arXiv subjects

Sukjoo Lee

Publications and source records attributed to Sukjoo Lee.

10 recordsLinked to original sources

arXiv2026

Symplectic leaves of meromorphic Hitchin systems

The moduli space of meromorphic Higgs bundles admits a Poisson structure due to the independent work of Bottacin and Markman. In this paper, we revisit the symplectic leaves of this Poisson structure for the tame case. We study the partial compactification of the restricted Hitchin map on the symplectic leaves to an algebraically completely integrable system. In particular, we show that such a partial compactification is realized by the moduli spaces of $\vec{\xi}$-parabolic Higgs bundles. These same moduli spaces also provide a symplectic resolution of the normalization of the closure of the corresponding symplectic leaves. Finally, we discuss connectedness results for the corresponding Betti moduli spaces under the tame non-abelian Hodge correspondence.

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arXiv2025

Fibers of Landau-Ginzburg models and rationality

In this article, we study how the rationality of a Fano threefold is reflected in its standard mirror Landau-Ginzburg model and its deformations. The main result is that a Fano threefold is rational if and only if the monodromy around every reducible fiber of its generic mirror Landau-Ginzburg model is unipotent.

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arXiv2025

On a conjecture of Hosono-Lee-Lian-Yau

We extend the mirror construction of singular Calabi-Yau double covers, introduced by Hosono, Lee, Lian, and Yau, to a broader class of singular Calabi-Yau $(\mathbb{Z}/2)^k$-Galois covers, and prove Hodge number duality for both the original and extended mirror pairs. A main tool in our approach is an analogue of the Cayley trick, which relates the de Rham complex of the branched covers to the twisted de Rham complex of certain Landau-Ginzburg models. In particular, it reveals direct relations between the Hodge numbers of the covers and the irregular Hodge numbers of the associated Landau-Ginzburg models. This construction is independent of mirror symmetry and may be of independent interest.

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arXiv2025

Relative spectral correspondence for parabolic Higgs bundles and Deligne--Simpson problem

In this paper, we generalize the spectral correspondence for parabolic Higgs bundles established by Diaconescu--Donagi--Pantev to the relative setting. We show that the relative moduli space of $\vec{\xi}$-parabolic Higgs bundles on a curve can be realized as the relative moduli space of pure dimension one sheaves on a family of holomorphic symplectic surfaces. This leads us to formulate the image of the relative moduli space under the Hitchin map in terms of linear systems on the family of surfaces. Then we explore the relationship between the geometry of these linear systems and the so-called $OK$ condition introduced by Balasubramanian--Distler--Donagi in the context of six-dimensional superconformal field theories. As applications, we obtain (a) the non-emptiness of the moduli spaces and (b) the Deligne--Simpson problem and its higher genus analogue. In particular, we prove a conjecture proposed by Balasubramanian--Distler--Donagi that the $OK$ condition is sufficient for solving the Deligne--Simpson problem.

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arXiv2024

Irregular Hodge numbers of stacky Clarke mirror pairs

We prove a duality between the graded pieces of the irregular Hodge filtration on the twisted cohomology for a large class of Clarke mirror pairs of stacky Landau-Ginzburg models. We use this to recover results of Batyrev--Borisov, generalize results of Ebeling-Gusein-Zade-Takahashi and Krawitz, and prove results similar to those of Gross-Katzarkov-Ruddat. We apply our results to prove a generalized version of a conjecture of Katzarkov-Kontsevich-Pantev for orbifold toric complete intersections with nef anticanonical divisors and orbifold Fano stacks, and we prove the Hodge number duality result for orbifold log Calabi-Yau complete intersections. Along the way, we study the behaviour of twisted cohomology under degeneration and prove that for certain degenerations of toric Landau--Ginzburg models, irregular Hodge numbers admit a tropical realization.

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arXiv2024

Generators for the cohomology of the moduli space of irregular parabolic Higgs bundles

We prove that the pure part of the cohomology ring of the moduli space of irregular $\underlineξ$-parabolic Higgs bundles is generated by the Künneth components of the Chern classes of a universal bundle and the Chern classes of the successive quotients of a universal flag of subbundles. As an application, in the regular full-flag case, we demonstrate a similar result for the cohomology ring of the moduli spaces of parabolic and strongly parabolic Higgs bundles.

