Arc-meromorphous functions

Abstract

We introduce arc-meromorphous functions, which are continuous functions representable as quotients of semialgebraic arc-analytic functions, and develop the theory of arc-meromorphous sheaves on Nash manifolds. Our main results are Cartan’s theorems A and B for quasi-coherent arc-meromorphous sheaves. 0. Introduction. In this note, building on the theory of arc-analytic functions initiated by the second named author [16], we introduce arcmeromorphous functions and arc-meromorphous sheaves on Nash manifolds. Arc-meromorphous functions are analogs for regulous and Nash regulous functions studied in [8] and [13], respectively. The term “regulous” is derived from “regular” and “continuous”, whereas “meromorphous” comes from “meromorphic” and “continuous”. Our theory of arc-meromorphous sheaves is developed in parallel to the theories of regulous sheaves [8] (see also the recent survey [14]) and Nash regulous sheaves [13]. It is established in [8] and [13] that Cartan’s theorems A and B hold for quasi-coherent regulous sheaves and quasi-coherent Nash regulous sheaves. Our main results are Theorem 2.4 (Cartan’s theorem A) and Theorem 2.5 (Cartan’s theorem B) for quasi-coherent arc-meromorphous sheaves. Recall that Cartan’s theorems A and B fail for coherent real algebraic sheaves [6, Example 12.1.5], [7, Theorem 1] and coherent Nash sheaves [11]. We refer to [6] for the general theory of semialgebraic sets, semialgebraic functions, and related concepts. Recall that a Nash manifold is an analytic submanifold X ⊂ Rn, for some n, which is also a semialgebraic set. A realvalued function on X is called a Nash function if it is both analytic and semialgebraic. By [22, Theorem VI.2.1, Remark VI.2.11], each Nash manifold is Nash isomorphic to a nonsingular algebraic set in Rm, for some m. 2020 Mathematics Subject Classification: 14P10, 14P20, 32B10, 58A07.