Title: "Polar homology"
Polar homology groups arise as holomorphic analogs of singular
homology groups in topology. Polar chains in a complex projective manifold
are complex subvarieties with meromorphic forms on them, while the
boundary operator is defined by taking the divisor of poles and the
Poincare residue on the divisor. Similarly, one can define, e.g.,
holomorphic analogs of the Gauss linking number or symplectic structure
on moduli spaces of flat connections on a Riemann surface.