I am sorry to make this a new question. I would have liked to leave a comment, but I suppose I don't have enough rep for that....
So, in the accepted answer to this question I don't understand why in the second term it is $\dim H^0(X,L\otimes \Lambda^{0,1})$ and not $L^{-1}$ with the same.thing. I suspect this is some Hodge theory thing that I am not recognizing, and it is probably completely trivial, but I would appreciate if someone could enlighten me.
Thank you.