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Bogomolov-Sommese vanishing and liftability for surface pairs in positive characteristic. (English) Zbl 1502.14046

The author studies the Bogomolov-Sommese vanishing theorem on a projective surface in positive characteristic. Let \((X, B)\) be a log canonical projective surface pair over an algebraically closed field of characteristic \(p\). Assume that \(\kappa(X, K_X+\lfloor B\rfloor)\ne 2\). The author shows that there is a positive integer \(p_0\) such that the following vanishing result holds true given that \(p>p_0\): \[ H^0(X, (\Omega_X^{[i]}(\log\lfloor B\rfloor)\otimes\mathcal{O}_X(-D))^{**})=0 \] for every \(\mathbb{Z}\)-divisor \(D\) on \(X\). Moreover, the optimal bounds of \(p_0\) are also given.
The key step of the proof is a generalization of Graf’s logarithmic extension theorem (Theorem 1.2 in [P. Graf, J. Lond. Math. Soc. (2) 104, No. 5, 2208–2239 (2021)]). The author shows that \(f_*(\Omega_Y^{[1]}(\log\lfloor B_Y\rfloor)\otimes\mathcal{O}_Y(-\lceil f^*D \rceil))^{**}\) is reflexive if \(p>5\), where \(D\) is a \(\mathbb{Z}\)-division on \(X\), \(f:Y\to X\) a birational morphism, \(Y\) is normal, and \(B_Y:=f^{-1}_*B+\mathrm{Exc}(f)\). This generalization enables the author to apply the logarithmic Akizuki-Nakano vanishing theorem (Corollary 3.8 in [N. Hara, Am. J. Math. 120, No. 5, 981–996 (1998; Zbl 0942.13006)]) for \(W_2\)-liftable pairs \((X, B)\).
In the paper, one can also find results on liftability of log resolutions of pairs and a Kawamata-Viehweg type vanishing theorem for \(\mathbb{Z}\)-divisors on a normal projective surface.
Reviewer: Fei Ye (New York)

MSC:

14F17 Vanishing theorems in algebraic geometry
14D15 Formal methods and deformations in algebraic geometry
14E30 Minimal model program (Mori theory, extremal rays)
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials

Citations:

Zbl 0942.13006
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References:

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