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Roman Fedorov

Publications and source records attributed to Roman Fedorov.

At least 19 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.

math.AG↗
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.

math.AG↗
arXiv2023

Unramified Grothendieck-Serre for isotropic groups

The Grothendieck-Serre conjecture predicts that every generically trivial torsor under a reductive group $G$ over a regular semilocal ring $R$ is trivial. We establish this for unramified $R$ granted that $G^{\mathrm{ad}}$ is totally isotropic, that is, has a "maximally transversal" parabolic $R$-subgroup. We also use purity for the Brauer group to reduce the conjecture for unramified $R$ to simply connected $G$--a much less direct such reduction of Panin had been a step in solving the equal characteristic case of Grothendieck-Serre. We base the group-theoretic aspects of our arguments on the geometry of the stack $\mathrm{Bun}_G$, instead of the affine Grassmannian used previously, and we quickly reprove the crucial weak $\mathbb{P}^1$-invariance input: for any reductive group $H$ over a semilocal ring $A$, every $H$-torsor $\mathscr{E}$ on $\mathbb{P}^1_A$ satisfies $\mathscr{E}|_{\{t = 0\}} \simeq \mathscr{E}|_{\{t = \infty\}}$. For the geometric aspects, we develop reembedding and excision techniques for relative curves with finiteness weakened to quasi-finiteness, thus overcoming a known obstacle in mixed characteristic, and show that every generically trivial torsor over $R$ under a totally isotropic $G$ trivializes over every affine open of $\mathrm{Spec}(R) \setminus Z$ for some closed $Z$ of codimension $\ge 2$.

math.AG↗
arXiv2023

Generically isotropic reductive group schemes are locally isotropic

Let $R$ be a semilocal geometrically factorial Noetherian domain of characteristic zero. We show that a reductive $R$-group scheme is isotropic if it is generically isotropic. We derive various consequences, in particular for the Grothenieck-Serre conjecture and for homotopic invariance of torsors.

math.AG↗
arXiv2022

Unramified Grothendieck-Serre for simply-connected group schemes satisfying an isotropy condition via unipotent chains

We prove a case of the Grothendieck-Serre conjecture: let $R$ be a Noetherian semilocal flat algebra over a Dedekind domain such that all fibers of $R$ are geometrically regular; let $G$ be a simply-connected reductive $R$-group scheme having a strictly proper parabolic subgroup scheme. Then a $G$-torsor over $R$ is trivial, provided that it is trivial over the total ring of fractions of $R$. We also simplify the proof of the conjecture in the quasi-split unramified case. The argument is based on the notion of a unipotent chain of torsors that we introduce. We also prove that if $R$ is a Noetherian normal domain and $G$ is as above, then for any generically trivial torsor over an open subset $U$ of the spectrum of $R$, there is a closed subset $Z$ of the spectrum of $R$ of codimension at least two such the torsor trivializes over every affine scheme that factors through $U-Z$.

math.AG↗
arXiv2021

Satake equivalence for Hodge modules on affine Grassmannians

For a reductive group $G$ we equip the category of $G_\mathcal{O}$-equivariant polarizable pure Hodge modules on the affine Grassmannian $\mathrm{Gr}_G$ with a structure of neutral Tannakian category. We show that it is equivalent to a twisted tensor product of the category of representations of the Langlands dual group and the category of pure polarizable Hodge structures.

math.AG↗
arXiv2021

On the purity conjecture of Nisnevich for torsors under reductive group schemes

Let $R$ be a regular semilocal integral domain containing an infinite field $k$. Let $f\in R$ be an element such that for all maximal ideals $\mathfrak m$ of $R$ we have $f\notin\mathfrak m^2$. Let $\mathbf G$ be a reductive group scheme over $R$. Under an isotropy assumption on $\mathbf G$ we show that a $\mathbf G$-torsor over the localization $R_f$ is trivial, provided it is rationally trivial. We show that it is not true without the isotropy assumption. Finally, if $\mathbf G$ is a commutative group scheme of multiplicative type and the regular semilocal ring contains a field of characteristic zero, we prove an analogue of Nisnevich purity conjecture for higher \'etale cohomology groups. The first statement is derived from its abstract version concerning presheaves of pointed sets satisfying some properties. The counterexample is constructed by providing a torsor over a local family of affine lines that cannot be extended to the family of projective lines. The latter is accomplished using the technique of affine Grassmannians.

math.AG↗
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.

math.AG↗
arXiv2018

On the Grothendieck-Serre Conjecture about principal bundles and its generalizations

Let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. Assuming that $G$ has an appropriate parabolic subgroup scheme, we prove the following statement. Given an affine $k$-scheme $W$, a principal $G$-bundle over $W\times_kU$ is trivial if it is trivial over the generic fiber of the projection $W\times_kU\to U$. We also simplify the proof of the Grothendieck-Serre conjecture: let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. A principal $G$-bundle over $U$ is trivial if it is trivial over the generic point of $U$. We generalize some other related results from the simple simply-connected case to the case of arbitrary reductive group schemes.

