An integral version of Zariski decompositions on normal surfaces (Q6564869)

Zariski decomposition is a fundamental tool in algebraic geometry. On surfaces, it has played significant roles in studying cohomology of divisors and other related problems. In this paper, the author investigates an integral version of Zariski decomposition on normal surfaces and its applications.\N\NA divisor \( D \) on a normal surface \( S \) is called \(\mathbb{Z}\)-positive if there is no effective negative-definite \(\mathbb{Z}\)-divisor \(B\) such that \(B-D\) is nef over \(B\), that is, \((B - D)C \ge 0\) for any irreducible component of \(B\). Typical examples of \(\mathbb{Z}\)-positive divisors include round-ups of nef divisors (Corollary 3.18) and numerically connected divisors (Proposition 3.21).\N\NThe main result (Theorem 3.5) reveals that a pseudo-effective divisor \( D \) on a normal proper surface \( S \) admits a unique decomposition \( D = P_{\mathbb{Z}} + N_{\mathbb{Z}} \), called the integral Zariski decomposition of \(D\), where \( P_{\mathbb{Z}} \) is a \(\mathbb{Z}\)-positive divisor and \( N_{\mathbb{Z}} \) is an effective negative-definite integral divisor such that \(-P_{\mathbb{Z}}\) is nef on \(N_{\mathbb{Z}}\).\N\NIntegral Zariski decompositions have been implicitly referenced in the literature. The author provides a detailed simple proof of the main result as well as the classical Zariski decomposition. The reader may find that the main result can also be proven constructively by applying the results from Section 3.2.\N\NA significant consequence of the integral Zariski decomposition is the following vanishing theorem (Proposition 4.10): \(H^1(S, \mathcal{O}_S(K_S + D)) = 0\) if \(D\) is big and \(\mathbb{Z}\)-positive, which generalizes Miyaoka's vanishing theorem on smooth surfaces to normal surfaces. A more general version of this vanishing theorem is presented in Theorem 4.1, where part (1), proved using Proposition 4.10, asserts that if \(D\) is a big divisor with the integral Zariski decomposition \(D = P_\mathbb{Z} + N_\mathbb{Z}\), then \[H^1(S, \mathcal{O}_S(K_S + D)) \cong H^1(N_\mathbb{Z}, \mathcal{L}_D), \] where \(\mathcal{L}_D\) is the cokernel of the natural homomorphism \(\mathcal{O}_S(K_S + P_\mathbb{Z}) \rightarrow \mathcal{O}_S(K_S + D)\). Theorem 4.1 generalizes several well-known vanishing theorems on surfaces and Remark 4.3 shows the details.\N\NBy applying the vanishing theorems, the author proves Reider-type theorems (Theorem 4.2 and Theorem 4.4) for normal surfaces and moreover extends Serrano-Paoletti's extension theorem to normal surfaces (Theorem 6.1).