Ramification in the cohomology of algebraic surfaces arising from ordinary double point singularities
A classical theorem in arithmetic geometry due to Serre-Tate (first proved by Néron-Ogg-Shafarevich in the curve case) states that an abelian variety has good reduction if and only if the action of the Inertia group on the first étale \(\ell\)-adic cohomology is trivial. One implication holding for all varieties alike, the converse is in general not true as can be seen already from curves of genus greater than one. There are, however, partial results, for instance for \(K3\) surfaces [\textit{C. Liedtke} and \textit{Y. Matsumoto}, Compos. Math. 154, No. 1, 1--35 (2018; Zbl 1386.14138); \textit{B. Chiarellotto} et al., Proc. Lond. Math. Soc. (3) 119, No. 2, 469--514 (2019; Zbl 1427.14075)] and for semistable surfaces in the realm of the weight-monodromy conjecture [\textit{M. Rapoport} and \textit{Th. Zink}, Invent. Math. 68, 21--101 (1982; Zbl 0498.14010)]. The present paper combines ideas from the previous two approaches to treat the following special situation: Let \(K\) be a complete field with respect to a discrete valuation such that the residue field \(k\) is algebraically closed of characteristric \(p>2\). Then there is a unique ramified quadratic extension \(L/K\); it corresponds to a unique quadratic character \(\psi\). Assume given a smooth surface \(X\) over \(K\) which possesses an integral model \(\mathcal X\) over the valuation ring \(R\) whose special fibre \(\mathcal X_k\) has at worst ordinary double points as singularities (the mildest singularity on a surface, given locally by \(xy=z^2\)). At a singular point \(P\in \mathcal X_k\), a formal neighbourhood in \(\mathcal X\) is isomorphic to \[ R[[x,y,z]/(z^2+xy+r) \] for some \(r\in \mathfrak p\), the maximal ideal of \(R\). Fix the \(\mathfrak p\)-adic order \(n_P\) of \(r\), i.e. \[ r\in \mathfrak p^{n_P}\setminus \mathfrak p^{n_P+1}. \] (There also is an intrinsic definition, so this is well-defined.) Let \(g\) be the number of singular points \(P\) such that \(n_P\) is odd: \[ g = \#\{P\in \mathcal X_k; 2\nmid n_P\}. \] Then the author proves the following result, assuming that \(p\nmid \ell\): Theorem.\(H^2(X_{\bar K}, \mathbb Q_\ell)^{I_K}\) has codimension \(g\) inside \(H^2(X_{\bar K}, \mathbb Q_\ell)\). Moreover, the quotient is \(\psi\)-isotypic. The proof proceeds by exhibiting a semi-stable model over \(K\), if all \(n_P\) are even, or over \(L\) in case some \(n_P\) is odd. This is achieved by successive explicit blow-ups (much in the spirit of [\textit{C. Liedtke} and \textit{Y. Matsumoto}, Compos. Math. 154, No. 1, 1--35 (2018; Zbl 1386.14138)], but less complicated since there are only ordinary double points). Then the author applies the Rapoport--Zink spectral sequence and appeals to known cases of the weight-monodromy conjecture due to \textit{M. Rapoport} and \textit{Th. Zink} [Invent. Math. 68, 21--101 (1982; Zbl 0498.14010)] in mixed characteristic and [\textit{T. Ito}, Am. J. Math. 127, No. 3, 647--658 (2005; Zbl 1082.14021)] in equal characteristic.
scientific article; zbMATH DE number 7137132