Complex sentence: With X. Dai and X. Ma, Liu established the asymptotic expansion of the Bergman kernel on a polarized compact Kähler manifold through heat kernel method and local index theory.
Split into simpler sentences: With X. Dai and X. Ma, Liu gave a proof of the asymptotic expansion of the Bergman kernel on a polarized compact Kähler manifold. The proof relies on heat kernel method and local index theory.