The origin of the high-temperature superconducting state observed in FeSe thin films, whose phase diagram displays no sign of magnetic order, remains a hotly debated topic. Here we investigate whether fluctuations arising due to the proximity to a nematic phase, which is observed in the phase diagram of this material, can promote superconductivity. We find that nematic fluctuations alone promote a highly degenerate pairing state, in which both s-wave and d-wave symmetries are equally favored, and Tc is consequently suppressed. However, the presence of a sizable spin-orbit coupling or inversion symmetry-breaking at the film interface lifts this harmful degeneracy and selects the s-wave state, in agreement with recent experimental proposals. The resulting gap function displays a weak anisotropy, which agrees with experiments in monolayer FeSe and intercalated Li1−x(OH)xFeSe.In most iron-based superconductors (FeSC), superconductivity is found in close proximity to a magnetically ordered state, suggesting that magnetic fluctuations play an important role in binding the Cooper pairs [1][2][3][4]. Indeed, the fact that the Fermi surface of these materials is composed of small hole pockets and electron pockets separated by the magnetic ordering vector led to the proposal of a sign-changing s +− wave state, in which the gap function has different signs in the hole and in the electron pockets. However, the recent observation of superconductivity over 70 K in monolayer FeSe brought new challenges to the field [5][6][7][8][9][10][11][12][13][14]. In contrast to the standard FeSC, no long-range magnetic order is observed in thin films or even bulk FeSe [15], and the Fermi surface of monolayer FeSe consists of electron pockets only [6,7,11,16]. Since T c in monolayer FeSe is the highest among all FeSC, the elucidation of its origin is a fundamental step in the search for higher T c in these systems.One of the proposed scenarios to explain the dramatic ten-fold increase of T c in monolayer FeSe with respect to the 8 K value in bulk FeSe [17] was the strong coupling to an optical phonon mode of the SrTiO 3 (STO) substrate [16,18,19], which is manifested by replica bands observed in ARPES [16]. Although such a coupling can certainly enhance T c [20][21][22][23][24], recent experiments indicate that the STO substrate may not be essential to achieve the high-T c state. In particular, T c up to 40 K was observed in electrostatically-gated films of FeSe with different thickness grown both on STO and MgO substrates [25]. Similar values of T c were reported in FeSe coated with potassium [26,27] and in the bulk sample Li 1−x (OH) x FeSe [28,29], which consists of intercalated FeSe layers. In common to all these systems is the fact that their Fermi surface consists of electron pockets only, suggesting that doping by negative charge carriers plays a fundamental role in stabilizing the high-T c state.Importantly, recent experiments in K-coated bulk FeSe [26] revealed that, besides shifting the chemical potential, electron-doping also su...