arXiv subjects

Hyeonjun Park

Publications and source records attributed to Hyeonjun Park.

15 recordsLinked to original sources

arXiv2026

Shifted symplectic rigidification

We construct shifted symplectic derived enhancements on rigidified moduli spaces of sheaves on Calabi-Yau varieties of dimension at least two. More generally, we prove that any $B\mathbb{G}_m$-action on a non-positively-shifted symplectic derived Artin stack is Hamiltonian. We provide a symplectic rigidification functor as the left adjoint to the trivial action functor in symplectic categories with Lagrangian correspondences. We also descend the Lagrangian correspondence of short exact sequences of sheaves to rigidified moduli spaces.

math.AG↗

arXiv2026

VCA: Vision-Click-Action Framework for Precise Manipulation of Segmented Objects in Target Ambiguous Environments

The reliance on language in Vision-Language-Action (VLA) models introduces ambiguity, cognitive overhead, and difficulties in precise object identification and sequential task execution, particularly in environments with multiple visually similar objects. To address these limitations, we propose Vision-Click-Action (VCA), a framework that replaces verbose textual commands with direct, click-based visual interaction using pretrained segmentation models. By allowing operators to specify target objects clearly through visual selection in the robot's 2D camera view, VCA reduces interpretation errors, lowers cognitive load, and provides a practical and scalable alternative to language-driven interfaces for real-world robotic manipulation. Experimental results validate that the proposed VCA framework achieves effective instance-level manipulation of specified target objects. Experiment videos are available at https://robrosinc.github.io/vca/.

cs.RO↗

arXiv2025

Period sheaves via perverse pullbacks

We construct period sheaves for Hamiltonian spaces, as conjectured in the work of Ben-Zvi, Sakellaridis and Venkatesh, using the perverse pullback functors introduced in the authors' previous work. We prove a dimensional reduction isomorphism (generalizing the results of Davison and Kinjo in cohomological Donaldson--Thomas theory) which implies that perverse pullbacks refine the ordinary pullback functors in constructible sheaf theory, and relate perverse pullbacks to microstalk functors. These results imply that our period sheaves recover the known constructions in the cotangent and Whittaker cases.

math.AG↗

arXiv2025

Perverse pullbacks

We define a new perverse t-exact pullback operation on derived categories of constructible sheaves which generalizes most perverse t-exact functors in sheaf theory, such as microlocalization, the Fourier-Sato transform and vanishing cycles. This operation is defined for morphisms of algebraic stacks equipped with a relative exact (-1)-shifted symplectic structure, and can be used to define cohomological Donaldson-Thomas invariants in a relative setting. We prove natural functoriality properties for perverse pullbacks, such as smooth and finite base change, compatibility with products and Verdier duality.

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arXiv2025

Cosection localization via shifted symplectic geometry

The purpose of this paper is to shed a new light on classical constructions in enumerative geometry from the view point of derived algebraic geometry. We first prove that the cosection localized virtual cycle of a quasi-smooth derived Deligne-Mumford stack with a $(-1)$-shifted closed $1$-form is equal to the virtual Lagrangian cycle of the degeneracy locus which is $(-2)$-shifted symplectic. We next establish a shifted analogue of the Lagrange multipliers method which gives us the quantum Lefschetz theorems as immediate consequences of the equality of virtual cycles. Lastly we study derived algebraic geometry enhancements of gauged linear sigma models which lead us to the relative virtual cycles in a general and natural form.

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arXiv2024

Proleptic Temporal Ensemble for Improving the Speed of Robot Tasks Generated by Imitation Learning

Imitation learning, which enables robots to learn behaviors from demonstrations by human, has emerged as a promising solution for generating robot motions in such environments. The imitation learning-based robot motion generation method, however, has the drawback of depending on the demonstrator's task execution speed. This paper presents a novel temporal ensemble approach applied to imitation learning algorithms, allowing for execution of future actions. The proposed method leverages existing demonstration data and pre-trained policies, offering the advantages of requiring no additional computation and being easy to implement. The algorithms performance was validated through real-world experiments involving robotic block color sorting, demonstrating up to 3x increase in task execution speed while maintaining a high success rate compared to the action chunking with transformer method. This study highlights the potential for significantly improving the performance of imitation learning-based policies, which were previously limited by the demonstrator's speed. It is expected to contribute substantially to future advancements in autonomous object manipulation technologies aimed at enhancing productivity.

