arXiv research

Smooth SE structures on Sasaki-joins and Bott orbifolds constructed.

We study the Bott-Chern cohomology of complex orbifolds obtained as quotient of a compact complex manifold by a finite group of biholomorphisms.

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.

In this work we prove a residue formula for Morita-Futaki-Bott invariant with respect any holomorphic vector fields with isolated (possibly degenerated) singularities in terms of Grothendieck's residues.

Study SO(3) vortices over orbifold Riemann surfaces and monopoles.

The paper finds CSC Sasaki metrics on specific 7-manifolds.

Symplectic coordinates found on projective structures on orbifolds.

Motivated by the moduli theory of taut contact circles on spherical 3-manifolds, we relate taut contact circles to transversely holomorphic flows. We give an elementary survey of such 1-dimensional foliations from a topological viewpoint. We describe a complex analogue of the classical Godbillon-Vey invariant, the so-c…

In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an a…

The theory of principal $G$-bundles over a Lie groupoid is an important one, unifying the various types of principal $G$-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal $G$-bundles. In this paper, we study the differential geometry of these objects, including connections …

The paper studies Morse theory for Lie algebra actions on Riemannian foliations.

Locally conformally Kahler (LCK) manifolds with potential are those which admit a Kahler covering with a proper, automorphic Kaehler potential. Existence of a potential can be characterized cohomologically as a vanishing of a certain cohomology class, called the Bott-Chern class. Compact LCK manifolds with potential ar…

Paper develops equivariant basic cohomology for Lie groupoids.

In this paper, we compute contact homology of some quasi-regular contact structures, which admit Hamiltonian actions of Reeb type of Lie groups. We will discuss the toric contact case, (where the torus is of Reeb type), and the case of homogeneous contact manifolds. In both of these cases the quotients by the Reeb acti…

In this paper, we consider generalized moment maps for Hamiltonian actions on $H$-twisted generalized complex manifolds introduced by Lin and Tolman \cite{Lin}. The main purpose of this paper is to show convexity and connectedness properties for generalized moment maps. We study Hamiltonian torus actions on compact $H$…

We propose a version of the Hodge conjecture in Bott-Chern cohomology and using results from characterizing real holomorphic chains by real rectifiable currents to provide a proof for this question. We define a Bott-Chern differential cohomology and use atomic section theory of Harvey and Lawson to construct refined Bo…

Formula derived for Bott-Chern classes in complex blow-ups.

In this paper, we develop twisted $K$-theory for stacks, where the twisted class is given by an $S^1$-gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure $K^i_α\otimes K^j_β\to K^{i+j}_{α+β}$ are derived. Our approach provides a uniform framework …

We derive a discrete analogue of Morse-Bott theory on CW complexes and use this discrete Morse-Bott function to do some Conley theory analysis. It turns out that our discrete Morse-Bott theory is indeed a generalization of Forman's discrete Morse theory.

Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.

A notion of geometric formality in the context of Bott-Chern and Aeppli cohomologies on a complex manifold is discussed. In particular, by using Aeppli-Bott-Chern-Massey triple products, it is proved that geometric Aeppli-Bott-Chern formality is not stable under small deformations of the complex structure.

New findings on complex manifold properties under deformations.

In studying the Bott-Chern and Aeppli cohomologies for q-complete manifolds, we introduce the class of cohomologically Bott-Chern q-complete manifolds.

In the present paper, we define Morse-Bott functions on manifolds with boundary which are generalizations of Morse functions and show Morse-Bott inequalities for these manifolds.

Study of deformed Bott-Chern cohomology on complex manifolds.

The Morse-Bott inequalities relate the topology of a closed manifold to the topology of the critical point set of a Morse-Bott function defined on it. The Morse-Bott inequalities are sometimes stated under incorrect orientation assumptions. We show that these assumptions are insufficient with an explicit counterexample…

Study on Kähler manifolds proves weak decompositions and relates harmonic forms.

Spin-structures on real Bott manifolds with Kähler structure are characterized.

Condition found for spinc structures on a specific type of manifold.

Unified Morse-Bott-Smale chain complex, resolves well-definedness issue.

The paper classifies Ricci solitons on specific Lorentzian Lie groups.

The Alexander polynomial is linked to Bott-Cattaneo-Rossi invariants via Chern-Simons theory.

Develops virtual Morse-Bott indices for four-manifolds, proving inequalities.

Research resolves sign conventions in Floer theory for Morse-Bott case.

Study bigraded formality and Aeppli-Bott-Chern-Massey products on complex manifolds.

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

The paper extends Bott-Chern Laplacian definition and explores its properties on almost Hermitian manifolds.

A real Bott manifold is the total space of iterated RP^1 bundles starting with a point, where each RP^1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z/2 coefficients are isomorphic. A real Bott manifold is a …

Lectures on polar actions and their properties in Riemannian geometry.

Study stabilizes Morse-Bott cohomology for equivariant manifolds.

Real Bott manifolds is a class of flat manifolds with holonomy group $\mathbb Z_2^k$ of diagonal type. In this paper we want to show how we can compute even Stiefel - Whitney classes on real Bott manifolds. This paper is an answer to the question of professor Masuda if is it possible to extend A. Gąsior "Spin-structure…

Study $L^2$ Hilbert complexes on complex manifolds.

Study new invariants in complex geometry using Bott-Chern hypercohomology.

Study primitive decompositions for harmonic forms on almost Kähler manifolds.

We show that a Bott-Morse foliation in dimension 3 admits a linear, singular, Poisson structure of rank 2 with Bott-Morse singularities. We provide the Poisson bivectors for each type of singular component, and compute the symplectic forms of the characteristic distribution.

We give the complete Bott-Chern-Aeppli cohomology for compact complex 3-folds in terms of Dolbeault, Frolicher, a bi-degree DeRham-like type of cohomology, $K^{p,q}$, defined as $$ K^{p,q}=\frac{ker( \partial ) \cap ker( {\bar{\partial}}) }{im( \partial )\cap ker( {\bar{\partial}} )+im( {\bar{\partial}})\cap ker( \part…

Two Morse-Bott volume forms are diffeomorphic if their cohomology classes are equal.

Only products of projective lines have vanishing Futaki invariants for all Kähler classes.