Kohn--Luttinger Superconductivity in Flat Chern Bands

Recent observations of superconductivity near correlated topological phases in flat bands suggest a facile link between flat-band geometry and electron pairing. In this work, we reveal a geometry-driven Kohn--Luttinger mechanism in which Landau-level-like form factors align the attractive lobe of the RPA-screened Coulomb interaction with the form-factor peak, generating an anomalously strong attractive channel near local band extrema. Using the Skyrmion lattice model as a minimal realization, we show that for spin-unpolarized pairing the form-factor magnitude enforces an emergent momentum-space translational symmetry and selects an extended-$s$ instability concentrated at small Fermi pockets, while for spin-polarized pairing the form-factor phase drives chiral $p$- and $f$-wave order without invoking spin fluctuations. The band-extrema enhancement persists in higher Landau-level analogs and survives finite-temperature screening and Berezinskii--Kosterlitz--Thouless phase fluctuations. Our work establishes quantum geometry as a key organizing principle for unconventional pairing in flat Chern bands.