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Register a new userSemistable modules over Lie algebroids in positive characteristic
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Authors
Adrian Langer
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We study Lie algebroids in positive characteristic and moduli spaces of their modules. In particular, we show a Langtons type theorem for the corresponding moduli spaces. We relate Langtons construction to Simpsons construction of gr-semistable Griffiths transverse filtration. We use it to prove a recent conjecture of Lan-Sheng-Zuo that semistable systems of Hodge sheaves on liftable varieties in positive characteristic are strongly semistable.
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We show a few basic results about moduli spaces of semistable modules over Lie algebroids. The first result shows that such moduli spaces exist for relative projective morphisms of noetherian schemes, removing some earlier constraints. The second result proves general separatedness Langton type theorem for such moduli spaces. More precisely, we prove S-completness of some moduli stacks of semistable modules. In some special cases this result identifies closed points of the moduli space of Gieseker semistable sheaves on a projective scheme and of the Donaldson--Uhlenbeck compactification of the moduli space of slope stable locally free sheaves on a smooth projective surface. The last result generalizes properness of Hitchins morphism and it shows properness of so called Hodge-Hitchin morphism defined in positive characteristic on the moduli space of Gieseker semistable integrable t-connections in terms of the p-curvature morphism. This last result was proven in the curve case by de Cataldo and Zhang using completely different methods.
We study geometric representation theory of Lie algebroids. A new equivalence relation for integrable Lie algebroids is introduced and investigated. It is shown that two equivalent Lie algebroids have equivalent categories of infinitesimal actions of Lie algebroids. As an application, it is also shown that the Hamiltonian categories for gauge equivalent Dirac structures are equivalent as categories.
We construct a family of homomorphisms between Weyl modules for affine Lie algebras in characteristic p, which supports our conjecture on the strong linkage principle in this context. We also exhibit a large class of reducible Weyl modules beyond level one, for p not necessarily small.
Let $G$ be a reductive group, and let $X$ be an algebraic curve over an algebraically closed field $k$ with positive characteristic. We prove a version of nonabelian Hodge correspondence for $G$-local systems over $X$ and $G$-Higgs bundles over the Frobenius twist $X$ with first order poles. To obtain a general statement of the correspondence, we introduce the language of parahoric group schemes to establish the correspondence.
We study the existence of Fuchsian differential equations in positive characteristic with nilpotent p-curvature, and given local invariants. In the case of differential equations with logarithmic local mononodromy, we determine the minimal possible degree of a polynomial solution.
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