cs.RO↗

arXiv2024

Shifted symplectic pushforwards

We introduce how to pushforward shifted symplectic fibrations along base changes. This is achieved by considering symplectic forms that are closed in a stronger sense. Examples include: symplectic zero loci and symplectic quotients. Observing that twisted cotangent bundles are symplectic pushforwards, we obtain an equivalence between symplectic fibrations and Lagrangians to critical loci. We provide two local structure theorems for symplectic fibrations: a smooth local structure theorem for higher stacks via symplectic zero loci and twisted cotangents, and an {é}tale local structure theorem for $1$-stacks with reductive stabilizers via symplectic quotients of the smooth local models. We resolve deformation invariance issue in Donaldson-Thomas theory of Calabi-Yau $4$-folds. Abstractly, we associate virtual Lagrangian cycles for oriented $(-2)$-symplectic fibrations as unique functorial bivariant classes over the exact loci. For moduli of perfect complexes, we show that the exact loci consist of deformations for which the $(0,4)$-Hodge pieces of the second Chern characters remain zero.

math.AG↗

arXiv2024

Cohomological Hall algebras for 3-Calabi-Yau categories

The aim of this paper is to construct the cohomological Hall algebras for $3$-Calabi--Yau categories admitting a strong orientation data. This can be regarded as a mathematical definition of the algebra of BPS states, whose existence was first mathematically conjectured by Kontsevich and Soibelman. Along the way, we prove Joyce's conjecture on the functorial behaviour of the Donaldson--Thomas perverse sheaves for the attractor Lagrangian correspondence of $(-1)$-shifted symplectic stacks. This result allows us to construct a parabolic induction map for cohomological Donaldson--Thomas invariants of $G$-local systems on $3$-manifolds for a reductive group $G$, which can be regarded as a $3$-manifold analogue of the Eisenstein series functor in the geometric Langlands program.

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arXiv2024

The stacky concentration theorem

We give a sufficient criterion for the Chow or algebraic bordism groups of an algebraic stack, localized at a set of Chern classes of line bundles, to be concentrated in some closed substack. This is a vast generalization of the torus fixed-point localization theorem in equivariant intersection theory, which is the special case of the stack quotient of a scheme $X$ by an action of a torus $T$. Taking on the one hand an algebraic stack in place of $X$, we deduce a generalization of torus localization to algebraic stacks. Taking on the other hand any algebraic group $G$ instead of $T$, we obtain a localization theorem in $G$-equivariant intersection theory.

math.AG↗

arXiv2024

Counting surfaces on Calabi-Yau 4-folds II: $\mathrm{DT}$-$\mathrm{PT}_0$ correspondence

This is the second part in a series of papers on counting surfaces on Calabi-Yau 4-folds. In this paper, we introduce $K$-theoretic $\mathrm{DT}, \mathrm{PT}_0, \mathrm{PT}_1$ invariants and conjecture a $\mathrm{DT}$-$\mathrm{PT}_0$ correspondence. For certain tautological insertions, we derive Lefschetz principles in both the compact and toric case allowing reductions to 3-dimensional $\mathrm{DT}, \mathrm{PT}$ invariants. We also develop a topological vertex and conjecture a $\mathrm{DT}$-$\mathrm{PT}_0$ vertex correspondence. These methods enable us to verify our conjectures in several examples.

math.AG↗

arXiv2022

Counting surfaces on Calabi-Yau 4-folds I: Foundations

This is the first part in a series of papers on counting surfaces on Calabi-Yau 4-folds. Besides the Hilbert scheme of 2-dimensional subschemes, we introduce \emph{two} types of moduli spaces of stable pairs. We show that all three moduli spaces are related by GIT wall-crossing and parametrize stable objects in the bounded derived category. We construct \emph{reduced} Oh-Thomas virtual cycles on the moduli spaces via Kiem-Li cosection localization and prove that they are deformation invariant along Hodge loci. As an application, we show that the variational Hodge conjecture holds for any family of Calabi-Yau 4-folds supporting a non-zero reduced virtual cycle.

