A remark on the virtual homotopical dimension of some moduli spaces of stable Riemann surfaces

A remark on an algebraic Riemann-Roch formula for flat bundles

We propose an algebraic Riemann-Roch formula for moving flat bundles on contant families in characteristic zero with values in the ring of algebraic differential characters. This formula lifts Grothendieck-Riemann-Roch formula in the Chow groups and Bismut-Lott formula in the odd dimensional Betti cohomology with $\C/\Z$ coefficients over the field of complex numbers.

A remark on the field of moduli of Riemann surfaces

Let $S$ be a closed Riemann surface of genus $g\geq 2$ and let ${\rm Aut}(S)$ be its group of conformal automorphisms. It is well known that if either: (i) ${\rm Aut}(S)$ is trivial or (ii) $S/{\rm Aut}(S)$ is an orbifold of genus zero with exactly three cone points, then $S$ is definable over its field of moduli ${\mathcal M}(S)$. In the complementary situation, explicit examples for which ${\mathcal M}(S)$ is not a field of definition are known. We provide upper bounds for the minimal degree extension of ${\mathcal M}(S)$ by a field of definition in terms of the quotient orbifold $S/{\rm Aut…

Remarks on groups of bundle automorphisms over the Riemann sphere

A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff-Grothendieck bundle splitting $\bigoplus_{i=1}^{r} \mathcal(m_{i})$ over $\mathbb{C}\mathbb{P}^{1}$ is provided, in terms of its action on a suitable space of generalized flags in the fibers over a finite subset $S\subset\mathbb{C}\mathbb{P}^{1}$. The relevance of such characterization derives from the possibility of constructing geometric models for diverse moduli spaces of stable objects in genus 0, such as parabolic bundles, parabolic Higgs bundles, and logarithmic connections, as collection…

Some remarks on the Schweitzer complex

We prove that the Schweitzer complex is elliptic and its cohomologies define cohomological functors. As applications, we obtain finite dimensionality, a version of Serre duality, restrictions of the behaviour of cohomology in small deformations, and an index formula which turns out to be equivalent to the Hirzebruch-Riemann-Roch relations.

超 Riemann 曲面模空间的高阶 Weil-Petersson 体积

Higher Weil-Petersson volumes of the moduli space of super Riemann surfaces

本文将 Stanford-Witten 关于超 Riemann 曲面模空间 Weil-Petersson 体积的递归公式推广到包含高次 κ 类的情形。

A remark on the jet bundles over the projective line

This is a footnote of a recent interesting work of Cohen, Manin and Zagier, where they, among other things, produce a natural isomorphism between the sheaf of (n-1)-th order jets of the n-th tensor power of the tangent bundle of a Riemann surface equipped with a projective structure and the sheaf of differential operators of order n (on the trivial bundle) with vanishing 0-th order part. We give a different proof of this result without using the coordinates, and following the idea of this proof we prove: Take a line bundle L with $L^2 = T$ on a Riemann surface equipped with a projective struct…