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Register a new userIndigenous bundles with nilpotent $p$-curvature
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We study indigenous bundles in characteristic p>0 with nilpotent p-curvature, and show that they correspond to so-called deformation data. Using this equivalence, we translate the existence problem for deformation data into the existence of polynomial solutions of certain differential equations with additional properties. As in application, we show that P^1 minus four points is hyperbolically ordinary (in the sense of Mochizuki. We also give a concrete application to existence of deformation data with fixed local invariants.
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We use Scholzes framework of diamonds to gain new insights in correspondences between $p$-adic vector bundles and local systems. Such correspondences arise in the context of $p$-adic Simpson theory in the case of vanishing Higgs fields. In the present paper we provide a detailed analysis of local systems on diamonds for the etale, pro-etale, and the $v$-topology, and study the structure sheaves for all three topologies in question. Applied to proper adic spaces of finite type over $mathbb{C}_p$ this enables us to prove a category equivalence between $mathbb{C}_p$-local systems with integral models, and modules under the $v$-structure sheaf which modulo each $p^n$ can be trivialized on a proper cover. The flexibility of the $v$-topology together with a descent result on integral models of local systems allows us to prove that the trivializability condition in the module category may be checked on any normal proper cover. This result leads to an extension of the parallel transport theory by Deninger and the second author to vector bundles with numerically flat reduction on a proper normal cover.
Given three arbitrary vector bundles on the Fargues-Fontaine curve where one of them is assumed to be semistable, we give an explicit and complete criterion in terms of Harder-Narasimha polygons on whether there exists a short exact sequence among them. Our argument is based on a dimension analysis of certain moduli spaces of bundle maps and bundle extensions using Scholzes theory of diamonds.
Let T -> S be a finite flat morphism of degree two between regular integral schemes of dimension at most two (and with 2 invertible), having regular branch divisor D. We establish a bijection between Azumaya quaternion algebras on T and quadric surface bundles over S with simple degeneration along D. This is a manifestation of the exceptional isomorphism A_1^2 = D_2 degenerating to the exceptional isomorphism A_1 = B_1. In one direction, the even Clifford algebra yields the map. In the other direction, we show that the classical algebra norm functor can be uniquely extended over the discriminant divisor. Along the way, we study the orthogonal group schemes, which are smooth yet nonreductive, of quadratic forms with simple degeneration. Finally, we provide two surprising applications: constructing counter-examples to the local-global principle for isotropy, with respect to discrete valuations, of quadratic forms over surfaces; and a new proof of the global Torelli theorem for very general cubic fourfolds containing a plane.
We study arithmetic of the algebraic varieties defined over number fields by applying Lagrange interpolation to fibrations. Assuming a conjecture of M. Stoll, we show, for Ch^atelet surface bundles over curves, that the violation of Hasse principle being accounted for by the Brauer-Manin obstruction is not invariant under an arbitrary finite extension of the ground field.
We investigate degenerations of syzygy bundles on plane curves over $p$-adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the special fiber consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fiber is a syzygy bundle and which become trivial when restricted to each projective line in the special fiber. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamental open problem in $p$-adic Simpson theory to determine the category of Higgs bundles corresponding to continuous representations of the etale fundamental group of a curve. Faltings $p$-adic Simpson correspondence and work of Deninger and the second author shows that bundles with Higgs field zero and potentially strongly semistable reduction fall into this category. Hence the results in the present paper determine a class of syzygy bundles on plane curves giving rise to a $p$-adic local system. We apply our methods to a concrete example on the Fermat curve suggested by Brenner and prove that this bundle has potentially strongly semistable reduction.
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