Enriques manifolds
Final Report Abstract
Algebraic geometry is the part of pure mathematics that deals with geometric object that – at least in principal – may be described via algebraic equations fi (x1 , . . . , xn ) = 0, 1 ≤ i ≤ m. Which concrete form the equations take is usual of secondary relevance. Highly important, however, is the domain of numbers from which the coefficients of the equations may be choosen. This is already intersting for fields. A very deep open question is whether one even may choose the ring of integers Z = {. . . , −2, −1, 0, 1, 2, . . .}. Such a situation describes geometric objects that admit incarnations over each field, in a continuous way. This seems to be highly exceptional, although it is difficult to understand why such a principal should hold. By deep results of Abrashkin and Fontaine there are no families of abelian varieties over the ring R = Z. This also precludes the existence of other families, for examples smooth curves C of genus g ≥ 1. In light of the classification of algebraic surfaces, it is natural to ask which kind of families of surfaces exist over R = Z. In this project we showed that for the so-called Enriques surfaces this is impossible. For this we developed several methods and results that provide insight to geometry in positive or mixed characteristic.
Publications
- 2017: Enriques surfaces with normal K3-like coverings. J. Math. Soc. Japan
Schröer
(See online at https://doi.org/10.2969/jmsj/83728372) - 2018: On equivariant formal deformation theory. Rend. Circ. Mat. Palermo. 26, 1113–1122
Schröer and Takayama
(See online at https://doi.org/10.1007/s12215-017-0322-x) - 2019: Numerically trivial dualizing sheaves. Dissertation. Heinrich–Heine-Universität, Düsseldorf
Schell
- 2019: The torsion structure of wild elliptic fibers. Dissertation. Heinrich–Heine-Universität, Düsseldorf
Zimmermann
- 2021: The torsion in the cohomology of wild elliptic fibers. J. Pure Appl. Algebra 225
Zimmermann
(See online at https://doi.org/10.1016/j.jpaa.2020.106522)
