arXiv research

Study slopes of direct images in complex manifolds, proving a Mehta-Ramanathan type theorem.

Develops a new method to study algebraic tangent cones of sheaves using valuations.

The paper establishes lower bounds on Yang-Mills functionals for fibrations.

This paper divides into two parts. Let $(X,ω)$ be a compact Hermitian manifold. Firstly, if the Hermitian metric $ω$ satisfies the assumption that $\partial\overline{\partial}ω^k=0$ for all $k$, we generalize the volume of the cohomology class in the Kähler setting to the Hermitian setting, and prove that the volume is…

We study the Yang-Mills flow on a holomorphic vector bundle E over a compact Kahler manifold X . Along a solution of the flow, we show the curvature $iΛF(A_t)$ approaches in $L^2$ an endomorphism with constant eigenvalues given by the slopes of the quotients from the Harder-Narasimhan filtration of E. This proves a sha…

This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this …

This survey intends to present the basic notions of Geometric Invariant Theory (GIT) through its paradigmatic application in the construction of the moduli space of holomorphic vector bundles. Special attention is paid to the notion of stability from different points of view and to the concept of maximal unstability, r…

We give a generalisation of the theory of optimal destabilizing 1-parameter subgroups to non-algebraic complex geometry. Consider a holomorphic action $G\times F\to F$ of a complex reductive Lie group $G$ on a finite dimensional (possibly non-compact) Kähler manifold $F$. Using a Hilbert type criterion for the (semi)st…

Inspired by Katz-Mazur theorem on crystalline cohomology and by Eskin-Kontsevich-Zorich's numerical experiments, we conjecture that the polygon of Lyapunov spectrum lies above (or on) the Harder-Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons a…

Paper studies convergence of Yang-Mills-Higgs flow on Kähler manifolds.

The paper studies obstructions to solutions of the Wess-Zumino-Witten equation and its generalizations.

Study describes splitting and filtration of Hodge bundle on quadratic differentials.

We show that the ``classical'' Harder-Narasimhan filtration associated to a non semistable vector bundle $E$ can be viewed as a limit object for the action of the gauge group in the direction of an optimal destabilizing vector. This vector appears as an extremal value of the so called "maximal weight function". We give…

It is shown that the singular set for the Yang-Mills flow on unstable holomorphic vector bundles over compact Kaehler manifolds is completely determined by the Harder-Narasimhan-Seshadri filtration of the initial holomorphic bundle. We assign a multiplicity to irreducible top dimensional components of the singular set …

Joyce's criterion for sLag smoothings extended to non-compact, non-transverse intersections.

The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebra…

In this paper we study the analytic tangent cones of admissible Hermitian-Yang-Mills connections near a homogeneous singularity of a reflexive sheaf, and relate it to the Harder-Narasimhan-Seshadri filtration. We also give an algebro-geometric characterization of the bubbling set. This strengthens our previous result.

In this paper, we study the asymptotic behavior of the Hermitian-Yang-Mills flow on a reflexive sheaf. We prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the Harder-Narasimhan-Seshadri filtration, this answers a question by Bando and Siu.

In this paper, we study the curvature estimate of the Hermitian-Yang-Mills flow on holomorphic vector bundles. In one simple case, we show that the curvature of the evolved Hermitian metric is uniformly bounded away from the analytic subvariety determined by the Harder-Narasimhan-Seshadri filtration of the holomorphic …

The slope conjecture proposed by Garoufalidis asserts that the Jones slopes given by the sequence of degrees of the colored Jones polynomials are boundary slopes. We verify the slope conjecture for graph knots, i.e. knots whose Gromov volume vanish.

Study slopes in 3-manifolds, proving conjectures about knots.

Using the Hatcher-Oertel algorithm for finding boundary slopes of Montesinos knots, we prove the Slope Conjecture and the Strong Slope Conjecture for a family of 3-tangle pretzel knots. More precisely, we prove that the maximal degrees of the colored Jones polynomial of such knots determine a boundary slope as predicte…

Constructs Lefschetz fibrations with slopes near 2.

The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies th…

Paper analyzes origami slope gaps and their distribution, finding a unique pattern.

Study provides concrete examples of knot slopes.

2-level SLOPE improves high-dimensional inference with fewer hyperparameters.

Concerning the set of exceptional surgery slopes for a hyperbolic knot, Lackenby and Meyerhoff proved that the maximal cardinality is 10 and the maximal diameter is 8. Their proof is computer-aided in part, and both bounds are achieved simultaneously. In this note, it is observed that the diameter bound 8 implies the m…

A slope $p/q$ is a characterising slope for a knot $K$ in $S^3$ if the oriented homeomorphism type of $p/q$-surgery on $K$ determines $K$ uniquely. We show that when $K$ is a hyperbolic knot its set of characterising slopes contains all but finitely many slopes $p/q$ with $q \geq 3$. We prove stronger results for hyper…

New research shows that many slopes are characterizing for satellite knots.

The Slope Conjecture relates the degree of the colored Jones polynomial to the boundary slopes of a knot. We verify the Slope Conjecture and the Strong Slope Conjecture for Montesinos knots $M(\frac{1}{r},\frac{1}{s-\frac{1}{u}},\frac{1}{t} )$ with $r,u,t$ odd, $s$ even and $u\leq-1$, $r<-1<1<s,t$.

New slopes identified for torus knots, improving previous results.

A non-trivial slope $r$ on a knot $K$ in $S^3$ is called a characterizing slope if whenever the result of $r$-surgery on a knot $K'$ is orientation preservingly homeomorphic to the result of $r$-surgery on $K$, then $K'$ is isotopic to $K$. Ni and Zhang ask: for any hyperbolic knot $K$, is a slope $r = p/q$ with $|p| +…

New knots found with specific slope properties.

New rules reduce SLOPE model fitting time by screening out irrelevant variables.

Let X be a norm curve in the SL(2,C)-character variety of a knot exterior M. Let t = || b || / || a || be the ratio of the Culler-Shalen norms of two distinct non-zero classes a, b in H_1(\partial M, Z). We demonstrate that either X has exactly two associated strict boundary slopes \pm t, or else there are strict bound…

The (Strong) Slope Conjecture relates the degree of the colored Jones polynomial of a knot to certain essential surfaces in the knot complement. We verify the Slope Conjecture and the Strong Slope Conjecture for 3-string Montesinos knots satisfying certain conditions.

Let r_m and r_M be the least and greatest finite boundary slopes of a hyperbolic knot K in S^3. We show that any cyclic surgery slopes of K must lie in the interval (r_m - 1/2, r_M + 1/2).

Study Mazur doubles of knots and their relation to the Slope Conjecture.

The study confirms conjectures about slopes of knots using knot Floer homology.

Proves existence of Lefschetz fibrations with arbitrary slopes.

The paper examines slopes and their norms in exceptional Dehn fillings.

This paper completes proofs for left orderable slopes of double twist knots.

A slope $\frac pq$ is called a characterizing slope for a given knot $K_0$ in $S^3$ if whenever the $\frac pq$-surgery on a knot $K$ in $S^3$ is homeomorphic to the $\frac pq$-surgery on $K_0$ via an orientation preserving homeomorphism, then $K=K_0$. In this paper we try to find characterizing slopes for torus knots $…

Efficient packages solve SLOPE problem in multiple languages.

New invariant slope for links helps complete signature formula.

Formula found for energy slope in complex geometry.

No characterizing slopes for multi-component links.