arXiv subjects

Jieao Song

Publications and source records attributed to Jieao Song.

8 recordsLinked to original sources

arXiv2026

Some divisors in the moduli space of Debarre-Voisin varieties

Let $V_{10}$ be a 10-dimensional complex vector space and let $σ\in\bigwedge^3V_{10}^\vee$ be a non-zero alternating 3-form. One can define several associated degeneracy loci: the Debarre-Voisin hyperkähler variety $X_6^σ\subset\mathrm{Gr}(6,V_{10})$, the Peskine variety $X_1^σ\subset\mathbf{P}(V_{10})$, and the hyperplane section $X_3^σ\subset \mathrm{Gr}(3,V_{10})$. We prove that when smooth, the varieties $X_6^σ$, $X_1^σ$, and $X_3^σ$ share one common integral Hodge structure, and that $X_1^σ$ and $X_3^σ$ both satisfy the integral Hodge conjecture in all degrees. This is obtained as a consequence of a detailed analysis of the geometry of these varieties along three divisors in the moduli space. On one of the divisors, an associated K3 surface $S$ of degree 6 can be constructed geometrically and the Debarre-Voisin fourfold is shown to be isomorphic to a moduli space of twisted sheaves on $S$, in analogy with the case of cubic fourfolds containing a plane.

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arXiv2026

Coble type hypersurfaces and hyperkähler fourfolds

A classical result, already observed by Coble, asserts that a genus 2 Jacobian can be embedded in $\mathbf P^8$ as the singular locus of a unique cubic hypersurface; similarly, the Kummer of a genus 3 Jacobian is embedded in $\mathbf P^7$ as the singular locus of a unique quartic hypersurface. We present a precise analogue of these results in the context of hyperkähler fourfolds: for the general member in the 20-dimensional locally complete families of polarized hyperkähler fourfolds of $\mathrm{K3}^{[2]}$-type with squares 4 or 6 and divisibility 1, we establish the existence of a unique Coble type hypersurface. As a consequence, we describe several geometric aspects of the corresponding moduli spaces.

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arXiv2026

A special Debarre-Voisin fourfold

Consider the finite simple group $\mathbf{G}:=\mathrm{PSL}(2,\mathbf{F}_{11})$ of order 660, which has an irreducible representation $V_{10}$ of dimension 10. In this note, we study a special trivector ${σ_0}\in \bigwedge^3V_{10}^\vee$ that is $\mathbf{G}$-invariant. Following the construction of Debarre-Voisin, we obtain a smooth hyperkähler fourfold $X_6^{σ_0}\subset\mathrm{Gr}(6,V_{10})$ with many symmetries. We will also look at the associated Peskine variety $X_1^{σ_0}\subset \mathbf{P}(V_{10})$, which is highly symmetric as well and admits 55 isolated singular points. It will help us to better understand the geometry of the special Debarre-Voisin fourfold $X_6^{σ_0}$. We also discuss an application of this example to the global geometry of the moduli space of Debarre-Voisin fourfolds.

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arXiv2025

Projective models for Hilbert squares of $K3$ surfaces

For a very general polarized $K3$ surface $S\subset \mathbb{P}^g$ of genus $g\ge 5$, we study the linear system on the Hilbert square $S^{[2]}$ parametrizing quadrics in $\mathbb{P}^g$ that contain $S$. We prove its very ampleness for $g\geq 7$. In the cases of genus 7 or 8, we describe in detail the projective geometry of the corresponding embedding by making use of the Mukai model for $S$. In both cases, it can be realized as a degeneracy locus on an ambient homogeneous space, in a strikingly similar fashion. In consequence, we give explicit descriptions of its ideal and syzygies. Furthermore, we extract new information on the locally complete families, in a first step towards the understanding of their projective geometry.

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arXiv2023

Kernels of categorical resolutions of nodal singularities

In this paper we study derived categories of nodal singularities. We show that for all nodal singularities there is a categorical resolution whose kernel is generated by a $2$ or $3$-spherical object, depending on the dimension. We apply this result to the case of nodal cubic fourfolds, where we describe the kernel generator of the categorical resolution as an object in the bounded derived category of the associated degree six K3 surface. This paper originated from one of the problem sessions at the Interactive Workshop and Hausdorff School "Hyperkähler Geometry", Bonn, September 6-10, 2021.

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arXiv2022

Second Chern class and Fujiki constants of hyperkähler manifolds

We study characteristic classes on hyperkähler manifolds with a view towards the Verbitsky component. The case of the second Chern class leads to a conditional upper bound on the second Betti number in terms of the Riemann--Roch polynomial, which is also valid for singular examples. We discuss the general structure of characteristic classes and the Riemann--Roch polynomial on hyperkähler manifolds using among other things Rozansky--Witten theory.

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arXiv2022

On the image of the period map for polarized hyperkähler manifolds

The moduli space for polarized hyperkähler manifolds of $\mathrm{K3}^{[m]}$-type or $\mathrm{Kum}_m$-type with a given polarization type is not necessarily connected, which is a phenomenon that only happens for $m$ large. The period map restricted to each connected component gives an open embedding into the period domain, and the complement of the image is a finite union of Heegner divisors. We give a simplified formula for the number of connected components, as well as a simplified criterion to enumerate the Heegner divisors in the complement. In particular, we show that the image of the period map may be different when restricted to different components of the moduli space.

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arXiv2021

Hilbert schemes of K3 surfaces, generalized Kummer, and cobordism classes of hyper-Kähler manifolds

We prove that the complex cobordism class of any hyper-Kähler manifold of dimension $2n$ is a unique combination with rational coefficients of classes of products of punctual Hilbert schemes of $K3$ surfaces. We also prove a similar result using the generalized Kummer varieties instead of punctual Hilbert schemes. As a key step, we establish a closed formula for the top Chern character of their tangent bundles.

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