arXiv subjects

Qingyuan Bai

Publications and source records attributed to Qingyuan Bai.

4 recordsLinked to original sources

arXiv2026

Localizing invariants of constructible sheaves

Given an open-closed decomposition of the stratifying poset, we construct a new semi-orthogonal decomposition of the $\infty$-category of constructible sheaves on a stratified space admitting an exit-path $\infty$-category. From this we obtain a direct sum decomposition of the localizing invariants of the $\infty$-category of constructible sheaves. Since the $\ast$-pullback to the open stratum in the usual (recollement) semi-orthogonal decomposition is not strongly left adjoint, this splitting does not follow from pure sheaf theory considerations. Instead, the splitting crucially relies on the exodromy equivalence: it implies that on the level of constructible sheaves, the $\ast$-pullback to a closed stratum and the $!$-pushforward from an open stratum admit left adjoints. These new functors provide an additional semi-orthogonal decomposition (with the roles of open and closed reversed) in which the relevant functors are strongly left adjoint.

math.KT↗

arXiv2025

On the K-theory of algebraic tori

Given an algebraic torus $T$ over a field $F$, its lattice of characters $Λ$ gives rise to a topological torus $\mathfrak{T}(T)=Λ_{\mathbb R}/Λ$ with a continuous action of the absolute Galois group $G$. We construct a natural equivalence between the algebraic $K$-theory $K_{\ast}(T)$ and the equivariant homology $H^{G}_{\ast}(\mathfrak{T}(T);K_G(F))$ of the topological torus $\mathfrak{T}(T)$ with coefficients in the $G$-equivariant $K$-theory of $F$. This generalizes a computation of $K_0(T)$ due to Merkurjev and Panin. We obtain this equivalence by analyzing the motive $\mathbb{K}_{F}^{T}$ in the stable motivic category $\mathrm{SH}(F)$ of Voevodsky and Morel, where $\mathbb{K}_{F}$ is the motivic spectrum representing homotopy $K$-theory. We construct a natural comparison map $\mathfrak{F}\colon \mathbb{K}_{F}[BΛ] \to \mathbb{K}_{F}^{T}$ from the $\mathbb{K}_{F}$-homology of the étale delooping of $Λ$ to $\mathbb{K}_{F}^{T}$ as a special case of a motivic Fourier transform and prove that it is an equivalence by using a motivic Eilenberg--Moore formula for classifying spaces of tori.

math.KT↗

arXiv2025

Toric Mirror Symmetry for Homotopy Theorists

We construct functors sending torus-equivariant quasi-coherent sheaves on toric schemes over the sphere spectrum to constructible sheaves of spectra on real vector spaces. This provides a spectral lift of the toric homolgoical mirror symmetry theorem of Fang-Liu-Treumann-Zaslow (arXiv:1007.0053). Along the way, we obtain symmetric monoidal structures and functoriality results concerning those functors, which are new even over a field $k$. We also explain how the `non-equivariant' version of the theorem would follow from this functoriality via the de-equivariantization technique. As a concrete application, we obtain an alternative proof of Beilinson's linear algebraic description of quasi-coherent sheaves on projective spaces with spectral coefficients.

math.AG↗

arXiv2022

Topological Fukaya category and mirror symmetry for toric Calabi-Yau 3-orbifolds

We prove a version of homological mirror symmetry statement for toric Calabi-Yau $3$-orbifolds, thus extending arXiv:1604.06448 to the case of orbifolds under the mirror symmetry setting considered in arXiv:1604.07123. The B-model is the matrix factorization category for the toric Calabi-Yau $3$-orbifold with a superpotential; while the A-model is a topologically defined Fukaya-type category on its mirror curve.

math.AT↗