We study the Gieseker moduli space $\mathcal{M}_{X_0}$ of stable rank-$2$, degree-$1$ vector bundles on an irreducible curve $X_0$ with one node. Its normalization is a two-step blow-up of the moduli space of generalized parabolic bundles on the normalization $\tilde X_0$, and $\mathcal{M}_{X_0}$ is recovered by gluing along a natural involution on the boundary. From this we compute $\operatorname{Pic}(\mathcal{M}_{X_0})$ and, via a Harder--Narasimhan stratification, the virtual Poincar\'e polynomial of $\mathcal{M}_{X_0}$. Since the normalization morphism is small, the intersection cohomology of $\mathcal{M}_{X_0}$ equals the ordinary cohomology of the smooth normalization, which gives a closed formula in terms of the arithmetic genus.
Let $f:X\to S$ be a smooth projective family, and fix a semisimple flat bundle on a fiber $X_0$. Its isomonodromic deformation determines a holomorphic section $\sigma_{\mathrm{dR}}:S\to M_{\mathrm{dR}}(X/S)$ of the relative de Rham moduli space. Applying the relative non-abelian Hodge correspondence fiberwise gives a...
A smooth complete intersection $X\subset\mathbb{P}^{n+2}_{\mathbb{C}}$ of two quadrics admits an interpretation as a moduli space of bundles. For $n=2$, it parametrizes stable rank $2$ parabolic bundles on $\mathbb{P}^1$ with five marked points, while for $n=3$ it parametrizes stable rank $2$ bundles of odd degree with...
The new framework of semi-tautological extensions has allowed for explicit understanding of low-degree odd cohomology groups of moduli spaces of stable curves. Let $S^*_\omega(\overline{\mathcal M}_{g,n}) \subset H^*(\overline{\mathcal M}_{g,n})$ denote the semi-tautological extension generated by $H^{11}(\overline{\ma...
Amy Bradford, Aitor Iribar López, A. Kretschmer et al.· 0 citations
The flat DHS connection $\mathcal J_{\mathrm{DHS}}$ constructed in arXiv:2602.01461 is smooth on the configuration space of $n$ points on a fixed compact Riemann surface $\Sigma$ of arbitrary genus $h$, takes values in an infinite-dimensional Lie algebra $\hat{\mathfrak t}_{h,n}$ and is invariant under the modular grou...
Eric D'Hoker, Benjamin Enriquez, Oliver Schlotterer et al.· 0 citations
For a finite abelian group $G$ acting linearly on the plane let $\mathcal{X}$ be the associated quotient stack. We analyze a fine moduli space which parameterizes zero dimensional substacks of $\mathcal{X}$ whose corresponding subschemes of the plane have a given representation type, denoted by $\mathcal{X}^{[\rho]}$....
Let $\mathcal{C}$ be a differential $\mathbb{Z}/2$-graded category over $\mathbb{C}$. Its periodic cyclic homology $HH^{per}_*(\mathcal{C})$, when viewed as a vector bundle over the formal punctured disk, is equipped with a canonical connection $\nabla^{\mathcal{C}}_{\partial_t}$ called the Getzler-Gauss-Manin connecti...
Zi-Hong Chen· 0 citations
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