphic curve not isotopic in the projective plane to any real algebraic curve of the same degree? Here, we focus our study on…
In Dubouloz and Mangolte (Fake real planes: exotic affine algebraic models of $\mathbb{R}^{2}$, arXiv:1507.01574, 2015) we define and partially classify fake real planes, that is, minimal complex surfaces with conjugation whose real locus…
We adapt the theory of non-Archimedean uniformization to construct a smooth surface from a lattice in PGL3(Q2) that has nontrivial torsion. It turns out to be a fake projective plane, commensurable with Mumford's fake plane yet distinct…
Fundamental groups of fake projective planes fall into fifty distinct isomorphism classes, one for each complex conjugate pair. We prove that this is not the case for their algebraic fundamental groups: there are only forty-six isomorphism…
This paper is an addition to the book [54] on Compact projective planes. Such planes, if connected and finite-dimensional, have a point space of topological dimension 2, 4, 8, or 16, the classical example in the last case being the…
The purpose of the article is to explain a new method to establish the existence of an exceptional collection of length three for a fake projective plane M with non-trivial automorphism group, related to a conjecture of…
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…
A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and…