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Register a new userClassification of subbundles on the Fargues-Fontaine curve
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Authors
Serin Hong
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We completely classify all subbundles of a given vector bundle on the Fargues-Fontaine curve. Our classification is given in terms of a simple and explicit condition on Harder-Narasimhan polygons. Our proof is inspired by the proof of the main theorem in [Hon19], but also involves a number of nontrivial adjustments.
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We completely classify all quotient bundles of a given vector bundle on the Fargues-Fontaine curve. As consequences, we have two additional classification results: a complete classification of all vector bundles that are generated by a fixed number of global sections and a nearly complete classification of subbundles of a given vector bundle. For the proof, we combine the dimension counting argument for moduli of bundle maps developed in [BFH+17] with a series of reduction arguments based on some reinterpretation of the classifying conditions.
We completely classify the possible extensions between semistable vector bundles on the Fargues-Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder-Narasimhan polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholzes language of diamonds. This analysis reduces our main results to a somewhat involved combinatorial problem, which we then solve via a reinterpretation in terms of the euclidean geometry of Harder-Narasimhan polygons.
We explain how to construct a cohomology theory on the category of separated quasi-compact smooth rigid spaces over $mathbf{C}_p$ (or more general base fields), taking values in the category of vector bundles on the Fargues-Fontaine curve, which extends (in a suitable sense) Hyodo-Kato cohomology when the rigid space has a semi-stable proper formal model over the ring of integers of a finite extension of $mathbf{Q}_p$. This cohomology theory factors through the category of rigid analytic motives of Ayoub.
We define and study Harder-Narasimhan filtrations on Breuil-Kisin-Fargues modules and related objects relevant to p-adic Hodge theory.
In this short notes, we prove a stronger version of Theorem 0.6 in our previous paper arXiv:1709.01485: Given a smooth log scheme $(mathcal{X} supset mathcal{D})_{W(mathbb{F}_q)}$, each stable twisted $f$-periodic logarithmic Higgs bundle $(E,theta)$ over the closed fiber $(X supset D)_{mathbb{F}_q}$ will correspond to a $mathrm{PGL}_r(mathbb{F}_{p^f})$-crystalline representation of $pi_1((mathcal{X} setminus mathcal{D})_{W(mathbb{F}_q)[frac{1}{p}]})$ such that its restriction to the geometric fundamental group is absolutely irreducible.
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