arXiv research

The paper examines Kodaira dimensions of specific complex 4-manifolds with torsion first Chern class.

We study the class of compact complex manifolds whose first Chern class vanishes in the Bott-Chern cohomology. This class includes all manifolds with torsion canonical bundle, but it is strictly larger. After making some elementary remarks, we show that a manifold in Fujiki's class C with vanishing first Bott-Chern cla…

Study categorizes Vaisman manifolds with vanishing first Chern class and finds canonical metrics.

We prove that any compact complex homogeneous space with vanishing first Chern class after an appropriate deformation of the complex structure admits a homogeneous Calabi-Yau with torsion structure, provided that it also has an invariant volume form. A description of such spaces among the homogeneous C-spaces is given …

The study finds tight contact structures without fillings in high dimensions.

Study shows uniform bounds on torsion and curvature for Chern-Ricci flow solutions.

In this note, we report on a work jointly done with C. Simpson on a generalization of Reznikov's theorem which says that the Chern-Simons classes and in particular the Deligne Chern classes (in degrees $> 1$) are torsion, of a flat vector bundle on a smooth complex projective variety. We consider the case of a smooth q…

Study fractional structures on bundle gerbe modules using rational homotopy theory.

New findings on Chern flat metrics and their criticality.

Study on special Hermitian manifolds with specific connection properties.

We make a detailed study of the Heegaard Floer homology of the product of a closed surface Sigma_g of genus g with S^1. We determine HF^+ for this 3-manifold completely for the spin^c structure having trivial first Chern class, which for g>2 was previously unknown. We show that in this case HF^\infty is closely related…

Using different forms of the arithmetic Riemann-Roch theorem and the computations of Bott-Chern secondary classes, we compute the analytic torsion and the height of Hirzebruch surfaces.

We present a formula for the full Cheeger-Chern-Simons class of the tautological flat complex vector bundle of rank two over BSL(2,\C^δ). Our formula improves the formula by Dupont and Zickert, where the class is only computed modulo 2-torsion.

Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected and abelian. A rigorous definition of an abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. A symplectic abelian Chern-Simons partition functi…

Study local third Chern class for point singularities on threefolds.

We consider topological field theories that compute the Reidemeister-Milnor-Turaev torsion in three dimensions. These are the psl(1|1) and the U(1|1) Chern-Simons theories, coupled to a background complex flat gauge field. We use the 3d mirror symmetry to derive the Meng-Taubes theorem, which relates the torsion and th…

This paper introduces complex Chern-Simons bundles in families setting and proves their crystalline nature.

Study geometric formal metrics and Massey products on Kähler manifolds with torsion.

Derivative estimates for pluriclosed flow control curvature and torsion.

We calculate the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.

Let $X$ be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that $X$ admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if $X$…

In this note, we affirm the partial answer to the long open Conjecture which states that any closed embeddable strictly pseudoconvex CR $3$-manifold admits a contact form $θ$ with the vanishing CR $Q$-curvature. More precisely, we deform the contact form according to an CR analogue of $Q$%-curvature flow in a closed st…

We present two formulas for Chern classes of the tensor product of two vector bundles. In the first formula we consider a matrix containing Chern classes of the first bundle and we take a polynomial of this matrix with Chern classes of the second bundle as coefficients. The determinant of this expression equals the Che…

Formula for analytic torsion forms in fibrations by projective curves.

In this paper we provide examples of hypercomplex manifolds which do not carry HKT structure. We also prove that the existence of HKT structure is not stable under small deformations. Similarly we provide examples of compact complex manifolds with vanishing first Chern class which do not admit a Hermitian structure wit…

Study on spherical CR manifolds with non-trivial Chern classes.

New computations show symplectic groups and mapping class groups have different properties regarding torsion.

Develops method to compute Chern-Simons potentials from higher-dimensional Pontryagin densities.

The study characterizes compact homogeneous manifolds with Bismut parallel torsion.

We obtain a general lower bound for the number of fixed points of a circle action on a compact almost complex manifold $M$ of dimension $2n$ with nonempty fixed point set, provided the Chern number $c_1c_{n-1}[M]$ vanishes. The proof combines techniques originating in equivariant K-theory with celebrated number theory …

To each second-order ordinary differential equation $σ$ on a smooth manifold $M$ a $G$-structure $P^σ$ on $J^1(\mathbb{R},M)$ is associated and the Chern connection $\nabla ^σ$ attached to $σ$ is proved to be reducible to $P^σ$; in fact, $P^σ$ coincides generically with the holonomy bundle of $\nabla ^σ$. The cases of …

Uniform estimates prove convergence of Chern-Ricci flow on complex surfaces.

This paper studies torsion obstructions to complex sections on manifolds.

Extends Gauduchon's result to higher dimensions, showing balanced metrics.

We study the pseudoriemannian geometry of almost parahermitian manifolds, obtaining a formula for the Ricci tensor of the Levi-Civita connection. The formula uses the intrinsic torsion of an underlying SL(n,R)-structure; we express it in terms of exterior derivatives of some appropriately defined differential forms. As…

We construct a canonical noncommutative spectral triple for every oriented closed Riemannian manifold, which represents the fundamental class in the twisted K-homology of the manifold. This so-called "projective spectral triple" is Morita equivalent to the well-known commutative spin spectral triple provided that the m…

We show that from the asymptotic behavior of an evaluation of the colored Jones polynomial of the figure-eight knot we can extract the Chern--Simons invariant and the twisted Reidemeister torsion associated with a representation of the fundamental group of the knot complement to the two-dimensional complex special line…

In this note, we prove that on an $n$-dimensional compact toric manifold with positive first Chern class, the Kähler-Ricci flow with any initial $(S^1)^n$-invariant Kähler metric converges to a Kähler-Ricci soliton. In particular, we give another proof for the existence of Kähler-Ricci solitons on a compact toric manif…

We describe some relations between coefficients of irreducible components of the first Chern class [FP15] and birational germs introduced by Dloussky {Dl16] for intermediate Kato surfaces.

The paper establishes inequalities for Chern classes and numbers on polarized manifolds and nef vector bundles.

We prove that the first Chern form of the moduli space of polarized Calabi-Yau manifolds, with the Hodge metric or the Weil-Petersson metric, represent the first Chern class of the canonical extensions of the tangent bundle to the compactification of the moduli space with normal crossing divisors.

Flow smooths Chern-Ricci-flat metrics on Hermitian manifolds.

The Klein-Grifone approach to global Finsler geometry is adopted. A global existence and uniqueness theorem for Chern connection is formulated and proved. The torsion and curvature tensors of Chern connection are derived. Some properties and the Bianchi identities for this connection are investigated. A concise compari…

A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern--Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers.…

The paper uses symplectic homology to study 3D Besse manifolds with vanishing first Chern class.

In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Her…

Decomposes singular Kähler spaces with trivial first Chern class into simpler components.

We show that Chern-Weil theory for tensor bundles over manifolds is a consequence of the existence of natural closed differential forms on total spaces of torsion free connections on frame bundles.