Simplicial Volume and Scalar Curvature on Closed Kähler Surfaces

Abstract

Let $M$ be a closed Kähler surface. We prove that every Riemannian metric $g$ on $M$ with ${\mathrm{Sc}}_g\ge -{\lambda }^2$, where $\lambda \ge 0$, satisfies $$\Vert M\Vert \le \frac{27}{2}{\lambda }^4{\mathrm{vol}}_g(M).$$ 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.

23 pages. Comments are welcome!