arXiv research

New proofs of Donaldson-Uhlenbeck-Yau theorem using geodesic rays.

Proof of Donaldson's theorem using Seiberg-Witten equations for multiple spinors.

In this paper, we provide an alternative proof of Donaldson's almost-holomorphic section theorem and symplectic Lefschetz pencil theorem, through constructions of certain special kind of Donaldson-type sections of the line bundle based on properties of exponential sums.

A proof via the Seiberg-Witten moduli space of Donaldson's theorem on smooth 4-manifolds with definite intersection forms.

We use Donaldson invariants of regular surfaces with p_g >0 to make quantitative statements about modulispaces of stable rank 2 sheaves. We give two examples: a quantitative existence theorem for stable bundles, and a computation of the rank of the canonical holomorphic two forms on the moduli space. The results are in…

New approach to proving Chen-Donaldson-Sun theorem with examples.

This note contains two remarks about the application of the d-invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being smoothly slice. The second carries out a suggestion by Stefan Friedl to replace t…

The paper proves a Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles.

Survey on embedding 3-manifolds in definite 4-manifolds, focusing on Donaldson's theorem.

We consider a perturbed Hermitian-Einstein equation, which we call the Donaldson-Thomas equation, on compact Kähler threefolds. In arXiv:0805.2195, we analysed some analytic properties of solutions to the equation, in particular, we proved that a sequence of solutions to the Donaldson-Thomas equation has a subsequence …

Extends Donaldson's techniques to symplectic orbifolds, proving existence of sections and computing cohomology.

New findings on knot genera using advanced techniques.

We generalize Witten's conjectured formula relating Donaldson and Seiberg-Witten invariants to manifolds of non-simple type, via equivariant localization techniques. This approach does not use the theory of non-abelian monopoles, but works directly on the Donaldson-Witten and Seiberg-Witten moduli spaces. We give a for…

Study smooth embeddings of line configurations in complex projective plane.

Proves existence of Kähler-Einstein metrics in big cohomology classes.

In arXiv:0805.2192, we set up a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian-Einstein equation perturbed by Higgs fields, and call Donaldson-Thomas equation, to analytically approach the Donaldson-Thomas invariants. In this article, we consider the equation on compact Kähler t…

We extend the notion of basic classes (for the Donaldson invariants) to 4-manifolds with $b^+>1$ which are (potentially) not of simple type or satisfy $b_1 >0$. We also give a structure theorem for the Donaldson invariants of 4-manifolds with $b^+>1$, $b_1>0$ and of strong simple type.

Continuity of delta invariant leads to uniform Kähler-Einstein metrics.

The paper proves that rational concordance of double twist knots is reciprocal.

We discuss a conjecture of Donaldson on a version of Yau's Theorem for symplectic forms with compatible almost complex structures and survey some recent progress on this problem. We also speculate on some future possible directions, and use a monotonicity formula for harmonic maps to obtain a new local estimate in the …

Study shows surgeries on certain knots bound rational homology 4-balls.

Let L be a nonunimodular definite lattice. Using a theorem of Elkies we show that whether L embeds in the standard definite lattice of the same rank is completely determined by a collection of lattice correction terms, one for each metabolizing subgroup of the discriminant group. As a topological application this gives…

Uniform proof of Kähler-Einstein metrics with arbitrary polarizations.

Paper proves uniqueness of special Lagrangian pair in Calabi-Yau 3-fold.

Study of instantons and knot representations in 3-manifolds.

Study proves obstructions to equivariantly slice strongly negative amphichiral knots.

In this paper, we prove a generalized Donaldson-Uhlenbeck-Yau theorem on Higgs bundles over a class of non-compact Gauduchon manifolds.

Let M be a compact oriented even-dimensional manifold. This note constructs a compact symplectic manifold S of the same dimension and a map f from S to M of strictly positive degree. The construction relies on two deep results: the first is a theorem of Ontaneda that gives a Riemannian manifold N of tightly pinched neg…

We prove an analogue of the Kotschick-Morgan conjecture in the context of SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the SO(3)-monopole cobordism. The main technical difficulty in the SO(3)-monopole program relating the Seiberg-Witten and Don…

The paper proves a new version of dimensional reduction in cohomological Donaldson-Thomas theory.

