arXiv research

Study local third Chern class for point singularities on threefolds.

Here we are fixing an output of a trivial calculation based on Konsevich's differential 2-form for the Chern class of polygon bundle. As a result an interesting combinatorics and arithmetics jumps right out of a jukebox. The calculation gives very simple rational combinatorial characteristics (we call it "curvature") o…

We develop a theory of Cech-Bott-Chern cohomology and in this context we naturally come up with the relative Bott-Chern cohomology. In fact Bott-Chern cohomology has two relatives and they all arise from a single complex. Thus we study these three cohomologies in a unified way and obtain a long exact sequence involving…

Study shows compact Sasakian manifolds are locally Heisenberg up to deformation.

We prove an Atiyah-Bott-Berline-Vergne type localization formula for Killing foliations in the context of equivariant basic cohomology. As an application, we localize some Chern-Simons type invariants, for example the volume of Sasakian manifolds and secondary characteristic classes of Riemannian foliations, to the uni…

For elliptic principal bundles $π:X\ra B$ over Kähler manifolds it was shown by Blanchard that $X$ has a Kähler metric if and only both Chern classes (with real coefficients) of $π$ vanish. For some elliptic principal bundles, when the span of these Chern classes is 1-dimensional, it was shown by Vaisman that $X$ carry…

A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering, with the monodromy acting on this covering by homotheties. We define three cohomology invariants, the Lee class, the Morse-Novikov class, and the Bott-Chern class, of an LCK-structure. These invariants together play the same …

We prove that the compact Kaehler manifolds with first Chern class nonnegative that admit holomorphic parabolic geometries are the flat bundles of rational homogeneous varieties over complex tori. We also prove that the compact Kaehler manifolds with negative first Chern class that admit holomorphic cominiscule geometr…

Let X be a Hermitian locally symmetric space. We prove that every Chern class of X has a canonical lift to the cohomology of the Baily- Borel-Satake compactification X* of X and that the resulting Chern numbers satisfy the Hirzebruch proportionality formula with respect to the compact dual X^ of X. The same result hold…

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…

The paper addresses deformations of Kähler spaces with vanishing first Chern class.

For a smooth family F of admissible elliptic pseudodifferential operators with differential form coefficients associated to a geometric fibration of manifolds M--> B we show that there is a natural zeta-form z(F,s) and zeta-determinant- form det(F) in the de-Rham algebra of smooth differential forms, generalizing the c…

For a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern-Euler class and used by Sha to formulate a relative Poincaré-Hopf theorem, under the condition that the metric on the manifold is locall…

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…

Quantum phase diagrams for Chern topological insulators show jumps at critical loci.

New classes defined for manifold pseudogroups, linking to cohomology and bundle structures.

The paper explores moduli space of heterotic system using two deformation paths.

The Hopf surfaces provide a family of minimal non-Kähler surfaces of class VII on which little is known about the Chern-Ricci flow. We use a construction of Gauduchon-Ornea for locally conformally Kähler metrics on primary Hopf surfaces of class 1 to study solutions of the Chern-Ricci flow. These solutions reach a volu…

Generalizes double transgression formulas on complex manifolds.

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…

For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class $\mathcal C$, whose dimensio…

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…

Principal circle bundle over a PL polyhedron can be triangulated and thus obtains combinatorics. The triangulation is assembled from triangulated circle bundles over simplices. To every triangulated circle bundle over a simplex we associate a necklace (in combinatorial sense). We express rational local formulas for all…

Study local properties of Chern-scalar curvature through linearization stability.

Explicit formula for Bergman kernel of abelian varieties proved.

The paper finds Riemannian metric representatives for Stiefel-Whitney classes.

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…

In this paper, we prove the following two results: First, we study a class of conformally invariant operators $P$ and their related conformally invariant curvatures $Q$ on even-dimensional Riemannian manifolds. When the manifold is locally conformally flat(LCF) and compact without boundary, $Q$-curvature is naturally r…

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

The paper proves inequalities for orbifold second Chern classes in Fujiki's class.

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

Assuming local uniform bounds on the metric for a solution of the Chern-Ricci flow, we establish local Calabi and curvature estimates using the maximum principle.

Study on special Hermitian metrics on cohomogeneity one manifolds.

We develop a Chern-Weil theory for compact Lie group action whose generic stabilizers are finite in the framework of equivariant cohomology. This provides a method of changing an equivariant closed form within its cohomological class to a form more suitable to yield localization results. This work is motivated by our w…

We introduce certain relative differential characters which we call Cheeger-Chern-Simons characters. These combine the well-known Cheeger-Simons characters with Chern-Simons forms. In the same way as the Cheeger-Simons characters generalize Chern-Simons invariants of oriented closed manifolds, the Cheeger-Chern-Simons …

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

Short proof given for a complex formula in spin manifold geometry.

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

Local index theorem for chiral geometric operators proved using heat kernel.

We study the symplectic Bott-Chern cohomology by L.-S. Tseng and S.-T. Yau for solvmanifolds endowed with left-invariant symplectic structures. Our results are applicable to cohomology with values in local systems. Studying symplectic Bott-Chern cohomology of solvmanifolds with values in local systems, we give some rem…

Study second-Chern-Einstein metrics on 4D manifolds, finding Killing vector fields and examples.

The paper generalizes current constructions to cohesive modules and characteristic forms.

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

A version of smooth K-theory is constructed, which is adapted to the total Chern class instead of the Chern character (contrarily to previous theories). Some total Chern class morphism from this K-theory to Cheeger-Simons differential characters is constructed. This answers a question raised by U. Bunke.

Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler.

New $L_\infty$ liftings derived from Chern-Simons classes for coherent sheaves.

In this paper we give explicit formulas of differential characteristic classes of principal $G$-bundles with connections and prove their expected properties. In particular, we obtain explicit formulas for differential Chern classes, differential Pontryagin classes and differential Euler class. Furthermore, we show that…

We introduce the notion of a {\vartheta}-summable Fredholm module over a locally convex dg algebra Ω and construct its Chern character as a cocycle on the entire cyclic complex of Ω, extending the construction of Jaffe, Lesniewski and Osterwalder to a differential graded setting. Using this Chern character, we prove an…