相关论文: D\'eterminant des op\'erateurs de Cauchy-Riemann r…

This paper consists in a very brief English summary of the results appearing in French in two previous articles. We omit the proofs and focus on explaining our approach and theorems. This paper is not intended to be published. Questions are…

Let $(N,c_N)$ be a complex vector bundle equipped with a real structure over a real curve $(\Sigma,c_\Sigma)$ of genus $g \geq 0$. We compute the sign of the action of the automorphisms of $(N,c_N)$ lifting the identity of $\Sigma$ on the…

Let $(\Sigma,\tau)$ denote a Riemann surface of genus $g \geq 2$ equipped with an anti-holomorphic involution $\tau$. In this paper we study the topology of the moduli space $M(r,\xi)^\tau$ of stable Real vector bundles over $(\Sigma,\tau)$…

We give an explicit formula for the holonomy of the orientation bundle of a family of real Cauchy-Riemann operators. A special case of this formula resolves the orientability question for spaces of maps from Riemann surfaces with Lagrangian…

We study the moduli space of solutions to the Seiberg-Witten equations with $N$ spinors on a compact Riemann surface. These moduli spaces arise in a program to define a new enumerative invariant of 3-manifolds. They are also of independent…

Just as a symmetric surface with separating fixed locus halves into two oriented bordered surfaces, an arbitrary symmetric surface halves into two oriented symmetric half-surfaces, i.e. surfaces with crosscaps. Motivated in part by the…

We consider the moduli space of rank two, odd degree, semi-stable Real vector bundles over a real curve, calculating the singular cohomology ring in odd and zero characteristic for most examples.

We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for…

Motivated by counting pseudo-holomorphic curves in symplectic Calabi-Yau $3$-folds, this article studies a chamber structure in the space of real Cauchy-Riemann operators on a Riemann surface, and constructs three chambered invariants…

We show that mapping class groups associated to all types of real algebraic curves are virtual duality groups. We also deduce some results about the orbifold homotopy groups of the moduli spaces of real algebraic curves. We achieve these…

In this paper, we investigate the geometry of the moduli space of curves by using the curvature properties of direct image sheaves of vector bundles. We show that the moduli space $(M_g, \omega_{WP})$ of curves with genus $g>1$ has…

Moduli space of genus zero stable maps to the projective three-space naturally carries a real structure such that the fixed locus is a moduli space for real rational spatial curves with real marked points. The latter is a normal projective…

In this paper we define and study pseudoholomorphic vector bundles structures, particular cases of which are tangent and normal bundle almost complex structures. These are intrinsically related to the Gromov D-operator. As an application we…

There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface.…

We study closures of GL_2(R)-orbits on the total space of the Hodge bundle over the moduli space of curves under the assumption that they are algebraic manifolds. We show that, in the generic stratum, such manifolds are the whole stratum,…

Let $\mathcal{M}_{n,d}$ be the moduli space of semi-stable rank $n$, trace-free Higgs bundles with fixed determinant of degree $d$ on a Riemann surface of genus at least $3$. We determine the following automorphism groups of…

We define Hitchin's moduli space for a principal bundle $P$, whose structure group is a compact semisimple Lie group $K$, over a compact non-orientable Riemannian manifold $M$. We use the Donaldson-Corlette correspondence, which identifies…

The orientability problem in real Gromov-Witten theory is one of the fundamental hurdles to enumerating real curves. In this paper, we describe topological conditions on the target manifold which ensure that the uncompactified moduli spaces…

Given an automorphism of a smooth complex algebraic curve, there is an induced action on the moduli space of semi-stable rank 2 holomorphic bundles with fixed determinant. We give a complete description of the fixed variety in terms of…

If E is a C^\infty complex vector bundle on an oriented C^\infty manifold \Sigma, diffeomorphic to a circle, then the space of sections of E has a canonical polarization in the sense of Pressley and Segal and so one has its determinantal…