Special lifts of ordinary \(K3\) surfaces and applications (Q713626)

Let \(X\) be an ordinary \(K3\) surface over a finite field \(k\). The deformation functor of \(X\) admits a natural group structure. Using this group structure, the deformation of \(X\) over an arbitrary local ring corresponding to the identity element (resp. torsion elements) can be defined, and is called the canonical (resp. quasi-canonical) lift of \(X\). In this paper, the cohomological properties of these canonical and quasi-canonical lifts of \(X\) to a complete discrete valuation ring of characteristic zero are investigated. The results are formulated as follows: these lifts are characterized by the existence of a (lifted) Frobenius acting on the Betti cohomology of the lifts. For a perfect field \(k\) of characteristic \(p>0\) with countably many elements, \(W=W(k)\) denote the ring of Witt vectors of \(k\) and \(K\) the field of quotients, \(\bar{K}\) an algebraic closure of \(K\) with a fixed embedding \(\bar{K}\subset{\mathbb{C}}\). Theorem 1. Let \(\mathcal{X}\) be a formal lift of an ordinary \(K3\) surface over a finite field \(k\) to a complete discrete valuation \(W\)-algebra \(R\) with \([R:W]<\infty\). Then \(\mathcal{X}\) is quasi-canonical if and only if \(\mathcal{X}\) is projective and a power of the Frobenius morphism on \(X\) is liftable to \(H^2({\mathcal{X}}({\mathbb{C}}),{\mathbb{Q}})\). Let \(Z\) be a complex \(K3\) surface and let \(NS(Z)^{\perp}\) be the orthogonal complement of the Néron-Severi group \(NS(Z)\) in \(H^2(Z,{\mathbb{Z}}(1))\) with respect to the cup product pairing. Then \(NS(Z)^{\perp}\) is a sub-integral Hodge structure of \(H^2(Z,{\mathbb{Z}}(1))\). Let \({\mathbf{M}}(Z)=NS(Z)^{\perp}\otimes{\mathbb{Q}}\), and let \({\mathbf{Hdg}}_Z\) be the Hodge group associated to \(Z\) and \({\mathbf{E}}=\text{End}_{\mathbf{Hdg}_Z}{\mathbf{M}}(Z)\). The next result is the determination of the Hodge group of \(X\) and the endomorphism ring of \({\mathbf{M}}(Z)\). Theorem 2. Assume that \(k={\mathbb{F}}_q\), and let \(X\) be an ordinary \(K3\) surface defined over \(k\). Let \(L\) be a finite extension of \(K\) with ring of integers \(R\). Let \({\mathcal{X}}\) be a quasi-canonical lift of \(X\) and \({\mathcal{X}}^{\circ}={\mathcal{X}}\otimes_R L\). Let \({\mathbf{Hdg}}_X\) be the Hodge group of \({\mathbf{M}}(X_{\mathbb{C}}^{\circ})\) and \({\mathbf{E}}=\text{End}_{{\mathbf{Hdg}}_X} {\mathbf{M}}(X_{\mathbb{C}}^{\circ})\). Let \(G\) be the Zariski closure over \({\mathbb{Q}}\) of the cyclic group generated by \(\pi/q\) in \(GL({\mathbf{M}}(X_{\mathbb{C}}^{\circ}))\), where \(\pi\) is the lifted Frobenius on \(H^2(X^{\circ}({\mathbb{C}}),{\mathbb{Q}})\). Then \({\mathbf{E}}={\mathbb{Q}}[\pi]\) and \({\mathbf{Hdg}}_X=G\). Applications of these results are discussed. Theorem 3. Let \(A\) be a complex abelian surface of CM type and \(X=Km(A)\) be the Kummer surface associated to \(A\). Then the Hodge conjecture is true for any self-product \(X\times X\times\cdots\times X\) of \(X\). Theorem 4. Let \(X\) be a weighted \(K3\) surface defined over some number field \(F\). Then there exists a finite place \(\nu\) of \(F\) such that \(\bar{X}_{\nu}\) is an ordinary \(K3\) surface defined over a finite field \(k_{\nu}\) and \({\mathcal{X}}_{\nu}\) is a quasi-canonical lift of \(\bar{X}_{\nu}\). Then the Hodge conjecture is true for any self-product \(X\times X\times\cdots\times X\) of \(X\). These last two theorems are proved by noting that the subspace of the Hodge cycles in the tensor algebra of \({\mathbf{M}}(X_{\mathbb{C}}^{\circ})\) is generated by the graphs \(\Gamma_n\) of iterations \(\pi^n\) of the lifted Frobenius \(\pi\) in \({\mathbf{M}}((X_{\mathbb{C}}^{\circ})\).