= Betti and Hodge bounds for compact complex surfaces
{title2=$2h^{1,0}\leq b_1\leq h^{1,0}+h^{0,1}$}
Closed <holomorphic one-forms> inject into <de Rham cohomology>: an exact one would have a global holomorphic primitive and hence vanish. If a holomorphic class equals an antiholomorphic class, their difference is $df$, and $\partial\bar\partial f=0$; compact pluriharmonic rigidity makes both forms vanish. Their direct sum gives the lower bound. The <sheaf of closed holomorphic one-forms> exact sequence gives $0\to H^0(\Omega^1)\to H^1(\mathbb C)\to H^1(\mathcal O)$, yielding the upper bound and $h^{1,0}\leq h^{0,1}$. These inequalities do not require a <Kähler manifold>.
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