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Yoon-Joo Kim

Publications and source records attributed to Yoon-Joo Kim.

9 recordsLinked to original sources
arXiv2026

Obstructions for codimension one multiple fibers of Lagrangian and Calabi--Yau fibrations

We prove that compact hyper-K\"ahler manifolds with a Lagrangian fibration over a projective space have no multiple fibers in codimension one. This has several consequences for the structure of Lagrangian fibrations, including progress on Sawon's conjecture on general singular fibers, Kamenova--Lu anti-hyperbolicity, and the extension of the N\'eron model action to a big open subset of the base. We prove the same result for Calabi--Yau fibrations on simply-connected K-trivial varieties, including elliptic fibrations, with a single exceptional case: an odd-dimensional K-trivial variety with a single fiber of multiplicity 2 over the projective line. This exceptional case is realized by examples of Borisov--Nuer and their generalizations.

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arXiv2026

Meromorphic Group Actions and the Support Theorem for Lagrangian Fibrations

We prove several new results about Lagrangian fibrations on holomorphic symplectic complex spaces, under the assumption that the total space is K\"ahler (but possibly non-compact or singular) and that the base is a complex manifold. First, we construct a meromorphic action by a family of meromorphic groups. Second, we use this structure, together with Hodge-theoretic methods, to prove a version of Ng\^o's support theorem for Lagrangian fibrations. Along the way, we prove a freeness theorem for the cohomology of compact K\"ahler spaces equipped with a meromorphic group action.

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arXiv2026

Kodaira-type classification of singular fibers of some minimal abelian fibrations

Let $X \to S$ be a minimal abelian fibration of relative dimension $n$ over a curve. We classify all possible singular fibers $X_s$ having $(n-1)$-dimensional ``abelian variety parts''. This generalizes Kodaira's work on elliptic fibrations, and Matsushita and Hwang--Oguiso's work on Lagrangian fibrations into a single framework. The classification is divided into three parts: semistable, unstable, and multiple. Multiple fibers are again divided into three types: semistable-like, mixed, and unstable-like.

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arXiv2025

The Néron model of a higher-dimensional Lagrangian fibration

Let $π: X \to B$ be a projective Lagrangian fibration of a smooth symplectic variety $X$ to a smooth variety $B$. Denote the complement of the discriminant locus by $B_0 = B \setminus \operatorname{Disc}(π)$, its preimage by $X_0 = π^{-1}(B_0)$, and the complement of the critical locus by $X' = X \setminus \operatorname{Sing}(π)$. Under an assumption that the morphism $X' \to B$ is surjective, we construct (1) the Néron model of the abelian fibration $π_0 : X_0 \to B_0$ and (2) the Néron model of its automorphism abelian scheme $\operatorname{Aut}^{\circ}_{π_0} \to B_0$. Contrary to the case of elliptic fibrations, $X'$ may not be the Néron model of $X_0$; this is precisely because of the existence of flops in higher-dimensional symplectic varieties. Using such techniques, we analyze when $X' \to B$ is a torsor under a smooth group scheme and also revisit some known results in the literature.

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arXiv2024

Isotrivial Lagrangian fibrations of compact hyper-Kähler manifolds

This article initiates the study of isotrivial Lagrangian fibrations of compact hyper-Kähler manifolds. We present four foundational results that extend well-known facts about isotrivial elliptic fibrations of K3 surfaces. First, we prove that smooth fibers of an isotrivial Lagrangian fibration are isogenous to a power of an elliptic curve. Second, we exhibit a dichotomy between two types of isotrivial Lagrangian fibrations, which we call A and B. Third, we give a classification result for type A isotrivial Lagrangian fibrations. Namely, if a type A isotrivial Lagrangian fibration admits a rational section, then it is birational to one of two straightforward examples of isotrivial fibrations of hyper-Kähler manifolds of $\text{K3}^{[n]}$-type and $\text{Kum}_n$-type. Finally, we prove that a genericity assumption on the smooth fiber of an isotrivial Lagrangian fibration ensures that the fibration is of type A.

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arXiv2022

The dual Lagrangian fibration of known hyper-Kähler manifolds

Given a Lagrangian fibration $π: X \to \mathbb{P}^n$ of a compact hyper-Kähler manifold of $\text{K3}^{[n]}$, $\text{Kum}_n$, $\text{OG10}$ or $\text{OG6}$-type, we construct a natural compactification of its dual torus fibration. Specifically, this compactification is given by a quotient of $X$ by certain automorphisms acting trivially on the second cohomology and respecting the Lagrangian fibration. It is a compact hyper-Kähler orbifold with identical period mapping behavior as $X$.

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arXiv2021

The LLV decomposition of hyper-Kaehler cohomology

Looijenga--Lunts and Verbitsky showed that the cohomology of a compact hyper-Kähler manifold $X$ admits a natural action by the Lie algebra $\mathfrak{so} (4, b_2(X)-2)$, generalizing the Hard Lefschetz decomposition for compact Kähler manifolds. In this paper, we determine the Looijenga--Lunts--Verbitsky (LLV) decomposition for all known examples of compact hyper-Kähler manifolds, and propose a general conjecture on the weights occurring in the LLV decomposition, which in particular determines strong bounds on the second Betti number $b_2(X)$ of hyper-Kähler manifolds. Specifically, in the $K3^{[n]}$ and $\mathrm{Kum}_n$ cases, we give generating series for the formal characters of the associated LLV representations, which generalize the well-known Göttsche formulas for the Euler numbers, Betti numbers, and Hodge numbers for these series of hyper-Kähler manifolds. For the two exceptional cases of O'Grady we refine the known results on their cohomology. In particular, we note that the LLV decomposition leads to a simple proof for the Hodge numbers of hyper-Kähler manifolds of O'Grady 10 type. In a different direction, for all known examples of hyper-Kähler manifolds, we establish the so-called Nagai's conjecture on the monodromy of degenerations of hyper-Kähler manifolds. More consequentially, we note that Nagai's conjecture is a first step towards a more general and more natural conjecture, that we state here. Finally, we prove that this new conjecture is satisfied by the known types of hyper-Kähler manifolds.

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arXiv2020

A conjectural bound on the second Betti number for hyper-Kähler manifolds

In previous work, we noted that the known cases of hyper-Kähler manifolds satisfy a natural condition on the LLV decomposition of the cohomology; informally, the Verbitsky component is the dominant representation in the LLV decomposition. Assuming this condition holds for all hyper-Kähler manifolds, we obtain an upper bound for the second Betti number in terms of the dimension.

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