Serin Hong

Interests

Research

number theory, algebraic geometry, representation theory, p-adic geometry, Langlands program

Teaching

number theory, algebraic geometry, linear algebra

Courses

Scholarly Contributions

Journals/Publications

• Hong, S. (2024). On nonemptiness of Newton strata in the $B_\mathrm{dR}^+$-Grassmannian for $\mathrm{GL}_n$. Documenta Mathematica, 29(5), 1059–1084. doi:10.4171/DM/970
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  We study the Newton stratification in the $B_\mathrm{dR}^+$-Grassmannian for $\mathrm{GL}_n$ associated to an arbitrary (possibly nonbasic) element of $B(\mathrm{GL}_n)$. Our main result classifies all nonempty Newton strata in an arbitrary minuscule Schubert cell. For a large class of elements in $B(\mathrm{GL}_n)$, our classification is given by some explicit conditions in terms of Newton polygons. For the proof, we proceed by induction on n using a previous result of the author that classifies all extensions of two given vector bundles on the Fargues-Fontaine curve.
• Hong, S. (2023).

  Classification of quotient bundles on the Fargues–Fontaine curve

  . Selecta Mathematica, New Series, 29(20). doi:10.1007/s00029-022-00819-6
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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 subsheaves of a given vector bundle. For the proof, we combine the dimension counting argument for moduli of bundle maps developed in Birkbeck et al. (J Inst Math Jussieu 21:487–532, 2022) with a series of reduction arguments based on some reinterpretation of the classifying conditions.
• Hong, S. (2023).

  On certain extensions of vector bundles in p-adic geometry

  . Mathematical Research Letters, 30(5), 1463-1483. doi:10.4310/MRL.2023.v30.n5.a6
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  Given two arbitrary vector bundles on the Fargues-Fontaine curve, we give an explicit criterion in terms of Harder-Narasimhan polygons on whether they realize a semistable vector bundle as their extensions. Our argument is largely combinatorial and builds upon the dimension analysis of certain moduli spaces of bundle maps developed in [BFH+22].
• Birkbeck, C., Feng, T., Hansen, D. J., Hong, S., Li, Q., Wang, A., & Ye, L. (2022).

  Extensions of vector bundles on the Fargues-Fontaine curve

  . Journal of the Institute of Mathematics Jussieu, 21(2), 487-532. doi:10.1017/s1474748020000183
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  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 Scholze's 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.
• Hong, S. (2021).

  Classification of subbundles on the Fargues–Fontaine curve

  . Algebra & Number Theory, 15(5), 1127-1156. doi:10.2140/ant.2021.15.1127
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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.
• Hong, S. (2019).

  On the Hodge–Newton filtration for p-divisible groups of Hodge type

  . Mathematische Zeitschrift, 291, 473-497. doi:10.1007/s00209-018-2092-3
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  A p-divisible group, or more generally an F-crystal, is said to be Hodge-Newton reducible if its Hodge polygon passes through a break point of its Newton polygon. Katz proved that Hodge-Newton reducible F-crystals admit a canonical filtration called the Hodge-Newton filtration. The notion of Hodge-Newton reducibility plays an important role in the deformation theory of p-divisible groups; the key property is that the Hodge-Newton filtration of a p-divisible group over a field of characteristic p can be uniquely lifted to a filtration of its deformation. We generalize Katz's result to F-crystals that arise from an unramified local Shimura datum of Hodge type. As an application, we give a generalization of Serre-Tate deformation theory for local Shimura data of Hodge type. We also apply our deformation theory to study some congruence relations on Shimura varieties of Hodge type.
• Hong, S. (2018).

  Harris-Viehmann conjecture for Hodge-Newton reducible Rapoport-Zink spaces

  . Journal of the London Mathematical Society, 98(3), 733-752. doi:10.1112/jlms.12160
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  Rapoport–Zink spaces, or more generally local Shimura varieties, are expected to provide geometric realization of the local Langlands correspondence via their l-adic cohomology. Along this line is a conjecture by Harris and Viehmann, which roughly says that when the underlying local Shimura datum is not basic, the l-adic cohomology of the local Shimura variety is parabolically induced. We verify this conjecture for Rapoport–Zink spaces which are Hodge type and Hodge–Newton reducible. The main strategy is to embed such a Rapoport–Zink space into an appropriate space of EL type, for which the conjecture is already known to hold by the work of Mantovan.