Asked this on MSE but didn't get much attention.

Let $ S $ be a compact complex surface. Can anyone provide a proof of the fact that the Frolicher 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.

Asked this on MSE 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.