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arXiv2024

Fukaya categories of hyperplane arrangements

To a simple polarized hyperplane arrangement (not necessarily cyclic) $\mathbb{V}$, one can associate a stopped Liouville manifold (equivalently, a Liouville sector) $\left(M(\mathbb{V}),\xi\right)$, where $M(\mathbb{V})$ is the complement of finitely many hyperplanes in $\mathbb{C}^d$, obtained as the complexifications of the real hyperplanes in $\mathbb{V}$. The Liouville structure on $M(\mathbb{V})$ comes from a very affine embedding, and the stop $\xi$ is determined by the polarization. In this article, we study the symplectic topology of $\left(M(\mathbb{V}),\xi\right)$. In particular, we prove that their partially wrapped Fukaya categories are generated by Lagrangian submanifolds associated to the bounded and feasible chambers of $\mathbb{V}$. A computation of the Fukaya $A_\infty$-algebra of these Lagrangians then enables us to identity these wrapped Fukaya categories with the $\mathbb{G}_m^d$-equivariant hypertoric convolution algebras $\widetilde{B}(\mathbb{V})$ associated to $\mathbb{V}$. This confirms a conjecture of Lauda-Licata-Manion (arXiv:2009.03981) and provides evidence for the general conjecture of Lekili-Segal (arXiv:2304.10969) on the equivariant Fukaya categories of symplectic manifolds with Hamiltonian torus actions.

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arXiv2024

Mirror P=W conjecture and extended Fano/Landau-Ginzburg correspondence

The mirror P=W conjecture, formulated by Harder-Katzarkov-Przyjalkowski, predicts a correspondence between weight and perverse filtrations in the context of mirror symmetry. In this paper, we revisit this conjecture through the lens of mirror symmetry for a Fano pair $(X,D)$, where $X$ is a smooth Fano variety and $D$ is a simple normal crossing divisor. We introduce its mirror object as a multi-potential analogue of a Landau-Ginzburg (LG) model, which we call the hybrid LG model. This model is expected to capture the mirrors of all irreducible components of $D$. We study the topological aspects, particularly the perverse filtration, and the Hodge theory of hybrid LG models, building upon the work of Katzarkov-Kontsevich-Pantev. As an application, we discover an interesting upper bound on the multiplicativity of the perverse filtration for a hybrid LG model. Additionally, we propose a relative version of the homological mirror symmetry conjecture and explain how the mirror P=W conjecture naturally emerges from it.

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arXiv2023

Semi-stable degenerations of Calabi-Yau manifolds and mirror P=W conjecture

Mirror symmetry for a semi-stable degeneration of a Calabi-Yau manifold was first investigated by Doran-Harder-Thompson when the degeneration fiber is a union of two (quasi)-Fano manifolds. They propose a topological construction of a mirror Calabi-Yau that is a gluing of two Landau-Ginzburg models mirror to those Fano manifolds. We extend this construction to a general type semi-stable degeneration. As each component in the degeneration fiber comes with the simple normal crossing anti-canonical divisor, one needs the notion of a hybrid Landau-Ginzburg model, a multi-potential analogue of classical Landau-Ginzburg models. We show that these hybrid LG models can be glued to provide a topological mirror candidate of the Calabi-Yau which is also equipped with the fibration over $\mathbb{P}^N$. Furthermore, it is predicted that the perverse Leray filtration associated to this fibration is mirror to the monodromy weight filtration on the degeneration side. We explain how this can be deduced from the original mirror P=W conjecture.

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arXiv2022

Semi-polarized meromorphic Hitchin and Calabi-Yau integrable systems

It was shown by Diaconescu, Donagi and Pantev that Hitchin systems of type ADE are isomorphic to certain Calabi-Yau integrable systems. In this paper, we prove an analogous result in the setting of meromorphic Hitchin systems of type A which are known to be Poisson integrable systems. We consider a symplectization of the meromorphic Hitchin integrable system, which is a semi-polarized integrable system in the sense of Kontsevich and Soibelman. On the Hitchin side, we show that the moduli space of unordered diagonally framed Higgs bundles forms an integrable system in this sense and recovers the meromorphic Hitchin system as the fiberwise compact quotient. Then we construct a new family of quasi-projective Calabi-Yau threefolds and show that its relative intermediate Jacobian fibration, as semi-polarized integrable systems, is isomorphic to the moduli space of unordered diagonally framed Higgs bundles.

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