arXiv subjects

Olga Trapeznikova

Publications and source records attributed to Olga Trapeznikova.

5 recordsLinked to original sources

arXiv2026

Parabolic bundles and the intersection cohomology of moduli spaces of vector bundles on curves

The study of the intersection cohomology of moduli spaces of semistable bundles was initiated by Frances Kirwan in the 1980s. In this paper, we develop a self-contained, purely geometric framework to study the intersection cohomology of the moduli spaces of semistable degree-0 vector bundles of arbitrary rank on Riemann surfaces. Motivated by the 2015 work of Mozgovoy and Reineke, our approach applies the Beilinson-Bernstein-Deligne-Gabber Decomposition Theorem to the forgetful map from a parabolic moduli space. We give a detailed description of the topology of this map, the geometry of its fibers, and the precise structure of the relevant local systems, making systematic use of the multiplicative structures of the theory. As a global application of this machinery, we obtain a geometric proof of a recursive formula that reduces the calculation of the intersection Betti numbers of the degree-0 moduli spaces to the known formulas for the smooth, degree-1 moduli spaces.

math.AG↗

arXiv2026

Intersection theory on singular moduli spaces of vector bundles: a parabolic approach

We present explicit formulas for the intersection pairing in the intersection cohomology of the moduli space $M_0(r)$ of rank-$r$, degree-$0$ semistable bundles on a Riemann surface. The key idea is to realize this intersection cohomology as a canonical subspace of the cohomology of a smooth moduli space of parabolic bundles, where the pairing can be computed via the Hecke correspondence and the Jeffrey-Kirwan iterated residue formulas. This approach provides a simpler alternative to the blow-up construction of Jeffrey-Kirwan-Kiem-Woolf, yielding formulas for the intersection pairing on $M_0(r)$, for arbitrary $r$, with a clear geometric interpretation.

math.AG↗

arXiv2023

Intersection cohomology of type-A toric varieties

Type-A toric varieties may be obtained as GIT quotients with respect to a torus action with weights corresponding to roots of the group $SL(k)$ for some $k>1$. These varieties appear in various important applications, in particular, as normal cones to strata in moduli spaces of vector bundles. In this paper, we describe the intersection Betti numbers of these varieties, and those of some associated projective varieties. We present an elegant combinatorial model for these numbers, and, using the work of Hausel and Sturmfels, we show that the relevant intersection cohomology groups are endowed with a canonical product structure.

math.AG↗

arXiv2022

Tautological bundles on parabolic moduli spaces: Euler characteristics and Hecke correspondences

We calculate the Euler characteristic of associated vector bundles over the moduli spaces of stable parabolic bundles on smooth curves. Our method is based on a wall-crossing technique from Geometric Invariant Theory, certain iterated residue calculus and the tautological Hecke correspondence. Our work was motivated by the results of Teleman and Woodward on the index of K-theory classes on moduli stacks.

math.AG↗

arXiv2022

The parabolic Verlinde formula: iterated residues and wall-crossings

We give a new proof for the parabolic Verlinde formula in all ranks based on a comparison of wall-crossings in Geometric Invariant Theory and certain iterated residue functionals. On the way, we develop a tautological variant of Hecke correspondences, calculate the Hilbert polynomials of the moduli spaces, and present a new, transparent, local approach to the rho-shift problem of the theory.

math.AG↗