We extend Donaldson's diagonalization theorem to intersection forms with certain local coefficients, under some constraints. This provides new examples of non-smoothable topological 4-manifolds.

We establish the slice-ribbon conjecture for a large family of Montesinos' knots by means of Donaldson's theorem on the intersection forms of definite 4-manifolds.

The paper proves a theorem and characterizes connections over normal varieties.

Solved a conjecture about rational homology projective planes with quotient singularities.

We prove that any flat family $(\mathcal{ F}_u)_{u\in U}$ of rank 2 torsion-free sheaves on a Gauduchon surface defines a continuous map on the semi-stable locus $U^{\mathrm {ss}}:=\{u\in U \ |\ \mathcal{ F}_u\hbox{ is slope semi-stable}\}$ with values in the Donaldson-Uhlenbeck compactification of the corresponding in…

We generalise theorems of Cochran-Lickorish and Owens-Strle to the case of links with more than one component. This enables the use of linking forms on double branched covers, Heegaard Floer correction terms, and Donaldson's diagonalisation theorem to complete the table of unlinking numbers for nonsplit prime links wit…

This is a sequel of our paper [arXiv:1809.08425] on the Quot-scheme limit and variational properties of Donaldson's functional, which established its coercivity for slope stable holomorphic vector bundles over smooth projective varieties. Assuming that the coercivity is uniform in a certain sense, we provide a new proo…

In this paper, we use the affine Hermitian-Yang-Mills flow to prove a generalized Donaldson-Uhlenbeck-Yau theorem on flat Higgs bundles over a class of non-compact affine Gauduchon manifolds.

Let $X$ be a compact Hermitian surface, and $g$ be any fixed Gauduchon metric on $X$. Let $E$ be an Hermitian holomorphic vector bundle over $X$. On the bundle $E$, Donaldson's heat flow is gauge equivalent to a flow of holomorphic structures. We prove that this flow converges, in the sense of Uhlenbeck, to the double …

Let $H$ be a semisimple algebraic group. We prove the semistable reduction theorem for $μ$--semistable principal $H$--bundles over a {\it smooth projective variety $X$} defined over the field $\bc$. When $X$ is a {\it smooth projective surface} and $H$ is simple, we construct the algebro--geometric Donaldson--Uhlenbeck…

Study on stability of hyperkähler flow in 4-manifolds.

This paper describes how, in the case of algebraic surfaces, the well-known theorem of Donaldson-Uhlenbeck-Yau can be proved in a framework of generalized 'multiplier ideal sheaves', following the ideas of Siu. The key concept is that the destabilizing sheaf satisfies a differential inclusion relation. This relation is…

In this note, we explain that Ross-Thomas' result on the weighted Bergman kernels on orbifolds can be directly deduced from our previous result. This result plays an important role in the companion paper to prove an orbifold version of Donaldson Theorem.

Algorithm finds ribbon disks for alternating knots, resolving sliceness for most prime knots.

We apply Donaldson's theorem on the intersection forms of definite 4--manifolds to characterize the lens spaces which smoothly bound rational homology 4--dimensional balls. Our result implies, in particular, that every smoothly slice 2--bridge knot is ribbon, proving the ribbon conjecture for 2--bridge knots.

Auxiliary equations improve bounds in symplectic geometry.

We prove that the Calabi-Yau equation can be solved on the Kodaira-Thurston manifold for all given $T^2$-invariant volume forms. This provides support for Donaldson's conjecture that Yau's theorem has an extension to symplectic four-manifolds with compatible but non-integrable almost complex structures.

In this note, we study the long time existence of the Calabi flow on $X = \mathbb{C}^n/\mathbb{Z}^n + i\mathbb{Z}^n$. Assuming the uniform bound of the total energy, we establish the non-collapsing property of the Calabi flow by using Donaldson's estimates and Streets' regularity theorem. Next we show that the curvatur…