arXiv subjects

David Zhiyuan Bai

Publications and source records attributed to David Zhiyuan Bai.

3 recordsLinked to original sources

arXiv2026

Hilbert schemes of elliptic surfaces: group actions and derived categories

Let $X\to C$ be an elliptic surface with integral fibers and a section. The Hilbert scheme $X^{[n]}$ fibers over $C^{[n]}$. We construct a commutative group scheme over the entire base $C^{[n]}$ that embeds as an open subscheme of the Hilbert scheme, such that its action on itself extends to the entirety of $X^{[n]}$. We show that the action is $δ$-regular in the sense of Ngô. Using the derived McKay correspondence, we construct an exact autoequivalence of $D^b\operatorname{Coh}(X^{[n]})$ whose kernel is a maximal Cohen-Macaulay sheaf on the fiber product. We show that this Fourier-Mukai transform intertwines with our group action, i.e. theorem of the square holds. We also discuss the case without a section using the theory of Tate-Shafarevich twists.

math.AG↗

arXiv2026

Non-reduced components of global nilpotent cones

We determine the non-reduced components of global nilpotent cones in various cases of interest. In particular, under the appropriate coprimality conditions, we show: (1) the global nilpotent cone for an $L$-twisted $\operatorname{GL}_r$-Hitchin fibration associated to a curve $C$ of genus $g\ge 2$ is nowhere reduced, where $L$ is either the canonical bundle or has degree greater than $2g-2$; (2) the global nilpotent cone for a moduli space of one-dimensional sheaves on a K3, abelian, or del Pezzo surface is nowhere reduced; (3) suppose $\ell$ is a primitive, basepoint-free, big and nef class on a K3 surface, then a general fiber of a Beauville-Mukai system for the class $r\ell$ has primitive homology class if and only if $r=1$. Our methods include group scheme actions on Lagrangian fibrations, a GIT-stratification of global nilpotent cones of Hitchin fibrations, and deformation to the normal cone.

math.AG↗

arXiv2026

Topology of projective Tate-Shafarevich twists

A Tate-Shafarevich twist $X^ϕ\to B$ of a fibration $X\to B$ modifies it by a $1$-cocycle of flows of vector fields relative to the base, locally in the analytic topology. Saccà conjectured that the total spaces of two projective Lagrangian fibrations related by such a twist are deformation-equivalent. Assuming that the class of the twist is torsion (which is often equivalent to the twist being realizable in the étale topology), we show that there is an isomorphism $H^\ast(X;\mathbb Q)\cong H^\ast(X^ϕ;\mathbb Q)$ of graded vector spaces that respects (1) the Hodge structures and (2) the Hodge-Riemann pairing. Consequently, the rational Beauville-Bogomolov-Fujiki lattices of these two spaces are Hodge-similar. Assuming further that $B$ is smooth, and both the original fibration and its twist admit $C^\infty$-sections, we show Saccà's conjecture using the theory of degenerate twistor deformations.

math.AG↗