Related papers: The Weight Filtration on the Constant Sheaf on a P…

Overparameterized shallow neural networks admit substantial parameter redundancy: distinct parameter vectors may represent the same predictor due to hidden-unit permutations, rescalings, and related symmetries. As a result, geometric…

Let $M$ be a smooth algebraic variety of dimension $2(p+q)$ with an algebraic symplectic form and a compatible deformation quantization $\mathcal{O}_h$ of the structure sheaf. Consider a smooth coisotropic subvariety $j: Y \to M$ of…

In this paper, we consider a finiteness problem of saturated subsheaves of a hermitian locally free sheaf on an arithmetic variety. As an application, we could prove the unique existence of an arithmetic Harder-Narasimham filtration.

We prove that the Harder-Narasimhan filtration for an unstable finite dimensional representation of a finite quiver coincides with the filtration associated to the 1-parameter subgroup of Kempf, which gives maximal unstability in the sense…

Let $Z\to X$ be a closed immersion of smooth affine schemes over an arbitrary field $k$, and $X^h_Z$ denote the henselization of $X$ along $Z$. For each presheaf $E\colon \mathbf{SH}(k)\to \mathrm{Ab}^\mathrm{op}$ on the stable motivic…

We show that the complicated *-structure characterizing for positive q the U_qso(N)-covariant differential calculus on the non-commutative manifold R_q^N boils down to similarity transformations involving the ribbon element of a central…

We show that any commutative rationally ruled surface with a choice of anticanonical curve admits a 1-parameter family of noncommutative deformations parametrized by the Jacobian of the anticanonical curve, and show that many standard facts…

We study a one parameter degeneration of Calabi Yau threefolds whose central fiber contains a single ordinary double point. Using the nearby and vanishing cycle formalism, we construct a canonical perverse object on the singular fiber from…

Let R be a commutative ring, q a unit of R and P a multiplicatively antisymmetric matrix with coefficients which are integers powers of q. Denote by SE(q,P) the multiparameter quantum matrix bialgebra associated to q and P.Slightly…

We relate shuffle algebras, as defined by Nichols, Feigin-Odesskii and Rosso, to perverse sheaves on symmetric products of the complex line (i.e., on the spaces of monic polynomials stratified by multiplicities of roots). More precisely, we…

This expository monograph cuts a short path from the common, elementary background in geometry (linear algebra, vector bundles, and algebraic ideals) to the most advanced theorems about involutive exterior differential systems: (1) The…

We study locally trivial deformations of toric varieties from a combinatorial point of view. For any fan $\Sigma$, we construct a deformation functor $\mathrm{Def}_\Sigma$ by considering \v{C}ech zero-cochains on certain simplicial…

The purely mathematical root of the dequantization constructions is the quest for a sheafification needed for presheaves on a noncommutative space. The moment space is constructed as a commutative space, approximating the noncommutative…

We define canonical refinements of Harder-Narasimhan filtrations and stratifications in moduli theory, generalising and relating work of Haiden-Katzarkov-Kontsevich-Pandit and Kirwan. More precisely, we define a canonical stratification on…

Let X be a smooth algebraic variety over a field of characteristic 0. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf O_X. These are stack-like versions of usual deformations. We prove that…

Let $X$ be a smooth quasi-projective algebraic surface and let $\Delta_n$ the big diagonal in the product variety $X^n$. We study cohomological properties of the ideal sheaves $\mathcal{I}^k_{\Delta_n}$ and their invariants…

A Coxeter $n$-orbifold is an $n$-dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order $m$, whose neighborhood is locally modeled on ${\mathbb R}^n$ modulo the…

Current continuous generative models (e.g., Diffusion Models, Flow Matching) implicitly assume that locally consistent causal mechanisms naturally yield globally coherent counterfactuals. In this paper, we prove that this assumption fails…

Let SU_X(n,L) be the moduli space of rank n semistable vector bundles with fixed determinant L on a smooth projective genus g>1 curve X. Let SU_X^s(n,L) denote the open subset parameterizing stable bundles. We show that for small i, the…

Given a normal projective irreducible stack $\mathscr X$ over an algebraically closed field of characteristic zero we consider framed sheaves on $\mathscr X$, i.e., pairs $(\mathcal E,\phi_{\mathcal E})$, where $\mathcal E$ is a coherent…