Let (X,D) be a log pair, X nonsingular, D simple normal crossing. Does $ h^1(X,\omega_X(D))=h^0(X,\Omega_X^1(D))$ as the usual Hodge number. I wish this is NOT true, but I can’t find any counterexample. I want some example that $ h^1(X,\omega_X(D))=0$ but $ h^0(X,\Omega_X(D))\neq 0$ .
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