Gauss-Bonnet-Chern theorem on moduli space

In this paper, we study the Chern classes on the moduli space of polarized Calabi-Yau manifolds. We prove that the integrations of the invariants of the curvature of the Weil-Petersson metric are finite. In some special cases, they are even rational numbers.

In this paper, we proved that the Weil-Petersson volume of Calabi-Yau moduli is a rational number. We also proved that the integrations of the invariants of the Ricci curvature of the Weil-Petersson metric with respect to the Weil-Petersson volume form are all rational numbers.

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.

In this paper we proved that the Weil-Petersson volume of the Chern class of any order over the moduli space of Calabi-Yau manifolds is a rational number. We also found the necessary and sufficient condition of the incompleteness of Weil-Petersson metric in several variables case.

In a previous article, we generalised the classical four-dimensional Chern-Gauss-Bonnet formula to a class of manifolds with finitely many conformally flat ends and singular points, in particular obtaining the first such formula in a dimension higher than two which allows the underlying manifold to have isolated conical singularities. In the present article, we extend this result to all even dimensions $n\geq 4$ in the case of a class of conformally flat manifolds.

In this paper it is proved that the volumes of the moduli spaces of polarized CY manifolds with respect to the Weil-Petersson metrics are finite and they are rational numbers.

This expository paper contains a detailed introduction to some important works concerning the Gauss-Bonnet-Chern theorem. The study of this theorem has a long history dating back to Gauss's Theorema Egregium (Latin: Remarkable Theorem) and culminated in Chern's groundbreaking work [14] in 1944, which is a deep and wonderful application of Elie Cartan's formalism. The idea and tools in [14] have a great generalization and continue to produce important results till today. In th...

We prove a Gauss-Bonnet theorem for (finite coverings of) moduli spaces of Riemann surfaces endowed with the McMullen metric. The proof uses properties of an exhaustion of moduli spaces by compact submanifolds with corners and the Gauss-Bonnet formula of Allendoerfer and Weil for Riemannian polyhedra.

In this paper, we propose a sharp and quantitative criterion, which focuses solely on $Q$ curvature, to demonstrate the Chern-Gauss-Bonnet integral. In contrast to the previous results [4,5,10], we use a new approach that involves estimating the singular integral. Furthermore, we derive the asymptotic formula for the solution to the general $Q$ curvature equation.

This paper forms part of a larger work where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of "global conformal invariants"; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand.