Bott vanishing for elliptic surfaces
In the paper under review, the author studies the Bott vanishing for certain elliptic surfaces. Let us recall that the Bott vanishing holds for a smooth projective variety \(X\) if \[H^{j}(X, \Omega_{X}^{i} \otimes A) = 0,\] for all ample line bundles \(A\), all \(i\geq 0\) and all \(j\geq 0\). More precisely, the author explores the Bott vanishing for elliptic surfaces \(X\) with an elliptic fibration \(\pi : \, X \rightarrow \mathbb{P}^{1}\). It is worth emphasizing that not for all ample line bundles on elliptic surfaces the Bott vanishing can potentially hold. The main goal of the author is to determine, as explicitly as possible, which ample line bundles \(A\) have \(H^{j}(X, \Omega_{X}^{i}\otimes A) = 0\) for all \(i\geq 0\) and \(j>0\). The crucial issue is testing whether \(H^{1}(X, \Omega_{X}^{1} \otimes A) =0\). It turns out, by Proposition 3.4 therein, that one needs to check that \(H^{1}(X,A)=0\) and \(H^{1}(X, \Omega_{X}^{1} \otimes A) = 0\), and to check whether \(H^{1}(X, \Omega_{X}^{1} \otimes A)=0\) provided that \(A-K_{X}\) is big and nef. The main result of the paper under review can be formulated as follows. Main Theorem. Let \(X\) be a smooth complex projective surface with an elliptic fibration \(\pi : \, X \rightarrow \mathbb{P}^{1}\) such that all fibres are reduced. Let \(\beta = \chi(\mathcal{O}_{X})\). Assume that \(A\) is an ample line bundle on \(X\) with \(r:=A.E\), where \(E\) is a fibre of \(\pi\). When \(r=1\), then \(H^{1}(X, \Omega_{X}^{1} \otimes A) \neq 0\) if and only if \(A^{2} \leq 21\beta - 3\), or \(\pi\) has a fibre of type \(II\). When \(r\geq 2\), assume that \(A^{2}\geq 2r^{2}\beta +25r\beta - 4r - 2\beta\). Then \(H^{1}(X, \Omega_{X}^{1}\otimes A) \neq 0\) if and only if \(\pi\) has a fibre of type \(III\) when \(r=2\), or a fibre of type \(IV\) when \(r=3\). Corollary. Let \(X\) be a smooth complex projective surface with an elliptic fibration \(\pi : X \rightarrow \mathbb{P}^{1}\) and an ample line bundle \(A = A_{0} + mE\), where \(A_{0}\) is a section of \(\pi\) and \(E\) is a fibre of \(\pi\). When \(m > 11\beta - 1\), the Bott vanishing holds for \((X,A)\) if and only if \(\pi\) has no fibres of type \(II\).
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