The global monodromy property for \(K3\) surfaces allowing a triple-point-free model
Let \(K=\mathbb{C}((t))\) and let \(X\) be a Calabi-Yau variety over \(K\) of dimension \(n\). Let \(\omega\) be a holomorphic \(n\)-fold. Let \(\mathcal{M}_{\mathbb{C}}\) be the Grothendieck group of varieties over \(\mathbb{C}\) localized with respect to \(\mathbb{L}\), the class of \(\mathbb{A}^1_{\mathbb{C}}\). Let \(\sigma\) be a topological generator of \(\mathrm{Gal}(K^{\mathrm{alg}}/K)\). For an element \(x\in \mathbb{Q}/\mathbb{Z}\) denote with \(\tau(x)\) its order. \textit{L. H. Halle} and \textit{J. Nicaise} [Adv. Math. 227, No. 1, 610--653 (2011; Zbl 1230.11076)] introduced the \textit{global monodromy property} for \(X\) which holds if there is a finite subset \(S\) of \(\mathbb{Z}\times \mathbb{Z}_{>0}\) such that the motivic zeta function of \((X,\omega)\) is in \(\mathcal{M}_k\left[ T,\frac{1}{1-\mathbb{L}^\mu T^N}\right]_{(\mu,N)\in S}\) and such that for each \((\mu,N)\in S\) the cyclotomic polynomial \(\Phi_{\tau(a/b)}(t)\) divides the characteristic polynomial of the monodromy operator \(\sigma\) on \(H^i(X \times_K K^{alg},\mathbb{Q}_\ell)\) for some \(i\in \mathbb{N}\). The author shows the global monodromy property for every \(K3\) surface \(X/K\) which admits a minimal model over \(\mathbb{C}t\) such that special fiber is a so-called flowerpot degeneration. Furthermore she shows that this property holds for \(K3\) surfaces with certain chain degenerations.
scientific article; zbMATH DE number 6690678
scientific article