arXiv research

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…

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 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.

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.

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

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.

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…

In this note we give a simple, model-independent construction of Chern classes as natural transformations from differential complex K-theory to differential integral cohomology. We verify the expected behaviour of these Chern classes with respect to sums and suspension.

Constructs Chern-Weil classes for Cartan geometries.

We present a simplification of Neumann's formula for the universal Cheeger-Chern-Simons class of the second Chern polynomial. Our approach is completely algebraic, and the final formula can be applied directly on a homology class in the bar complex.

The paper extends Chern-Weil-Lecomte map to $L_{\infty}$-algebras.

We define and study the secondary Chern-Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study index for a vector field with non-isolated singularities on a submanifold. As an application, our studies give conceptual proofs of a classical result of Chern.

This paper calculates Dijkgraaf-Witten invariants from Chern classes.

We present a geometric approach, in the spirit of the Chern-Weil theory, for constructing cocycles representing the classes of the Hopf cyclic cohomology of the Hopf algebra H(n) relative to GL(n, R). This provides an explicit description of the universal Hopf cyclic Chern classes, which complements our earlier geometr…

Using the concept of a cohesive module defined by Block, we use the theory of superconnections in the sense of Quillen to construct natural superconnections on Hermitian cohesive modules. By the Chern-Weil construction, we obtain characteristic classes with values in Bott-Chern cohomology which refines the usual deRham…

We construct Chern-Weil classes on infinite dimensional vector bundles with structure group contained in the algebra $\cl[\leq 0](M, E)$ of non-positive order classical pseudo-differential operators acting on a finite rank vector bundle $E$ over a closed manifold $M$. Mimicking the finite dimensional Chern-Weil constru…

We provide a characterization of quotients of three-dimensional complex tori by finite groups that act freely in codimension one via a vanishing condition on the first and second orbifold Chern class. We also treat the case of actions free in codimension two, using instead the "birational" second Chern class, as we cal…

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…

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

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.

Develops equivariant Chern characters for coherent sheaves with group actions.

We prove explicit formulas for Chern classes of tensor products of vector bundles, with coefficients given by certain universal polynomials in the ranks of the two bundles.

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…

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…

The Chern-Simons forms for R-linear connections on Lie algebroids are considered. A generalized Chern-Simons formula for such R-linear connections is obtained. We it apply to define Chern character and secondary characteristic classes for R-linear connections of Lie algebroids.

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.

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

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

Study volumes of Bott-Chern classes on complex manifolds.

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

Study on contact manifolds reveals infinite isometry groups for large Chern classes.

The paper proves conditions for minimal compact Kähler manifolds with vanishing second Chern class.

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…

Let h be a Real bundle, in the sense of Atiyah, over a space X. This is a complex vector bundle together with an involution which is compatible with complex conjugation. We use the fact that BU is equipped with a structure of conjugation space, as defined by Hausmann, Holm, and Puppe, to construct equivariant Chern cla…

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…

Summing Hamiltonian manifolds with a common submanifold.

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…

We extend finite dimensional Chern-Simons theory to certain infinite dimensional principal bundles with connections, in particular to the frame bundle $FLM\to LM$ over the loop space of a Riemannian manifold $M$. Chern-Simons forms are defined roughly as in finite dimensions with the invariant polynomials replaced by a…

Paper investigates prescribing Chern scalar curvatures on specific manifolds.

The paper establishes inequalities for Chern classes and Riemann-Roch type inequalities for projective manifolds.

Some cohomology elements, called $ν$ classes, as a supergeneralization of universal Chern classes, are introduced for canonical super line bundles over $ν$ projective spaces, a novel supergeometric generalization of projective spaces. It is shown that these classes may be described by analytic representatives of elemen…

An odd Seiberg-Witten invariant imposes bounds on the signature of a closed, almost complex 4-manifold with vanishing first Chern class. This applies in particular to symplectic 4-manifolds of Kodaira dimension zero.