Let M be a closed Kähler surface. We prove that every Riemannian metric g on M with Scg≥−λ2
Nearby in the stack
, where
λ≥0
, satisfies
∥M∥≤227λ4volg(M).
This proves Gromov's quantitative scalar-curvature--simplicial-volume conjecture for closed Kähler surfaces. We also construct infinitely many non-Kähler symplectic 4-manifolds of general type with positive simplicial volume for which the same estimate holds.