HERMITIAN VECTOR BUNDLES OF RANK TWO AND ADJOINT SYSTEMS ON ARITHMETIC SURFACES

Let K be a number field and OK be its ring of integers. Let f : X→Spec(OK) be an arithmetic surface and let L̄ be an arithmetically nef hermitian line bundle over X. The hermitian structure on L̄ defines a natural structure of hermitian OK-module on H0(X; L̄ ⊗ ωX/OK). A closed point P∈X is said to be a fixed point for the adjoint system of L̄ if, for every D∈H0(X; L̄ ⊗ ωX/OK) such that ‖D‖sup⩽1 we have that D∣P=0. We prove that the existence of a fixed point for the adjoint system of L̄ imposes some (Arakelov) numerical condition on L̄. We prove also an arithmetic analogue of Cayley–Bacharach criterion for the existence of an hermitian vector bundle of rank two over X with prescribed arithmetic Chern classes and a section vanishing only on a fixed closed point. In the last part we apply this to find an arithmetic analogue of Reider’s Theorem on fixed points of the adjoint system of L̄.