Return to Revisions

This is actually a comment on Piotr Achinger's answer, but a bit too long for a comment. I don't think you need the spectral sequence, nor the differential $\partial_2$. From the exact sequence $0\rightarrow \mathbb{C}\rightarrow \mathcal{O}_X \rightarrow (\Omega ^1_X)_{d=0}\rightarrow 0\ $ you get an exact sequence $$0\rightarrow H^0(X,\Omega ^1_X)_{d=0}\rightarrow H^1(X,\mathbb{C})\rightarrow H^1(X,\mathcal{O}_X)\xrightarrow{d} H^1(X,(\Omega ^1_X)_{d=0})$$from which the inequality $b_1\leq h^{1,0}+h^{0,1}$ follows immediately.