Some examples of $$(p-1)$$ ( p - 1 ) -th Frobenius split p
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Archiv der Mathematik
Some examples of (p − 1)-th Frobenius split projectivized bundles He Xin
Abstract. We prove that the projectivized cotangent bundles of smooth quadrics of dimensions three and four are (p − 1)-th Frobenius split when p > 10. Besides, we show that the cotangent bundles of certain ordinary elliptic K3 surfaces are not Frobenius split. Mathematics Subject Classification. 14G17, 14M15, 14J60. Keywords. Frobenius splitting, Cotangent bundle, K3 surface.
1. Introduction. Let X be an algebraic variety defined over an algebraically closed field of characteristic p > 0. Then X is said to be Frobenius split or F -split for short if the morphism of OX -modules OX → FX∗ OX splits, where FX : X → X is the Frobenius morphism sending a local section of OX to its p-th power. When X is smooth, by the Grothendieck duality, we have an isomorphism ∼ H 0 (X, ω 1−p ). (1.1) ϕX : Hom(FX∗ OX , OX ) = X Definition 1.1. A smooth algebraic variety X is said to be (p − 1)-th F -split if −1 p−1 there exists a global section s of ωX such that ϕ−1 ) defines a Frobenius X (s splitting of X. The list of known F -split varieties includes many interesting members such as ordinary varieties with trivial canonical line bundles [8, Proposition 3.1(b)] and reductions of Fano varieties in characteristic 0 modulo sufficiently large primes. However, most algebraic varieties do not fall in this category and (p − 1)-th F -split varieties are even fewer. It should also be remarked that many (p − 1)-th F -split varieties are endowed with a group action, such as toric varieties and their cotangent bundles [1,19], complete flag varieties [4, Theorem 2.3.1], and toroidal spherical varieties [2].
H. Xin
Arch. Math.
The property of being F -split can be made use of to obtain vanishing results in various settings. A well-known result nowadays saying that ample line bundles on F -split varieties have vanishing higher cohomologies is proved in [13] where the notion of Frobenius splitting is introduced. Another remarkable result is a characteristic p proof of Dolbeault’s vanishing theorem [3] given in [11] by using the F -splitness of cotangent bundles of flag varieties. In spite of the notable results obtained ibid., a special case, i.e., the cotangent bundle of SLn /B, is proved to be F -split in an earlier paper [12]. Moreover, the construction ibid. is shown in [11, 7.1] and [4, Example 5.1.15] to yield a (p − 1)-th Frobenius splitting of the projectivized cotangent bundle of SLn /B. Besides projective spaces and SLn /B, n ≥ 2, it seems unknown whether the projectivized cotangent bundles of other flag varieties admit (p − 1)-th Frobenius splittings. One aim of this paper is to provide some examples of (p − 1)-th F -split projectivized bundles on low-dimensional flag varieties. Let Q3 and Q4 be the smooth quadrics of dimensions three and four, then we have the following Theorem 1.2. The projectivized cotangent bundles of Q3 and Q4 are (p − 1)-th F -split when p > 10. We will also show (see Theorem 2.4) that the projectivizations of pullbacks of t
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