arXiv research

The singular set of a foliation is always connected under certain conditions.

We construct Bott-type and stable equivariant Seiberg-Witten Floer homology and cohomology for rational homology spheres, and prove their diffeomorphism invariance.

We construct Bott-type and equivariant Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. This paper is a revised version of math.DG/9701010. Some typos are removed.

In this work we prove a Baum-Bott type residue theorem for flags of holomorphic foliations. We prove some relations between the residues of the flag and the residues of their correspondent foliations. We define the Nash residue for flags and we give a partial answer to the Baum-Bott type rationality conjecture in this …

We construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory: the singular version, the de Rham version and the Cartan version, with the first…

We use closed geodesics to construct and compute Bott-type Morse homology groups for the energy functional on the loop space of flat $n$-dimensional tori, $n\ge 1$, and Bott-type Floer cohomology groups for their cotangent bundles equipped with the natural symplectic structure. Both objects are isomorpic to the singula…

We prove residual formulas for vector fields defined on compact complex orbifolds with isolated singularities and give some applications of these on weighted projective spaces.

We study the algebraic properties of the generalized Futaki invariant of an almost Fano variety and prove that it is in fact a pushforward to a point of an appropriate equivariant Chow cohomology class of the variety. This allows us to use Bott-type formulae for calculating the invariant. We show this use on some examp…

Let M be a manifold carrying the action of a Lie group G, and A a Lie algebroid on M equipped with a compatible infinitesimal G-action. Out of these data we construct an equivariant Lie algebroid cohomology and prove for compact G a related localization formula. As an application we prove a Bott-type formula.

Proves a Baum--Bott formula for foliations by curves with logarithmic terms.

In this work we prove a Baum-Bott type formula for non-compact complex manifold of the form $\tilde{X}=X- \mathcal{D}$, where $X$ is a complex compact manifold and $\mathcal{D}$ is a normal crossing divisor on $X$. As applications, we provide a Poincaré-Hopf type Theorem and an optimal description for a smooth hypersur…

We study the Harvey-Lawson spark characters of level p on complex manifolds. Presenting Deligne cohomology classes by sparks of level $p$, we give an explicit analytic product formula for Deligne cohomology. We also define refined Chern classes in Deligne cohomology for holomorphic vector bundles over complex manifolds…

This paper studies torsion obstructions to complex sections on manifolds.

A celebrated result due to Poincaré affirms that a closed non-degenerate minimizing geodesic $γ$ on an oriented Riemannian surface is hyperbolic. Starting from this classical theorem, our first main result is a general instability criterion for timelike and spacelike closed semi-Riemannian geodesics on a (non)oriented …

Fewer obstructions for small graphs in knotless embedding.

3028 obstructions found for embedding without knots.

The paper develops obstructions for embedding 2D complexes into 4D space.

Global obstructions found for conformally Einstein metrics in 6D.

Complete surgery obstructions for manifolds with finite fundamental group, disproving a conjecture.

Given a polarized manifold there are obstructions for asymptotic Chow semistability described as integral invariants. One of them is an obstruction to the existence for the first Chern class of the polarization to admit a constant scalar curvature Kähler (cscK) metric. A natural question is whether or not the other obs…

The article classifies cubiquitous sublattices and applies them to branched covers.

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S^3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundam…

New obstructions found for smooth desingularization of compact Einstein orbifolds.

For $\ell >1$, we develop $L^{(2)}$-signature obstructions for $(4\ell-3)$-dimensional knots with metabelian knot groups to be doubly slice. For each $\ell>1$, we construct an infinite family of knots on which our obstructions are non-zero, but for which double sliceness is not obstructed by any previously known invari…

Study identifies obstructions for solving a 4th-order boundary problem.

Developing a singular dimension descent method for positive scalar curvature obstructions

Study Euler obstruction of 1-forms on determinantal singularities.

Proves Massey's theorems on complex structure obstructions.

The paper explores properties of CR hypersurfaces and their flatness.

Study on a specific obstruction in four-dimensional geometry.

Introduce generalized Ueda obstruction classes for line bundles and apply them to non-semi-positivity.

Surgery obstructions extended to integer homology spheres using Heegaard Floer homology.

Extended solitons show constant curvature on compact manifolds.

Study the topological information of map germs using Euler obstruction.

New obstructions show some 4-manifold homeomorphisms are pseudo-isotopic but not isotopic.

Geometrically, a new obstruction is found for 4-manifold realizations.

Study torsion obstructions to positive scalar curvature on manifolds.

New curvature obstruction for Killing vector fields on Lorentzian manifolds.

Develops formal moduli theory for splitting complex supermanifolds.

The paper shows examples of 2-complexes that can't be embedded in R^4, hiding obstructions in higher Milnor invariants.

We describe two simple obstructions to the existence of Ricci-flat Kahler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki-Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kahler-Einstein metrics on Fano orbifolds.…

Study pseudoisotopies in 4-manifolds, finding specific elements of obstruction.

Obstructs Legendrian knots from being slices of concordances using doubly slice genus.

Applying a local Gauss-Bonnet formula for closed subanalytic sets to the complex analytic case, we obtain characterizations of the Euler obstruction of a complex analytic germ in terms of the Lipschitz-Killing curvatures and the Chern forms of its regular part. We also prove analogous results for the global Euler obstr…

We examine surgery on a knot in $S^3$ to determine surgery obstructions to Seifert fibered integral homology spheres. We find such surgery obstructions using Heegaard Floer, Knot Floer homology and the mapping cone formula for computing Heegaard Floer homology of surgery on a knot. Here however, we take a different app…

Obstructions found for closed Fedosov star products on symplectic and Kähler manifolds.

Computes homology of an obstruction chain complex in grid homology.

There is an obstruction to the existence of Kähler -Einstein metrics which is used to define the GIT weight for K-stability, and it has been extended to various geometric problems. This survey paper considers such extended obstructions to the existence problem of Kähler -Ricci solitons, Sasaki-Einstein metrics and (con…