Globally F-regular type of moduli spaces
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Mathematische Annalen
Globally F-regular type of moduli spaces Xiaotao Sun1
· Mingshuo Zhou1
Received: 1 May 2019 / Revised: 11 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove that moduli spaces of semistable parabolic bundles and generalized parabolic sheaves with fixed determinant on a smooth projective curve are of globally F-regular type. Mathematics Subject Classification Algebraic Geometry · 14H60 · 14D20
1 Introduction Let X be a variety over a perfect field of characteristic p > 0 and F : X → X be the Frobenius morphism. Then X is called F-split (Frobenius split) if the natural homomorphism O X → F∗ O X is split. It is known that some important varieties are F-split. For example, flag varieties and their Schubert subvarieties (cf. [7,13]), product of two flag varieties for the same group G (cf. [8]) and cotangent bundles of flag varieties (cf. [4]). An example, which is more closer to this article, should be mentioned. Mehta–Ramadas proved in [6] that for a generic smooth projective curve C of genus g over an algebraically closed field of characteristic p ≥ 5, moduli spaces of semistable parabolic bundles of rank 2 on C are F-split. They also made a conjecture that moduli spaces of semistable parabolic bundles of rank 2 on any smooth curve C with a fixed determinant are F-split.
Communicated by Vasudevan Srinivas. Both authors are supported by the National Natural Science Foundation of China No.11831013 and No.11921001; Mingshuo Zhou is also supported by the National Natural Science Foundation of China No.11501154.
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Xiaotao Sun [email protected] Mingshuo Zhou [email protected]
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Center of Applied Mathematics, School of Mathematics, Tianjin University, No.92 Weijin Road, Tianjin 300072, People’s Republic of China
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X. Sun, M. Zhou
The notion of globally F-regular varieties was introduced by Smith in [16]. A variety X is called globally F-regular if, for any effective divisor D, the natural homomorphism O X → F∗e O X (D) is split for some integer e > 0. It is clear that a globally F-regular variety must be F-split. On the other hand, some well-known F-split varieties include toric varieties and Schubert varieties are proved ([5,16]) to be globally F-regular. Thus it is natural to extend Mehta–Ramadas conjecture: the moduli L spaces UC, ω of semistable parabolic bundles of rank r with a fixed determinant L on any smooth curves C (parabolic structures determined by a given data ) are globally F-regular varieties. However, this is still a difficult open problem, we will study its characteristic zero analogy in this article. A variety X over a field of characteristic zero is called of globally F-regular type (resp. F-split type) if its modulo p reduction X p is globally F-regular (resp. F-split) for a dense set of p. Projective varieties of globally F-regular type have remarkable geometric and cohomological properties (see Theorem 2.5). Let UC, ω be the moduli space of semistable parabolic bundles of rank r and degree d on a smooth curve C of g
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