Most of the pulsars that have been detected by the {em Fermi} gamma-ray observatory show similar spectral characteristics and cutoffs in a narrow range, around a few GeV. We discuss the possibility that the pulsed radiation comes from the equatorial current sheet which separates the two magnetic hemispheres right outside the light cylinder - a region which we will call the``near wind'' region. Synchrotron radiation from thermal particles in the current sheet in the near wind can dominate the gamma-ray emission, leading to cutoffs, luminosities and spectra that are similar to the ones observed by the LAT.