Holomorphic linking numbers

The Gauss linking number of two curves in the three-space has a complex counterpart. In the talk we define the holomorphic linking number for complex curves in complex three-folds. Moreover, one can define "polar homology" groups of complex projective manifolds by regarding meromorphic forms on their submanifolds as a complex analogue of orientation, and taking the residue as the boundary operator. We also discuss gauge-theoretic aspects of the above correspondence, and, in particular, its relation to the holomorphic Chern--Simons theory.