  In this paper we are begining to explore the complex counterpart of the
Chern-Simon-Witten theory. We define the complex analogue of the Gauss linking
number for complex curves embedded in a Calabi-Yau threefold using the formal
path integral that leads to a rigorous mathematical expression. We give an
analytic and geometric interpretation of our holomorphic linking following the
parallel with the real case. We show in particular that the Green kernel that
appears in the explicit integral for the Gauss linking number is replaced by
the Bochner-Martinelli kernel. We also find canonical expressions of the
holomorphic linking using the Grothendieck-Serre duality in local cohomology,
the latter admits a generalization for an arbitrary field.
