arXiv research

Continuity shown for Yang-Mills flow on semistable bundles.

Study of holomorphic distributions on projective 3-space, focusing on stable tangent sheaves.

Link slope stability to Donaldson's functional via Quot-scheme limits.

Compact surfaces' quotient schemes lack semipositive Ricci curvature.

Let $X$ be a compact connected Riemann surface of genus $g$, with $g \geq 2$, and let ${\mathcal O}_X$ denote the sheaf of holomorphic functions on $X$. Fix positive integers $r$ and $d$ and let ${\mathcal Q}(r,d)$ be the Quot scheme parametrizing all torsion coherent quotients of ${\mathcal O}^{\oplus r}_X$ of degree …

Paper generalizes balanced metrics existence to singular cases using Quot-scheme limit.

Let $S^n(X)$ be the $n$-fold symmetric product of a compact connected Riemann surface $X$ of genus $g$ and gonality $d$. We prove that $S^n(X)$ admits a Kähler structure such that all the holomorphic bisectional curvatures are nonpositive if and only if $n < d$. Let ${\mathcal Q}_X(r,n)$ be the Quot scheme parametrizin…

New proof of Donaldson-Uhlenbeck-Yau theorem using variational approach.

Let X be a compact connected Riemann surface of genus at least two. Fix positive integers r and d. Let Q denote the Quot scheme that parametrizes the torsion quotients of {\mathcal O}^{\oplus r}_X of degree d. This Q is also the moduli space of vortices for the standard action of U(r) on {\mathbb C}^r. The group \text{…

The study explores how different Grothendieck topologies and functors between categories preserve locality.

3-manifold groups are rigid under certain conditions.

Shows natural quasi-Poisson structure on multiplicative Grothendieck-Springer space.

Proves Grothendieck-Teichmüller group acts on specific mapping class groups.

Proves a theorem for complex flat vector bundles using differential forms.

We give a direct proof that the Freed-Lott differential analytic index is well defined and a condensed proof of the differential Grothendieck-Riemann-Roch theorem. As a byproduct we also obtain a direct proof that the R/Z analytic index is well defined and a condensed proof of the R/Z Grothendieck-Riemann-Roch theorem.

Develops equivariant Chern characters for coherent sheaves with group actions.

Study on string links invariant under associator choice and Grothendieck--Teichmüller group action.

3-manifold groups are uniquely identifiable via their profinite completions.

We identify the Grothendieck group of the tangle Floer dg algebra with a tensor product of certain $U_q(gl(1|1))$ representations. Under this identification, up to a scalar factor, the map on the Grothendieck group induced by the tangle Floer dg bimodule associated to a tangle agrees with the Reshetikhin-Turaev homomor…

Polytopes in R^n with integral vertices form a monoid under the Minkowski sum, and the Grothendieck construction gives rise to a group. We show that every symmetric polytope is a norm in this group for every n.

Grothendieck's Esquisse d'un programme is often referred to for the ideas it contains on dessins d'enfants, the Teichm{ü}ller tower, and the actions of the absolute Galois group on these objects or their etale fundamental groups. But this program contains several other important ideas. In particular, motivated by surfa…

Study automorphism groups of braid groups with 4 or more strings.

The aim of this note is to take benefit of the foam nature of the Khovanov-Kuperberg algebras to compute the Grothendieck groups of their categories of finitely generated projective modules. The computation relies on the Hattori-Stallings trace and some geometrical properties of foams in a solid torus.

Study of orbifold Chern character using superconnections.

In this paper we give a simplified proof of the flat Grothendieck-Riemann-Roch theorem. The proof makes use of the local family index theorem and basic computations of the Chern-Simons form. In particular, it does not involve any adiabatic limit computation of the reduced eta-invariant.

In his 1944 paper Veränderliche Riemannsche Flächen , Teichmüller defined a structure of complex manifold on the set of isomorphism classes of marked closed Riemann surfaces of genus g. The complex manifold he obtained is the space called today Teichmüller space. In the same paper, Teichmüller introduced the so-called …

Explains historical connections between vector bundle splitting and Riemann-Hilbert problems.

This paper attempts to relate some ideas of Grothendieck in his Esquisse d'un programme and some of the recent results on 2-dimensional topology and geometry. Especially, we shall discuss the Teichmüller theory, the mapping class groups, $SL(2, \bold C)$ representation variety of surface groups, and Thurston's theory o…

Paper constructs Chern character for coherent sheaves.

We use the Grauert--Grothendieck complex on differentiable spaces to study basic relative forms on the inertia space of a compact Lie group action on a manifold. We prove that the sheaf complex of basic relative forms on the inertia space is a fine resolution of Bryliski's sheaf of functions on the inertia space.

We construct master spaces for oriented torsion free sheaves coupled with morphisms into a fixed reference sheaf. These spaces are projective varieties endowed with a natural $\C^*$-action. The fixed point set of this action contains the moduli space of semistable oriented torsion free sheaves and the quot scheme assoc…

Researchers prove Lie algebras of differential operators and Grothendieck constructions coincide.

The paper studies the commutator subgroups of twin groups and their properties.

The paper proves stabilization in hypersurface sections using Grothendieck rings and probabilistic methods.

Desingularizes singular spaces using sheaves and groupoids.

Computes second homology groups of orbifold groups, proving profinite rigidity and Grothendieck pairs.

Integrally splits L-spectra of integers into simpler components.

The study explores rigidity in groups and their products, finding uncountable families of non-isomorphic subgroups.

Study of Fukaya category on surfaces with mapping class group action.

Calculates characteristic classes of flat vector bundles on complex fibers.

The purpose of this paper is to give a proof of the real part of the Riemann-Roch-Grothendieck theorem for complex flat vector bundles at the differential form level in the even dimensional fiber case. The proof is, roughly speaking, an application of the local family index theorem for a perturbed twisted spin Dirac op…

In the present paper we discuss an independent on the Grothendieck-Sato isomorphism approach to the Riemann-Roch-Hirzebruch formula for an arbitrary differential operator. Instead of the Grothendieck-Sato isomorphism, we use the Topological Quantum Mechanics (more or less equivalent to the well-known constructions with…

We prove that the Grothendieck-Springer simultaneous resolution viewed as a correspondence between the adjoint quotient of a Lie algebra and its maximal torus is Lagrangian in the sense of shifted symplectic structures. As Hamiltonian spaces can be interpreted as Lagrangians in the adjoint quotient, this allows one to …

Geometric models for representations up to homotopy using simplicial vector bundles.

The paper shows plentiful non-homotopy finite Poincaré duality spaces.

In this paper we give a proof of an index theorem by Bismut. As a consequence we obtain another proof of the Grothendieck-Riemann-Roch theorem in differential cohomology.

SDP achieves exponential error rate in cluster estimation for Stochastic Block Model.

We review some ideas of Grothendieck and others on actions of the absolute Galois group Γ Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichm{ü}ller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation…