Title:

# Monodromy Criterion for the Good Reduction of Surfaces

Hernandez Mada, Genaro (2015) Monodromy Criterion for the Good Reduction of Surfaces. PhD thesis, Concordia University.

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## Abstract

Let p>3 be a prime number and K a finite extension of Q_p. We consider a proper and smooth surface XK over K, with a semistable model X over the ring of integers OK of K. In this thesis, we give a criterion for the good reduction of XK for the case of K3 surfaces, in terms of the monodromy operator in the second De Rham cohomology group.
We don't use trascendental methods nor p-adic Hodge Theory as in other works concerning this problem. Instead, we first get a p-adic version of the Clemens-Schmid exact sequence and use it to study the degree of nilpotency of the monodromy operator N on the log-crystalline cohomology group of the special fiber Xs of the semistable model X.
By the work of Nakkajima, we can assume that Xs is a combinatorial K3 surface. Then, we prove that Xs is of type I iff N=0; Xs if of type II iff N is not zero and N^2=0; Xs is of type III iff N^2 is not zero. In particular, this implies that XK has good reduction if and only if the monodromy operator on its second De Rham cohomology is zero.
Finally, we also give some ideas on how to address the same problem for the case of Enriques surfaces. In particular, we prove that we are reduced to the case of K3 surfaces.

Divisions: Concordia University > Faculty of Arts and Science > Mathematics and Statistics Thesis (PhD) Hernandez Mada, Genaro Concordia University Ph. D. Mathematics 3 July 2015 Iovita, Adrian Monodromy, Good Reduction, K3 Surfaces 980454 GENARO HERNANDEZ MADA 28 Oct 2015 12:42 18 Jan 2018 17:51

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