Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2108.11029

The Deligne-Simpson problem for connections on $\mathbb{G}_m$ with a maximally ramified singularity

Maitreyee C. Kulkarni·Neal Livesay·Jacob P. Matherne·Bach Nguyen·Daniel S. Sage
View original source ↗Read source document ↗Export metadata ↓

Abstract

The classical additive Deligne-Simpson problem is the existence problem for Fuchsian connections with residues at the singular points in specified adjoint orbits. Crawley-Boevey found the solution in 2003 by reinterpreting the problem in terms of quiver varieties. A more general version of this problem, solved by Hiroe, allows additional unramified irregular singularities. We apply the theory of fundamental and regular strata due to Bremer and Sage to formulate a version of the Deligne-Simpson problem in which certain ramified singularities are allowed. These allowed singular points are called toral singularities; they are singularities whose leading term with respect to a lattice chain filtration is regular semisimple. We solve this problem in the important special case of connections on $\mathbb{G}_m$ with a maximally ramified singularity at $0$ and possibly an additional regular singular point at infinity. We also give a complete characterization of all such connections which are rigid, under the additional hypothesis of unipotent monodromy at infinity.

Explore related subjects

math.AGmath.DGmath.RT

Keep this discovery

BibTeXRIS

Maitreyee C. Kulkarni, Neal Livesay, Jacob P. Matherne, Bach Nguyen, Daniel S. Sage. 2021-08-25. The Deligne-Simpson problem for connections on $\mathbb{G}_m$ with a maximally ramified singularity. https://doi.org/10.1016/j.aim.2022.108596

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Perverse Euler Characteristics of Hermitian Locally Symmetric Spaces

We prove that finite-volume locally Hermitian symmetric spaces of noncompact type have nonnegative perverse Euler characteristics. To show this, we obtain a nefness result for the logarithmic cotangent bundle of a smooth toroidal compactification. Combining this with a positivity criterion for Euler characteristics of perverse sheaves, we deduce the nonnegativity result. We further prove that the inequality is strict for perverse sheaves with full support. As applications, we get nonnegativity results for perverse Euler characteristics on various moduli spaces.

math.AG↗
arXiv2026

Coupled Pklt Tuples and Varieties of Pklt Type

We introduce asymptotic multiplier ideal sheaves and log canonical thresholds associated with tuples of pseudoeffective divisors on a projective klt pair. We prove that the threshold of a coupled potentially klt tuple is computed by a quasi-monomial valuation. For varieties of potentially klt type, we prove that every big divisor admits a birational Zariski decomposition with semiample positive part. We also prove finite generation of multisection rings of big divisors and give a criterion for a variety of potentially klt type to be a Mori dream space.

math.AG↗
arXiv2026

Graded Betti numbers of general curves of large degree

Let $C$ be a smooth projective complex curve of genus $g$ and gonality $k$, and $L$ be a very ample line bundle on $C$. When $L$ has sufficiently large degree, the vanishing and nonvanishing of the Koszul cohomology groups $K_{p,q}(C,L)$ have been determined previously, but the exact values of the graded Betti numbers $\kappa_{p,q}(C, L)$ remain largely unknown. In this paper, we give explicit closed formulas for all graded Betti numbers $\kappa_{p,q}(C, L)$ when the Brill--Noether locus $W_k^1(C)$ has the expected dimension and $H^1(C, L \otimes \omega_C^{-1})=0$. Consequently, we determine the complete Betti table for a general curve when $\deg L \geq 4g-3$ or when $\deg L \geq 3g-3$ and $L$ is general. We also explicitly compute the Boij--S\"{o}derberg coefficient of the section ring $R(C, L)$ governing asymptotic purity, and show eventual monotonicity of the remaining coefficients: they decrease for hyperelliptic curves and increase under a natural generic reducedness assumption on the relevant Brill--Noether loci.

math.AG↗