Globally valued function fields: existential closure

The material presented in these notes forms part of a joint research project on the logic of fields with many valuations, connected by a product formula; we define such structures and name them globally valued fields (GVFs). This text aims primarily at a proof that the canonical GVF structure on k(t) is existentially closed. This can be read as saying that a variety with a distinguished curve class is a good approximation for a formula in the language of GVFs, in the same way that a variety is close to a formula for the theory ACF of algebraically closed fields. The text is based on classes given by the second author in Spring 2015 Jerusalem and in Spring 2016 in Paris (FSMP). It has a large expository element, especially concerning the algebraic geometry needed for this. This material is based on (short initial sections of) [12], [23], [17], [25], [3] and [27]. However in a number of cases, modifications of results there were needed. One critical statement is a strong form of BDPP duality ([2]), where elements of the interior of the dual cone to the effective cone are shown to have n − 1’st roots in some limit of blowups; the usual version in the literature presents them only as convex combinations of powers of ample divisors. We work in arbitrary characteristic, whereas some of the literature restricts to characteristic zero. Since these notes were first written (and perhaps before), many of these improvements have undoubtedly appeared in the literature; we would be grateful for further references. In these notes, we restrict attention to the function field case. Thus our globally valued fields should properly be called ‘purely non-archimedean globally valued fields’. There are nevertheless some applications to number fields, see Corollary 2.3. Globally valued fields are defined in §1 via globalizing measures (up to renormalization), and also via an explicit axiomatization in an appropriate language for real-valued logic. The equivalence of the two was proved in notes for the previous semester; in the function field case, it also follows from the geometric characterization of quantifier-free types in § 11.