Nonabelian Hodge theory in positive characteristic via exponential twisting

Lan, Guitang; Sheng, Mao; Zuo, Kang

doi:10.4310/mrl.2015.v22.n3.a12

Cited by 19 publications

(31 citation statements)

References 5 publications

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“…When D = ∅, this was first proved in [OV,thm 2.8]. A simplified construction of the correspondence in the last case can be found in [LSZ2].…”

Section: Proof Of Theoremmentioning

confidence: 86%

Kodaira–Saito vanishing via Higgs bundles in positive characteristic

Arapura

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. The goal of this paper is to give a new proof of a special case of the Kodaira-Saito vanishing theorem for a variation of Hodge structure on the complement of a divisor with normal crossings. The proof does not use the theory of mixed Hodge modules, but instead reduces it to a more general vanishing theorem for semistable nilpotent Higgs bundles, which is then proved by using some facts about Higgs bundles in positive characteristic.In 1990, Saito [S1, prop 2.33] gave a far reaching generalization of Kodaira's vanishing theorem using his theory of mixed Hodge modules. A number of interesting applications have been found in recent years; we refer to Popa's survey [P2] for a discussion of these, and to [P1, Sl] for a discussion of the theorem itself. We would like to explain the easiest, but still important, case of the theorem where the mixed Hodge module is of the form Rj * V (in standard "non perverse" notation), where V is a polarized variation of pure Hodge structure on the complement j : U → X of a divisor with simple normal crossings D. Let us also assume that V has unipotent local monodromies around components of D. By Deligne [D], the flat vector bundle O U ⊗ V has a canonical extension V such that the original connection extends to a logarithmic connection ∇ : V → Ω 1 X (log D) ⊗ V with nilpotent residues. By a theorem of Schmid [Sc], V has a filtration F by subbundles extending the Hodge filtration. This induces a filtration on the de Rham complexwhich, for us, starts in degree 0. Saito's theorem tell us that if L is an ample line bundle and i > dim X, thenMore generally, this holds when V is replaced by an admissible variation of mixed Hodge structure.The first goal of this paper is to give a short proof of this special case by reduction to characteristic p > 0. Illusie [I] had previously given a proof by such a reduction, when V arises geometrically from a semistable map of varieties. Our proof, however, is different and it works even for nongeometric cases (and this seems important for certain applications, e.g. to Shimura varieties [Su]). We first replace the variation of Hodge structure V by the vector bundle E = Gr F (V ), together with the so called Higgs field θ = Gr F (∇). By work of Simpson, this pair is semistable. In addition, the rational Chern classes c i (E) = 0, and θ is nilpotent. In fact, our main result is Partially supported by the NSF .

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“…When D = ∅, this was first proved in [OV,thm 2.8]. A simplified construction of the correspondence in the last case can be found in [LSZ2].…”

Section: Proof Of Theoremmentioning

confidence: 86%

Kodaira–Saito vanishing via Higgs bundles in positive characteristic

Arapura

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…In the work of [OV07], Ogus and Vologodsky established such a correspondence for nilpotent objects. More precisely, under the assumption that a lifting of X /S modulo p 2 exists, where X is the Frobenius twist of X, they construct a Cartier transform from the category of modules with flat connections nilpotent of exponent 1 ≤ p to the category of Higgs modules nilpotent of exponent ≤ p. There is an alternative approach to this result in [LSZ15]. A generalisation of this work to higher level arithmetic differential operators (in the sense of [Ber96]) is given in [GLQ10].…”

Section: Introductionmentioning

confidence: 99%

Birational characterization of Abelian varieties and ordinary Abelian varieties in characteristic p>0

Hacon¹,

Patakfalvi²,

Zhang³

2019

Duke Math. J.

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Let X/k be an abelian variety over an algebraically closed field k of characteristic p > 0. In this paper, using the Azumaya property of the sheaf of crystalline differential operators and the Morita equivalence, we show that etale locally over the Hitchin base, the moduli stack of Higgs bundles on the Frobenius twist X is equivalent to that of local systems on X. We follow the approach of [Gro16].

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“…Let C −1 1 be the inverse Cartier transform of Ogus-Vologodsky from the category of nilpotent logarithmic Higgs module of exponent ≤ p − 1 to the category of nilpotent logarithmic flat module of exponent ≤ p − 1 with respect to the chosen W 2 -lifting (X 2 , D 2 ). We refer the reader to [LSZ0] for an elementary approach to the construction of the inverse Cartier/Cartier transform in the case D 1 = ∅ and the Appendix in a special log case. A Higgs-de Rham flow over X 1 is a diagram of the following form: where the initial term (E 0 , θ 0 ) is a nilpotent graded Higgs bundle with exponent ≤ p − 1; for i ≥ 0, F il i is a Hodge filtration on the flat bundle C −1 1 (E i , θ i ) of level ≤ p−1; for i ≥ 1, (E i , θ i ) is the graded Higgs bundle associated with the de Rham bundle (C −1 1 (E i−1 , θ i−1 ), F il i−1 ).…”

Section: Introductionmentioning

confidence: 99%

Uniformization of p-adic curves via Higgs–de Rham flows

Lan

Sheng

Yang

et al. 2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. Let k be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve X 1 defined over k, there exists a lifting X of the curve to the ring W (k) of Witt vectors as well as a lifting of the Higgs bundle to a periodic Higgs bundle over X/W (k). These liftings give rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group π 1 (X K ) of the generic fiber of X. This curve X and its associated representation is in close relation to the canonical curve and its associated canonical crystalline representation in the p-adic Teichmüller theory for curves due to S. Mochizuki. Our result may be viewed as an analogue of the Hitchin-Simpson's uniformization theory of hyperbolic Riemann surfaces via Higgs bundles.

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Nonabelian Hodge theory in positive characteristic via exponential twisting

Cited by 19 publications

References 5 publications

Kodaira–Saito vanishing via Higgs bundles in positive characteristic

Kodaira–Saito vanishing via Higgs bundles in positive characteristic

Birational characterization of Abelian varieties and ordinary Abelian varieties in characteristic p>0

Uniformization of p-adic curves via Higgs–de Rham flows

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