We combine linear and non-linear optical spectroscopy at 4K with ab initio calculations to study the electronic bandstructure of MoSe2 monolayers. In 1-photon photoluminescence excitation (PLE) and reflectivity we measure a separation between the A-
and B-exciton emission of 220 meV. In 2-photon PLE we detect for the A- and B-exciton the 2p state 180meV above the respective 1s state. In second harmonic generation (SHG) spectroscopy we record an enhancement by more than 2 orders of magnitude of the SHG signal at resonances of the charged exciton and the 1s and 2p neutral A- and B-exciton. Our post-Density Functional Theory calculations show in the conduction band along the $K-Gamma$ direction a local minimum that is energetically and in k-space close to the global minimum at the K-point. This has a potentially strong impact on the polarization and energy of the excitonic states that govern the interband transitions and marks an important difference to MoS2 and WSe2 monolayers.