math.AG↗
arXiv2018

Partial Fourier--Mukai transform for integrable systems with applications to Hitchin fibration

Let X be an abelian scheme over a scheme B. The Fourier--Mukai transform gives an equivalence between the derived category of X and the derived category of the dual abelian scheme. We partially extend this to certain schemes X over B (which we call degenerate abelian schemes) whose generic fiber is an abelian variety, while special fibers are singular. Our main result provides a fully faithful functor from a twist of the derived category of Pic$^τ$(X/B) to the derived category of X. Here Pic$^τ$(X/B) is the algebraic space classifying fiberwise numerically trivial line bundles. Next, we show that every algebraically integrable system gives rise to a degenerate abelian scheme and discuss applications to Hitchin systems.

math.AG↗
arXiv2017

Two circles and only a straightedge

We answer a question of David Hilbert: given two circles it is not possible in general to construct their centers using only a straightedge. On the other hand, we give infinitely many families of pairs of circles for which such construction is possible.

math.MG↗
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 $-\infty$.

math.AG↗
arXiv2016

Affine Grassmannians of group schemes and exotic principal bundles over A^1

Let G be a simple simply-connected group scheme over a regular local scheme U. Let E be a principal G-bundle over A^1_U trivial away from a subscheme finite over U. We show that E is not necessarily trivial and give some criteria of triviality. To this end we define affine Grassmannians for group schemes and study their Bruhat decompositions for semi-simple group schemes. We also give examples of principal G-bundles over A^1_U with split G such that the bundles are not isomorphic to pull-backs from U.

math.AG↗
arXiv2015

Variations of Hodge structures for hypergeometric differential operators and parabolic Higgs bundles

Consider the holomorphic bundle with connection on $\mathbb P^1-\{0,1,\infty\}$ corresponding to the regular hypergeometric differential operator \[ \prod_{j=1}^h(D-α_j)-z\prod_{j=1}^h(D-β_j), \qquad D=z\frac{d}{dz}. \] If the numbers $α_i$ and $β_j$ are real and for all $i$ and $j$ the number $α_i-β_j$ is not integer, then the bundle with connection is known to underlie a complex polarizable variation of Hodge structures. We calculate some Hodge invariants for this variation, in particular, the Hodge numbers. From this we derive a conjecture of Corti and Golyshev. We also use non-abelian Hodge theory to interpret our theorem as a statement about parabolic Higgs bundles.

math.AG↗
arXiv2015

Mountain Peak Detection in Online Social Media

We present a system for the classification of mountain panoramas from user-generated photographs followed by identification and extraction of mountain peaks from those panoramas. We have developed an automatic technique that, given as input a geo-tagged photograph, estimates its FOV (Field Of View) and the direction of the camera using a matching algorithm on the photograph edge maps and a rendered view of the mountain silhouettes that should be seen from the observer's point of view. The extraction algorithm then identifies the mountain peaks present in the photograph and their profiles. We discuss possible applications in social fields such as photograph peak tagging on social portals, augmented reality on mobile devices when viewing a mountain panorama, and generation of collective intelligence systems (such as environmental models) from massive social media collections (e.g. snow water availability maps based on mountain peak states extracted from photograph hosting services).

cs.CV↗
arXiv2015

Estimating snow cover from publicly available images

In this paper we study the problem of estimating snow cover in mountainous regions, that is, the spatial extent of the earth surface covered by snow. We argue that publicly available visual content, in the form of user generated photographs and image feeds from outdoor webcams, can both be leveraged as additional measurement sources, complementing existing ground, satellite and airborne sensor data. To this end, we describe two content acquisition and processing pipelines that are tailored to such sources, addressing the specific challenges posed by each of them, e.g., identifying the mountain peaks, filtering out images taken in bad weather conditions, handling varying illumination conditions. The final outcome is summarized in a snow cover index, which indicates for a specific mountain and day of the year, the fraction of visible area covered by snow, possibly at different elevations. We created a manually labelled dataset to assess the accuracy of the image snow covered area estimation, achieving 90.0% precision at 91.1% recall. In addition, we show that seasonal trends related to air temperature are captured by the snow cover index.

cs.MM↗
arXiv2015

SnowWatch: Snow Monitoring through Acquisition and Analysis of User-Generated Content

We present a system for complementing snow phenomena monitoring with virtual measurements extracted from public visual content. The proposed system integrates an automatic acquisition and analysis of photographs and webcam images depicting Alpine mountains. In particular, the technical demonstration consists in a web portal that interfaces the whole system with the population. It acts as an entertaining photo-sharing social web site, acquiring at the same time visual content necessary for environmental monitoring.

cs.CV↗