Betti and Hodge bounds for compact complex surfaces ($2h^{1,0}\le b_1\le h^{1,0}+h^{0,1}$)

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 $h^{1,0}\le h^{0,1}$. These inequalities do not require a Kähler manifold.