Non-Collapsible Dual Complexes and Fake del Pezzo Surfaces

Abstract

We propose the new construction of complex surfaces with h 1,0 = h 2,0 = 0 from smoothings of normal crossing surfaces with noncollapsible dual complexes and carry it out for the simplest case of the duncehat complex, obtaining the surface with h 1,1 = 9 (presumably Barlow surface).

A Surface Degeneration With Non-Collapsible Contractible Dual Complex

“…This paper is a cleaned up and improved version of author's preprint [14]. It features more straightforward version of the main theorem and various improvements and clarifications.…”

“…The general fiber of this family has h 1,0 = h 2,0 = 0 and h 1,1 = 9, we conjecture that it is in the deformation class of the Barlow surface. This text is a complete rewrite of author's previous work [14].…”