|DATE||May 23 (Thu), 2019|
|INSTITUTE||Tata Institute of Fundamental Reseach|
|TITLE||The geometry of 2-adically uniformised fake projective planes II|
Fake projective planes are surfaces of general type with the same Betti numbers as the projective planes. The first such surface was constructed by Mumford using p-adic uniformisation with p=2. Subsequently, two other such surfaces were constructed by Ishida and Kato using groups discovered by Cartwright, Mantero, Steger and Zappa. A few years later all such surfaces over the complex numbers were classified by Prasad--Yeung and Cartwright--Steger. In these talks I will explain the construction method of Mumford and then discuss how the explicit geometry of the special fibre of 2-adically uniformised fake projective planes---already used in the case of Mumford's surface by Ishida for other purposes---can be used to study the geometry of these surfaces. In particular, I will discuss how one may compute the cohomology of line bundles on such fake projective planes and applications to exceptional collections and bicanonical maps.