WEEKLY ACTIVITIES

Time: 9:00 -- 10:00, January 10, 2024.

Venue: Room 612, A6, Institute of Mathematics, VAST

Abstract: A smooth compact complex surface with the same Betti numbers as the complex projective plane P^2 is either P^2 or is called a fake projective plane(FPP). Indeed, such a surface has Chern numbers c_2 =3, c_1^2 = 9, Picard number= 1; thus, its canonical class K is either ample or anti-ample, and in the latter case it is isomorphic to P^2. In other words, an FPP is a surface of general type with geometric genus 0 and K^2 = 9. Furthermore, it can be uniformized by the unit complex 2-ball, by Aubin and Yau, and its fundamental group is a co-compact arithmetic subgroup of PU(2, 1) by Klingler. The existence of such a surface was first proved by Mumford in 1979, via the 2-adic uniformization method. Algebraic varieties are not always described via polynomial equations: sometimes they are constructed via uniformization: this means, as quotients of bounded symmetric domains, via the action of discontinuous groups. General theorems (as Kodaira's) imply the algebraicity of these quotient complex manifolds. The problem concerning the algebro-geometrical properties of such varieties constructed via uniformization and especially the description of their projective embeddings (and the corresponding polynomial equations) lies at the crossroads of several allied fields: the theory of arithmetic groups and division algebras, complex algebraic and differential geometry, use of group symmetries, and topological and homological tools in the study of quotient spaces. Of particular importance are the so-called ball quotients, especially in dimension 2, since they yield the surfaces with the maximal possible canonical volume K^2 for a fixed value of the geometric genus p_g. In this talk I will report recent progress on FPPs, such as their derived categories, canonical maps and their equations.