arXiv research

Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.

Study smooth embeddings of line configurations in complex projective plane.

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

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

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.

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…

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

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

Detects exotic surfaces without smooth invariants, providing first example of knotted RP².

Paper proves conditions for rational homology complex projective planes with singularities.

We give a complete classification of the spherical 3-manifolds that bound smooth rational homology 4-balls. Furthermore, we determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. To this end, we use constraints for 3-manifolds to bound rational homology bal…

Let p and q be distinct integers greater than one. We show that the 2-component pretzel link P(p,q,-p,-q) is not slice, even though it has a ribbon mutant, by using 3-fold branched covers and an obstruction based on Donaldson's diagonalization theorem. As a consequence, we prove the slice-ribbon conjecture for 4-strand…

We compute the unknotting number of two infinite families of pretzel knots, $P(3,1,\dots,1,b)$ (with $b$ positive and odd and an odd number of 1s) and $P(3,3,3c)$ (with $c$ positive and odd). To do this, we extend a technique of Owens using Donaldson's diagonalization theorem, and one of Traczyk using the Jones polynom…

We establish an inequality which gives strong restrictions on when the standard definite plumbing intersection lattice of a Seifert fibered space over $S^2$ can embed into a standard diagonal lattice, and give two applications. First, we answer a question of Neumann-Zagier on the relationship between Donaldson's theore…

Diagonalizes tautological classes of definite 4-manifolds.

We show that, if a rational homology 3-sphere $Y$ bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by $Y$. To this end, we make use of constraints on definite…

Using Gauge theoretical techniques employed by Lisca for 2-bridge knots and by Greene-Jabuka for 3-stranded pretzel knots, we show that no member of the family of Montesinos knots M(0;[m_1+1,n_1+2],[m_2+1,n_2+2],q), with certain restrictions on m_i, n_i, and q, can be (smoothly) slice. Our techniques use Donaldson's di…

Study on nonorientable 4-genus of double twist knots.

This paper contains the results of efforts to determine values of the smooth and the topological slice genus of 11- and 12-crossing knots. Upper bounds for these genera were produced by using a computer to search for genus one concordances between knots. For the topological slice genus further upper bounds were produce…

Sharp bounds on K-semistable Fano varieties for low dimensions.

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

Self dual symmetric R-spaces have special curves, called circles, introduced by Burstall, Donaldson, Pedit and Pinkall in 2011, whose definition does not involve the choice of any Riemannian metric. We characterize the elements of the big transformation group G of a self dual symmetric R-space M as those diffeomorphism…

We investigate rational homology cobordisms of 3-manifolds with non-zero first Betti number. This is motivated by the natural generalization of the slice-ribbon conjecture to multicomponent links. In particular we consider the problem of which rational homology $S^1\times S^2$'s bound rational homology $S^1\times D^3$'…

In this note, using the recent compactness results of Tian and Chen-Donaldson-Sun, we prove the K-semistable version of Yau-Tian-Donaldson correspondence for Fano manifolds.

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…

The paper solves a uniform Yau-Tian-Donaldson conjecture for toric manifolds.

We prove the local existence of unique smooth solutions of the Donaldson geometric flow on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. It is a semiflow on the Besov space $B^{1,p}_2(M, Λ^2)$ for $p > 4$. The Donaldson geometric flow was introduced by Simon Dona…

Defines knot concordance invariant using instanton homology and Donaldson invariants.

We analyze the u-plane contribution to Donaldson invariants of a four-manifold X. For $b_2^+(X)>1$, this contribution vanishes, but for $b_2^+=1$, the Donaldson invariants must be written as the sum of a u-plane integral and an SW contribution. The u-plane integrals are quite intricate, but can be analyzed in great det…

Effective Yau-Tian-Donaldson conjecture for spherical varieties.

Formula proves invariant matches for smooth and orbifold test configurations.

Proves Yau-Tian-Donaldson conjecture for toric manifolds and bundles.

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.

Let $X$ be a complex four-dimensional compact Calabi-Yau manifold equipped with a Kähler form $ω$ and a holomorphic four-form $Ω$. Under certain assumptions, we define Donaldson-Thomas type deformation invariants by studying the moduli space of the solutions of Donaldson-Thomas equations on the given Calabi-Yau manifol…

We prove an estimate for Donaldson's $Q$-operator on a prequantized compact symplectic manifold. This estimate is an ingredient in the recent result of Keller and Lejmi about a symplectic generalization of Donaldson's lower bound for the $L^2$-norm of the Hermitian scalar curvature.

In this paper, we solve a problem of Kobayashi posed in \cite{Ko4} by introducing a Donaldson type functional on the space $F^+(E)$ of strongly pseudo-convex complex Finsler metrics on $E$ -- a holomorphic vector bundle over a closed Kähler manifold $M$. This Donaldson type functional is a generalization in the complex…

The paper studies cohomological Donaldson-Thomas theory for local systems on a 3-torus.

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 …

Gradient estimate proved for Donaldson's equation on Kähler manifolds.

This paper connects complex hyperkähler structures to Donaldson-Thomas invariants.

Given a complex 4-fold $X$ with an (Calabi-Yau 3-fold) anti-canonical divisor $Y$, we study relative Donaldson-Thomas invariants for this pair, which are elements in the Donaldson-Thomas cohomologies of $Y$. We also discuss gluing formulas which relate relative invariants and $DT_{4}$ invariants for Calabi-Yau 4-folds.

This paper solves matrix blind joint block diagonalization with noise.

Diagonalizes metrics of 3D Lorentzian manifolds.

Study Ricci vector fields on 2D space with diagonal metrics.

Paper calculates Donaldson-Thomas invariants for a specific category.

The Donaldson metric is a metric on the space of symplectic two-forms in a fixed cohomology class. It was introduced in [2]. We compute the associated Levi-Civita connection, describe it's geodesics and compute the formula for the covariant Hessian of an energy functional on the space of symplectic structures in a fixe…

Develops a novel stochastic algorithm for diagonal estimation of large matrices.

Solves Yau-Tian-Donaldson conjecture for smooth projective varieties.