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Qingchun Ji

Publications and source records attributed to Qingchun Ji.

12 recordsLinked to original sources
arXiv2026

Surjectivity of real-linear Cauchy--Riemann operators: from the minimal Harder--Narasimhan slope to automatic transversality

This paper relates the minimal Harder--Narasimhan slope to the surjectivity of real-linear Cauchy--Riemann operators. We establish a conformally invariant $L^2$ criterion and an asymptotic slope criterion, which yield higher-rank automatic transversality criteria for pseudoholomorphic curves beyond the classical rank-one framework. Applications to pseudoholomorphic spheres in $S^6$ provide quantitative $L^2$ obstructions to the integrability of almost complex structures.

math.SG↗
arXiv2025

Formally Integrable Structures I. Resolution of Solution Sheaves

This is the first of a series of papers on the $L^2$-theory for formally integrable structures. It is devoted to constructing a resolution of the solution sheaf for a class of overdetermined systems introduced by L. H{ö}rmander. A sufficient condition for global exactness is obtained, which leads to gluing techniques for local solutions formulated as Cousin type problems. In addition, we also prove the local solvability of the Treves complex for formally integrable structures with vanishing Levi forms, including Levi flat structures as special cases. To the best of the authors' knowledge, nothing more than the elliptic case is known about the local $L^2$-solvability of the Treves complex in the Levi flat case.

math.AP↗
arXiv2025

Formally Integrable Structures II. Division Problem

We formulate a division problem for a class of overdetermined systems introduced by L. H{ö}rmander, and establish an effective divisibility criterion. In addition, we prove a coherence theorem which extends Nadel's coherence theorem from complex structures to elliptic systems of partial differential equations.

math.CV↗
arXiv2025

Levi Flat Structures via Structure Sheaves: Differential Complexes, Convexity, and Global Solvability

This paper investigates Levi flat structures from the perspective of structure sheaves. We employ formal integrability to construct a class of differential complexes, thereby providing a resolution for the structure sheaf and a global realization of the Treves complex. Drawing inspiration from Morse theory and Grauert's convexity, we introduce notions of convexity and positivity that fully exploits Levi flatness, which ensures the global exactness of the differential complex and demonstrates Sobolev regularity in the compact case. As applications, we establish the global solvability of the Treves complex for Levi flat structures, together with results on singular cohomology and the extension problem for canonical forms in the elliptic case.

math.CV↗
arXiv2024

Basic (Dolbeault) Cohomology of Foliated Manifolds with Boundary

In this paper, we develop $L^2$ theory for Riemannian and Hermitian foliations on manifolds with basic boundary. We establish a decomposition theorem, various vanishing theorems, a twisted duality theorem for basic cohomologies and an extension theorem for basic forms of induced Riemannian foliation on the boundary. We prove the complex analogues for Hermitian foliations. To show the Dolbeault decomposition of basic forms, we extend Morrey's basic estimate to foliation version. We also investigate the global regularity for $\bar{\partial}_B$-equations.

math.DG↗
arXiv2019

Eigenvalue estimates via Hömander's $L^2$-method

Under various elliptic boundary conditions, we obtain lower eigenvalue estimates for Dirac operators by using Hormander's weighted $L^2$-technique. Lower bounds in terms of the volume of the underlying manifolds are also deduced from the sharp Sobolev inequality due to Li and Zhu(\cite{LZ}).

math.DG↗
arXiv2018

Holomorphic Curves into Algebraic Varieties Intersecting Divisors in Subgeneral Position

Recently, there are many developments on the second main theorem for holomorphic curves into algebraic varieties intersecting divisors in general position or subgeneral position. In this paper, we refine the concept of subgeneral position by introducing the notion of the index of subgeneral position. With this new notion we give some surprising improvement of the previous known second main theorem type results. Moreover, via the analogue between Nevanlinna theory and Diophantine approximation, the corresponding Schmidt's subspace type theorems are also established in the final section.

math.CV↗
arXiv2016

L2-estimates on p-convex Riemannian manifolds

In this paper, we establish various L2-estimates for the exterior differential operator on p-convex Riemannian manifolds in the sense of Harvey and Lawson. As geometric applications, we prove vanishing and finiteness results for the de Rham cohomology groups.

math.DG↗
arXiv2016

Solvability of Dirac type equations

This paper develops a weighted $L^2$-method for the (half) Dirac equation. For Dirac bundles over closed Riemann surfaces, we give a sufficient condition for the solvability of the (half) Dirac equation in terms of a curvature integral. Applying this to the Dolbeault-Dirac operator, we establish an automatic transversality criteria for holomorphic curves in Kähler manifolds. On compact Riemannian manifolds, we give a new perspective on some well-known results about the first eigenvalue of the Dirac operator, and improve the estimates when the Dirac bundle has a $Z_2$-grading. On Riemannian manifolds with cylindrical ends, we obtain solvability in the $L^2$-space with suitable exponential weights while allowing mild negativity of the curvature.

math.DG↗
arXiv2011

Division Theorems for Exact Sequences

Under certain integrability and geometric conditions, we prove division theorems for the exact sequences of holomorphic vector bundles and improve the results in the case of Koszul complex. By introducing a singular Hermitian structure on the trivial bundle, our results recover Skoda's division theorem for holomorphic functions on pseudoconvex domains in complex Euclidean spaces.

math.DG↗
arXiv2011

Division Theorems for the Koszul Complex

We establish a twisted version of Skoda's estimate for the Koszul complex from which we get division theorems for the Koszul complex. This generalizes Skoda's division theorem. We also show how to use Skoda triples to produce division theorems for the Koszul complex.

math.CV↗