Asked this on [MSE](https://math.stackexchange.com/questions/4421900/frolicher-spectral-sequence-of-a-surface) but didn't get much attention.

Let $ S $ be a compact complex surface. Can anyone provide a proof of the fact that the Frölicher spectral sequence of $ S $ degenerates at $ E_1 $? 

Note, no assumptions on $ S $ being Kahler are made. I think the only hard part here is to show that $$ \partial : E_1^{0,1} = H^1 (S, \mathcal{O}_S ) \rightarrow E_1^{1,1} = H^1 (S, \Omega_S ) $$ vanishes but I'm not able to do this.