  A combination of strong Cooper pairing and weak superconducting fluctuations
is crucial to achieve and stabilize high-Tc superconductivity. We demonstrate
that a coexistence of a shallow carrier band with strong pairing and a deep
band with weak pairing, together with the Josephson-like pair transfer between
the bands to couple the two condensates, realizes an optimal multicomponent
superconductivity regime: it preserves strong pairing to generate large gaps
and a very high critical temperature but screens the detrimental
superconducting fluctuations, thereby suppressing the pseudogap state.
Surprisingly, we find that the screening is very efficient even when the
inter-band coupling is very small. Thus, a multi-band superconductor with a
coherent mixture of condensates in the BCS regime (deep band) and in the
BCS-BEC crossover regime (shallow band) offers a promising route to higher
critical temperatures.
