arXiv subjects

Alexander Soibelman

Publications and source records attributed to Alexander Soibelman.

7 recordsLinked to original sources

arXiv2026

Motivic invariants of moduli stacks of Higgs bundles and bundles with connections: results and speculations

We review some results and techniques from our papers devoted to the computation of motivic classes of stacks of parabolic Higgs budles and bundles with connections on a curve. In the last section we present some directions for future work, as well as some speculations. The latter include a generalization of the P=W conjecture inspired by the work of Maxim Kontsevich and the third author on the Riemann--Hilbert correspondence for complex symplectic manifolds as well as our running project on the motivic classes of the moduli stacks of nilpotent pairs on the formal disk and geometric Satake correspondence for double affine Grassmannians.

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arXiv2024

Motivic classes of irregular Higgs bundles and irregular connections on a curve

Let $X$ be a smooth projective curve over a field of characteristic zero and let $\mathcal D$ be an effective divisor on $X$. We calculate motivic classes of various moduli stacks of parabolic vector bundles with irregular connections on $X$ and of irregular parabolic Higgs bundles on $X$ with poles bounded by $\mathcal D$ and with fully or partially fixed formal normal forms. Along the way, we obtain several results about irregular connections and irregular parabolic Higgs bundles. In particular, we give a criterion for the existence of a connection on a higher level parabolic bundle and also develop homological algebra for irregular connections and irregular parabolic Higgs bundles. We also simplify our previous results in the regular case by re-writing the formulas for motivic classes in terms of the HLV generating function.

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arXiv2021

Lecture Notes on Quiver Representations and Moduli Problems in Algebraic Geometry

These lecture notes consist of an introduction to moduli spaces in algebraic geometry, with a strong emphasis placed on examples related to the theory of quiver representations. The goal is to provide the background necessary to understand the construction of the moduli spaces of stable and semistable quiver representations due to King, Nakajima's quiver varieties, as well as some 'quiver versions' of existing moduli spaces of algebra-geometric objects.

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arXiv2020

Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve

Let $X$ be a smooth projective curve over a field of characteristic zero and let $D$ be a non-empty set of rational points of $X$. We calculate the motivic classes of moduli stacks of semistable parabolic bundles with connections on $(X,D)$ and motivic classes of moduli stacks of semistable parabolic Higgs bundles on $(X,D)$. As a by-product we give a criteria for non-emptiness of these moduli stacks, which can be viewed as a version of the Deligne-Simpson problem.

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arXiv2017

Motivic classes of moduli of Higgs bundles and moduli of bundles with connections

Let X be a smooth projective curve over a field of characteristic zero. We calculate the motivic class of the moduli stack of semistable Higgs bundles on X. We also calculate the motivic class of the moduli stack of vector bundles with connections by showing that it is equal to the class of the stack of semistable Higgs bundles of the same rank and degree zero. We follow the strategy of Mozgovoy and Schiffmann for counting Higgs bundles over finite fields. The main new ingredient is a motivic version of a theorem of Harder about Eisenstein series claiming that all vector bundles have approximately the same motivic class of Borel reductions as the degree of Borel reduction tends to $-\infty$.

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arXiv2014

The very good property for parabolic vector bundles over curves

The purpose of this note is to extend Beilinson and Drinfeld's "very good" property to moduli stacks of parabolic vector bundles on curves of genuses $g = 0$ and $g = 1$. Beilinson and Drinfeld show that for $g > 1$ a trivial parabolic structure is sufficient for the moduli stacks to be "very good". We give a necessary and sufficient condition on the parabolic structure for this property to hold in the case of lower genus.

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arXiv2014

The moduli stack of parabolic bundles over the projective line, quiver representations, and the Deligne-Simpson problem

In "Quantization of Hitchin's Integrable System and Hecke Eigensheaves", Beilinson and Drinfeld introduced the "very good" property for a smooth complex equidimensional stack. They prove that for a semisimple complex group G, the moduli stack of G-bundles over a smooth complex projective curve X is "very good", as long as X has genus g > 1. In the case of the projective line, when g = 0, this is not the case. However, the result can sometimes be extended to the projective line by introducing additional parabolic structure at a collection of marked points and slightly modifying the definition of a "very good" stack. We provide a sufficient condition for the moduli stack of parabolic vector bundles over the complex projective line to be very good. We then use this property to study the space of solutions to the Deligne-Simpson problem.

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