Algebraicity of exterior Cauchy transforms of algebraic ovals: a homological formulation

Abstract

Let $Ω\subset\C$ be a bounded domain whose boundary is an oval of a real algebraic curve. We study when the exterior Cauchy transform \[ \ct_Ω(z)=\frac1π\int_Ω\frac{dA(ζ)}{z-ζ} \] is algebraic. The boundary formula identifies this transform with a Cauchy-type integral on the normalization $X$ of the relevant irreducible component of the Schwarz correspondence $P(z,w)=0$. The main point is that $X$ is fixed while only the divisor $π^{-1}(z)$ of moving poles varies. Thus the natural monodromy is point-pushing on a punctured fixed surface and becomes trivial on absolute homology after the moving punctures are filled; in particular, the usual Picard--Lefschetz transvection picture does not produce absolute cycles in this problem. The principal theorem is a residue criterion: if the lifted boundary is separating, i.e., if it bounds an integral two-chain on $X$ disjoint from the fixed polar divisor, then the exterior Cauchy transform is algebraic and is given by an explicit residue sum with chain multiplicities. This implies, in particular, algebraicity for every smooth oval on a rational real algebraic curve and for separating ovals in positive genus. We also record the corresponding complete-real-locus statement for dividing real curves, with the necessary affine-plane caveats. Nonseparating ovals are treated as a conjectural period problem: we formulate a period-rank test which can detect possible Abelian contributions. The examples include the ellipse, the nodal cubic logarithm, a smooth Weierstrass cubic as an elliptic-period test case, and a conditional positive-genus construction illustrating algebraic transforms beyond quadrature domains.