Exceptional collections and the bicanonical map of Keum's fake projective planes

Abstract

We apply the recent results of Galkin et al. [GKMS15] to study some geometrical features of Keum's fake projective planes. Among other things, we show that the bicanonical map of Keum's fake projective planes is always an embedding. Moreover, we construct a nonstandard exceptional collection on the unique fake projective plane $X$ with $H_1(X;\mathbb{Z})=(\mathbb{Z}/2\mathbb{Z}{)}^4$.

arXiv commentSection 3 is expanded. Added acknowledgements. To appear in Commun. Contemp. Math