Positivity on subvarieties and vanishing of higher cohomology
The article under review studies the relationship between the line bundle with partial positivity and vanishing of their higher cohomology. Let \(L\) be a Cartier divisor \(L\) over a complex projective scheme \(X\). If the restriction of \(L\) on \(E_1 \cap\cdots \cap E_q\) is ample, where \(E_i\) is a general element of \(|A_j|\) for some very ample Cartier divisor \(A_j\) on \(X\), the author proved the following vanishing theorem: \[ H^i (X, \mathcal{F}\otimes \mathcal{O}_X (mL +N))=0 \qquad\text{for all } i>q , \] where \(\mathcal{F}\) is an arbitrary coherent sheaf on \(X\), \(m > m(L, \mathcal{F})\) and \(N\) is an arbitrary nef divisor. In the case \(X\) is smooth and the restriction of \(L\) on \(E_1 \cap\cdots \cap E_q\) is big and nef, the author also proved a Kawamata-Viehweg type vanishing theorem: \[ H^i (X, K_X\otimes \mathcal{O}_X (L))=0 \qquad\text{for all } i>q . \] The proofs of these vanishing theorems uses a standard complex and some induction arguments. Thus the proof is purely algebraic. Compare to many other vanishing theorems, a remarkable feature of this article is that the divisor \(L\) here is not necessarily pseudo-effective.
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