  The issue of confinement and bose condensation is studied for gauge models of
high-Tc superconductors. First the Abelian-Higgs model in (2+1)D, i.e.,
XY-model coupled to lattice gauge field $a_{\mu}$ with coupling $g$, is studied
taking into account both the instantons and vortices. This model corresponds to
integer filling of the bosons, and can be mapped to a dual superconductor. Our
main resut is that the instantons introduce a term which couples linearly to
the dual superconductor order parameter, and tend to pin its phase. As a result
the vortex condensation always occurs due to the instantons, and the Meissner
effect for the gauge field $a_\mu$ is absent, although $a_{\mu}$ is massive.
This state is essentially the same as the confining phase of the pure gauge
model. Away from integer filling, a ``magnetic field'' $\mu$ (the chemical
potential of the bosons) is applied to this dual superconductor. Then the Higgs
phase revives in the case of weak $g$ and large $x$, where vortices do not
condense in spite of the instantons. In the opposite case, i.e., strong $g$ and
small $x$, phase separation occurs, forming either microscopic patches or
macroscopic stripe domains of the Mott insulating state.
