Sharp systolic inequalities for Kähler manifolds with positive scalar curvature attain equality on CP^n with Fubini-Study metric and imply Gromov's rational-essentialness conjecture.
Compact K\"ahler contact manifolds
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We prove that a non-projective compact K\"ahler contact manifold is of the form $\mathbb{P} T_Y$, where $Y$ is a compact K\"ahler manifold.
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Sharp systolic inequalities for K\"ahler manifolds
Sharp systolic inequalities for Kähler manifolds with positive scalar curvature attain equality on CP^n with Fubini-Study metric and imply Gromov's rational-essentialness conjecture.