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Let $X$ be a regular projective (complex) surface and $S$ be a finite set of closed points on $X$. Denote by $j:X\backslash S \to X$ the open immersion. Is $H^0(R^1j_*\mathcal{O}_{X \backslash S}) \cong H^1(j_*\mathcal{O}_{X\backslash S})$?

EDIT: Assume further that $H^1(\mathcal{O}_X)=0$.

Let $X$ be a regular projective (complex) surface and $S$ be a finite set of closed points on $X$. Denote by $j:X\backslash S \to X$ the open immersion. Assume further that $H^1(\mathcal{O}_X)=0$. Is $H^0(R^1j_*\mathcal{O}_{X \backslash S})=0$?

Let $X$ be a regular projective (complex) surface and $S$ be a finite set of closed points on $X$. Denote by $j:X\backslash S \to X$ the open immersion. Is $H^0(R^1j_*\mathcal{O}_X) \cong H^1(j_*\mathcal{O}_X)$?

EDIT: Assume further that $H^1(\mathcal{O}_X)=0$.

Let $X$ be a regular projective (complex) surface and $S$ be a finite set of closed points on $X$. Denote by $j:X\backslash S \to X$ the open immersion. Is $H^0(R^1j_*\mathcal{O}_{X \backslash S}) \cong H^1(j_*\mathcal{O}_{X\backslash S})$?

EDIT: Assume further that $H^1(\mathcal{O}_X)=0$.

Let $X$ be a regular projective (complex) surface and $S$ be a finite set of closed points on $X$. Denote by $j:X\backslash S \to X$ the open immersion. Is $H^0(R^1j_*\mathcal{O}_X) \cong H^1(j_*\mathcal{O}_X)$?

Let $X$ be a regular projective (complex) surface and $S$ be a finite set of closed points on $X$. Denote by $j:X\backslash S \to X$ the open immersion. Is $H^0(R^1j_*\mathcal{O}_X) \cong H^1(j_*\mathcal{O}_X)$?

EDIT: Assume further that $H^1(\mathcal{O}_X)=0$.