Related papers: Johnson's homomorphisms and the Arakelov-Green fun…

Let \Sigma be a compact orientable surface with genus g and n boundary components B = (B_1,..., B_n). Let c = (c_1,...,c_n) in [-2,2]^n. Then the mapping class group MCG of \Sigma acts on the relative SU(2)-character variety X_c :=…

Let G_2 be the exceptional Lie group of automorphisms of the complex Cayley algebra and C be a generic, smooth, connected, projective curve over $\mathbb{C}$ of genus at least 2. For a complex Lie group G, let H^0(M(G),L^k) be the space of…

Let $G$ be a connected reductive affine algebraic group defined over the complex numbers, and $K\subset G$ be a maximal compact subgroup. Let $X , Y$ be irreducible smooth complex projective varieties and $f: X \rightarrow Y$ an algebraic…

We express the first jet bundle of curves in Euclidean space as homogeneous spaces associated to a Galilean-type group. Certain Cartan connections on a manifold with values in the Lie algebra of the Galilean group are characterized as…

We define the equivariant Chern-Schwartz-MacPherson class of a possibly singular algebraic variety with a group action over the complex number field (or a field of characteristic 0). In fact, we construct a natural transformation from the…

We produce an equality between the Gromov-Witten invariants of the moduli space M of rank two odd degree stable vector bundles over a Riemann surface $\Sigma$ and the Donaldson invariants of the algebraic surface $\Sigma \times P^1$. We…

A formula for the first Chern class of the Verlinde bundle over the moduli space of smooth genus g curves is given. A finite-dimensional argument is presented in rank 2 using geometric symmetries obtained from strange duality, relative…

We consider closed manifolds that admit a metric locally isometric to a product of symmetric planes. For such manifolds, we prove that the Euler characteristic is an obstruction to the existence of flat structures, confirming an old…

We use recent progress on Chern-Simons gauge theory in three dimensions [18] to give explicit, closed form formulas for the star product on some functions on the affine space ${\mathcal A}(\Sigma)$ of (smooth) connections on the trivialized…

Let $f:\mathcal{X}\to S$ be a proper holomorphic submersion of complex manifolds and $G$ a complex reductive linear algebraic group with Lie algebra $\mathfrak{g}$. Assume also given a holomorphic principal $G$-bundle $\mathcal{P}$ over…

We consider the moduli space of flat $SO(2n+1)$-connections (up to gauge transformations) on a Riemann surface, with fixed holonomy around a marked point. There are natural line bundles over this moduli space; we construct geometric…

Let $\widetilde{\cal M}_{g,n}$, for $2g-2+n>0$, be the moduli stack of $n$-pointed, genus $g$, stable complex curves of compact type. Various characterizations and properties are obtained of both the algebraic and topological fundamental…

Let $\MS_g$ be the moduli space of stable holomorphic vector bundles of rank 2 and fixed determinant of odd degree over a smooth complex projective curve of genus $g$. This paper proves various properties of the rational cohomology ring…

We show that a flat principal bundle with compact connected structure group and its adjoint bundles of Lie groups have the same cohomology as the trivial bundle, which is done by proving they satisfy the condition for the Leray-Hirsch…

For a class of closed manifolds N, we construct a family of functions on the Hamiltonian group G of the cotangent bundle T*N. These restrict to homogeneous quasi-morphisms on the subgroup generated by Hamiltonians with support in a given…

We give a presentation of the cohomology ring of spatial polygon spaces $M(r)$ with fixed side lengths $r \in \mathbb R^n_+$. These spaces can be described as the symplectic reduction of the Grassmaniann of 2-planes in $\mathbb C^n$ by the…

In this note, we investigate the Chern classes of flat bundles in the arithmetic Deligne Cohomology, introduced by Green-Griffiths, Asakura-Saito. We show nontriviality of the Chern classes in some cases and the proof also indicates that…

We consider the moduli spaces of flat $SL(n, C)$-bundles on Riemann surfaces with one puncture when we fix the conjugacy class ${\cal C}$ of the monodromy transformation around the puncture. We show that under a certain condition on the…

Let G be a Lie goup, let M and N be smooth connected G-manifolds, let f be a smooth G-map from M to N, and let P denote the fiber of f. Given a closed and equivariantly closed relative 2-form for f with integral periods, we construct the…

We give a differential geometric construction of the holomorphic family of Higgs bundle moduli spaces over a curve C as a fibration over Teichm\"uller space. The method uses a function f defined on the character variety, essentially the…