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Hanfei Guo

Publications and source records attributed to Hanfei Guo.

8 recordsLinked to original sources

Derived Torelli for Kuznetsov components of cubic fourfolds

We prove that every orientation-preserving Hodge isometry between the integral Mukai lattices of the Kuznetsov components of two smooth cubic fourfolds is induced by an exact equivalence. The proof uses Markman--Mehrotra's comonad reconstruction method to recover both Kuznetsov components from a common Fourier--Mukai comonad on a moduli space of stable objects. As an application, we prove that the Kuznetsov components of two smooth cubic fourfolds are equivalent whenever their Fano varieties of lines are derived equivalent.

math.AG↗

Stable atomic vector bundles are 1-obstructed

We prove that every stable atomic (twisted) vector bundle on a projective hyper-Kähler manifold is 1-obstructed. In the appendix, we construct a simple atomic object that is not 1-obstructed, providing a counterexample to Beckmann's conjecture.

math.AG↗

Receptive-field-constrained stimulus optimization for human early and intermediate visual cortex

An ongoing challenge in sensory neuroscience is to characterize the feature dimensions encoded by cortical populations. Recent approaches probe feature selectivity in a data-driven way, by synthesizing a most-exciting-input (MEI) for a target neural population. While this approach has been successfully applied to human higher visual cortex using fMRI data, generating MEIs for early- and mid-level retinotopic visual areas requires additional modeling constraints due to small receptive field sizes. To address this challenge, we introduce two novel MEI generation frameworks, Receptive Field Diffusion for Visual Exploration (RF-DiVE) and Receptive Field Gradient Optimization (RF-GO). Both methods use a population receptive field (pRF)-constrained voxelwise encoding model; RF-DiVE combines this with a pretrained latent diffusion model, while RF-GO uses regularized gradient ascent. When applied to single voxels in retinotopically defined areas V1-hV4, using data from the Natural Scenes Dataset, we obtain MEIs that exhibit consistent structure within the pRF, suggesting selectivity for local features like contour, color, and texture. We systematically compare MEIs generated by RF-DiVE and RF-GO using two encoding backbones, performing in-silico validation of predicted responses to MEIs using independent encoding models. Across all methods and all visual areas, MEIs elicit higher model-predicted responses than the most activating natural images. We further find that the choice of generation framework and encoding backbone differentially affects MEI properties, including their visual appearance, structural interpretability, and cross-model generalizability. These results offer a new approach for performing data-driven characterization of spatial and feature selectivity across human visual cortex.

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Atomic sheaves on hyper-Kähler manifolds via Bridgeland moduli spaces

In this paper, we provide new examples of 1-obstructed and atomic sheaves on an infinite series of locally complete families of projective hyper-Kähler manifolds. More precisely, (1) we prove that the fixed loci of the natural anti-symplectic involutions on the moduli spaces of stable objects in the Kuznetsov component $\mathcal{K}u(X)$ of a Gushel--Mukai fourfold $X$ are 1-obstructed Lagrangian submanifolds, (2) we construct a family of immersed atomic Lagrangian submanifolds on each moduli space of stable objects in $\mathcal{K}u(X)$, and (3) we construct non-rigid projectively hyperholomorphic twisted bundles on any hyper-Kähler manifold of $\mathrm{K3^{[n]}}$-type for infinitely many $n$. Additionally, we discuss examples of atomic Lagrangian submanifolds satisfying $b_1=20$ in a family of hyper-Kähler manifolds of $\mathrm{K3^{[2]}}$-type, as well as atomic sheaves supported on non-atomic Lagrangians.

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Lagrangian families of Bridgeland moduli spaces from Gushel-Mukai fourfolds

Let $X$ be a very general Gushel-Mukai (GM) variety of dimension $n\geq 4$, and let $Y$ be a smooth hyperplane section. There are natural pull-back and push-forward functors between the semi-orthogonal components (known as the Kuznetsov components) of the derived categories of $X$ and $Y$. In this paper, we prove that the Bridgeland stability of objects is preserved by both pull-back and push-forward functors. We then explore various applications of this result, such as constructing an $8$-dimensional smooth family of Lagrangian subvarieties for each moduli space of stable objects in the Kuznetsov component of a general GM fourfold and proving the projectivity of the moduli spaces of semistable objects of any class in the Kuznetsov component of a general GM threefold, as conjectured by Perry, Pertusi, and Zhao.

math.AG↗

Double EPW cubes from twisted cubics on Gushel-Mukai fourfolds

In this paper, we conduct the first systematic investigation of twisted cubics on Gushel-Mukai (GM) fourfolds. We then study the double EPW cube, a 6-dimensional hyperkähler manifold associated with a general GM fourfold $X$, through the Bridgeland moduli space, and show that it is the maximal rationally connected (MRC) quotient of the Hilbert scheme of twisted cubics on $X$. We also prove that a general double EPW cube admits a covering by Lagrangian subvarieties constructed from the Hilbert schemes of twisted cubics on GM threefolds, which provides a new example for a conjecture of O'Grady.

math.AG↗

Conics on Gushel-Mukai fourfolds, EPW sextics and Bridgeland moduli spaces

We identify the double dual EPW sextic $\widetilde{Y}_{A^{\perp}}$ and the double EPW sextic $\widetilde{Y}_A$, associated with a very general Gushel-Mukai fourfold $X$, with the Bridgeland moduli spaces of stable objects of character $Λ_1$ and $Λ_2$ in the Kuznetsov component $\mathcal{K}u(X)$. This provides an affirmative answer to a question of Perry-Pertusi-Zhao. As an application, we prove a conjecture of Kuznetsov-Perry for very general Gushel-Mukai fourfolds.

math.AG↗

A moduli theoretic approach to Lagrangian subvarieties of hyperkähler varieties: Examples

We propose two conjectures on a moduli theoretic approach to constructing Lagrangian subvarieties of hyperkähler varieties arising from the Kuznetsov components of cubic fourfolds or Gushel--Mukai fourfolds. Then we verify the conjectures in several cases, recovering classical examples. As a corollary, we confirm a conjecture of O'Grady in several instances on the existence of Lagrangian covering families for hyperkähler varieties.

math.AG↗