Special Year Learning Seminar

Every o-minimal structure determines a collection of "tame" or "definable" subsets of $bbR^n$. We can then ask about the fragment of complex geometry present in the structure: Which holomorphic functions are definable, and which spaces are cut out by definable holomorphic functions? These "definable analytic spaces" have proven increasingly important in recent years, particularly for their applications in Hodge theory. It is now becoming clear that definable analytic geometry is an interesting subject in its own right, parallel to algebraic or analytic geometry and interpolating between them. There is a robust notion of definable coherent sheaf which possesses many expected properties, and there is even a GAGA theorem relating definable and algebraic sheaves.

A natural question is whether definable coherent sheaves ever satisfy a vanishing theorem: Do there exist spaces analogous to affine schemes or Stein spaces on which all coherent cohomology vanishes? In the usual structure $bbR_{an,exp}$, such spaces appear rare. But we show that in the smaller structure $bbR_{an}$ they are abundant, at least in dimension one. Time permitting, we discuss the situation in higher dimensions and some partial results.