Simplicial Volume and Scalar Curvature on Closed K\"ahler Surfaces

Let $M$ be a closed K"ahler surface. We prove that every Riemannian metric $g$ on $M$ with ${\mathrm{Sc}}_g\ge -{\lambda }^2$, where $\lambda \ge 0$, satisfies $$\parallel M\parallel \le \frac{27}{2}{\lambda }^4{\mathrm{vol}}_g(M)\mathrm{.}$$ This proves Gromov’s quantitative scalar-curvature–simplicial-volume conjecture for closed K"ahler surfaces. We also construct infinitely many non-K"ahler symplectic 4-manifolds of general type with positive simplicial volume for which the same estimate holds.