  We construct the analytic lattice cohomology associated with the analytic
type of any complex normal surface singularity. It is the categorification of
the geometric genus of the germ, whenever the link is a rational homology
sphere. It is the analytic analogue of the topological lattice cohomology,
associated with the link of the germ whenever it is a rational homology sphere.
This topological lattice cohomology is the categorification of the
Seiberg--Witten invariant, and conjecturally it is isomorphic with the Heegaard
Floer cohomology.
  We compare the two lattice cohomologies: in some simple cases they coincide,
but in general, the analytic cohomology is sensitive to the analytic structure.
We expect a deep connection with deformation theory. We provide several basic
properties and key examples, and we formulate several conjectures and problems.
  This is the initial article of a series, in which we develop the analytic
lattice cohomology of singularities.
