  Recent measurements of the anisotropy of the upper critical field Bc2 on MgB2
single crystals have shown a puzzling strong temperature dependence. Here, we
present a calculation of the upper critical field based on a detailed modeling
of bandstructure calculations that takes into account both the unusual Fermi
surface topology and the two gap nature of the superconducting order parameter.
Our results show that the strong temperature dependence of the Bc2 anisotropy
can be understood as an interplay of the dominating gap on the sigma-band,
which possesses a small c-axis component of the Fermi velocity, with the
induced superconductivity on the pi-band possessing a large c-axis component of
the Fermi velocity. We provide analytic formulas for the anisotropy ratio at
T=0 and T=Tc and quantitatively predict the distortion of the vortex lattice
based on our calculations.
