Concerning characterizations of boundaries of holomorphic 1 -chains within complex surfaces.

Abstract

Let <italic>X</italic> be a complex manifold or an analytic variety and let gamma be a closed, oriented, C2 real 1-chain in <italic>X</italic>. We say gamma is the boundary of an holomorphic 1-chain within <italic>X</italic> if there is a holomorphic 1-chain <italic>V</italic> in <italic>X</italic>\spt gamma such that <italic> b</italic>[<italic>V</italic>] = [gamma], in the sense of currents, and spt <italic>V</italic> &Sub; <italic>X</italic>. We produce a family of new characterizations for boundaries of holomorphic 1-chains within <blkbd>CP2</blkbd> , related to a previous characterization involving shockwaves. We show that some of these characterizations are computationally accessible and may be tractably tested. By employing birational maps, we further characterize the boundaries of holomorphic 1-chains within <a><ac><blkbd>C</blkbd></ac><ac>&d4;</ac></a>x<a><ac><blkbd> C</blkbd></ac><ac>&d4;</ac></a> and <blkbd>C</blkbd>x<a><ac><blkbd>C</blkbd></ac><ac>&d4;</ac></a> . By a separate approach, we produce a distinctly new vein of characterizations for boundaries of holomorphic 1-chains within <blkbd>CP2</blkbd> and <blkbd>C</blkbd>x<a><ac><blkbd>C</blkbd></ac><ac>&d4;</ac></a> .