The light Higgs window in the 2HDM at GigaZ

The sensitivity to a light Higgs boson in the general 2HDM (II), with a mass below 40 GeV, is estimated for an future e+e- linear collider operating with very high luminosity at the Z peak (GigaZ). We consider a possible Higgs boson production via the Bjorken process, the (hA) pair production, the Yukawa process Z -> b {\bar b} h(A), -> tau {\bar tau} h(A), and the decay Z ->h(A)+gamma. Although the discovery potential is considerably extended compared to the current sensitivities, mainly from LEP, the existence of a h or A even with a mass of a few GeV cannot be excluded with two billion Z decays. The need to study the very light Higgs scenario at a linear e+e- collider running at several hundred GeV and the LHC is emphasised.