math.AG↗

arXiv2022

Virtual localization revisited

Let $T$ be a split torus acting on an algebraic scheme $X$ with fixed locus $Z$. Edidin and Graham showed that on localized $T$-equivariant Chow groups, (a) push-forward $i_*$ along $i : Z \to X$ is an isomorphism, and (b) when $X$ is smooth the inverse $(i_*)^{-1}$ can be described via Gysin pullback $i^!$ and cap product with $e(N)^{-1}$, the inverse of the Euler class of the normal bundle $N$. In this paper we show that (b) still holds when $X$ is a quasi-smooth derived scheme (or Deligne-Mumford stack), using virtual versions of the operations $i^!$ and $(-)\cap e(N)^{-1}$. As a corollary we prove the virtual localization formula $[X]^{vir} = i_* ([Z]^{vir} \cap e(N^{vir})^{-1})$ of Graber-Pandharipande without global resolution hypotheses and over arbitrary base fields. We include an appendix on fixed loci of group actions on (derived) stacks which should be of independent interest.

math.AG↗

arXiv2021

Virtual pullbacks in Donaldson-Thomas theory of Calabi-Yau 4-folds

Recently, Oh and Thomas constructed algebraic virtual cycles for moduli spaces of sheaves on Calabi-Yau 4-folds. The purpose of this paper is to provide a virtual pullback formula between these Oh-Thomas virtual cycles. We find a natural compatibility condition between 3-term symmetric obstruction theories that induces a virtual pullback formula. There are two types of applications. Firstly, we introduce a Lefschetz principle in Donaldson-Thomas theory, which relates the tautological DT4 invariants of a Calabi-Yau 4-fold with the DT3 invariants of its divisor. As corollaries, we prove the Cao-Kool conjecture on the tautological Hilbert scheme invariants for very ample line bundles and the Cao-Kool-Monavari conjecture on the tautological DT/PT correspondence for line bundles with Calabi-Yau divisors when the tautological complexes are vector bundles. Secondly, we present a correspondence between the Oh-Thomas virtual cycles on the moduli spaces of pairs and the moduli spaces of sheaves by combining the virtual pullback formula and a pushforward formula for virtual projective bundles. As corollaries, we prove the Cao-Maulik-Toda conjecture on the primary PT/GV correspondence for irreducible curve classes and the Cao-Toda conjecture on the primary JS/GV correspondence under the coprime condition, assuming the Cao-Maulik-Toda conjecture on the primary Katz/GV correspondence. Moreover, we also prove tautological versions of these two correspondences.

math.AG↗

arXiv2021

Virtual intersection theories

We construct virtual fundamental classes in all intersection theories including Chow theory, K-theory and algebraic cobordism for quasi-projective Deligne-Mumford stacks with perfect obstruction theories and prove the virtual pullback formula, the virtual torus localization formula and cosection localization principle.

math.AG↗

arXiv2020

Localizing virtual cycles for Donaldson-Thomas invariants of Calabi-Yau 4-folds

Recently Oh-Thomas constructed a virtual cycle $[X]^{\mathrm{vir}}\in A_*(X)$ for a quasi-projective moduli space $X$ of stable sheaves or complexes over a Calabi-Yau 4-fold against which DT4 invariants may be defined as integrals of cohomology classes. In this paper, we prove that the virtual cycle localizes to the zero locus $X(σ)$ of an isotropic cosection $σ$ of the obstruction sheaf $Ob_X$ of $X$ and construct a localized virtual cycle $[X]^{\mathrm{vir}}_{\mathrm{loc}}\in A_*(X(σ))$. This is achieved by further localizing the Oh-Thomas class which localizes Edidin-Graham's square root Euler class of a special orthogonal bundle. When the cosection $σ$ is surjective so that the virtual cycle vanishes, we construct a reduced virtual cycle $[X]^{\mathrm{vir}}_{\mathrm{red}}$. As an application, we prove DT4 vanishing results for hyperkähler 4-folds. All these results hold for virtual structure sheaves and K-theoretic DT4 invariants.

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