Moduli spaces of parabolic U(p,q)U(p,q)-Higgs bundles

Using the $L^2$-norm of the Higgs field as a Morse function, we count the number of connected components of the moduli space of parabolic $U(p,q)$-Higgs bundles over a Riemann surface with a finite number of marked points, under certain genericity conditions on the parabolic structure. This space is homeomorphic to the moduli space of representations of the fundamental group of the punctured surface in $U(p,q)$, with fixed compact holonomy classes around the marked points. We apply our results to the study of representations of the fundamental group of elliptic surfaces of general type.Comment: 46 pages, no figures. Corrected typos, added remarks. To appear in "Quarterly Journal of Mathematics

Moduli spaces of framed GG--Higgs bundles and symplectic geometry

Let $X$ be a compact connected Riemann surface, $D\, \subset\, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x\, \subsetneq\, G_x$ a Zariski closed subgroup for every $x\, \in\, D$. A framed principal $G$--bundle is a pair $(E_G,\, \phi)$, where $E_G$ is a holomorphic principal $G$--bundle on $X$ and $\phi$ assigns to each $x\, \in\, D$ a point of the quotient space $(E_G)_x/H_x$. A framed $G$--Higgs bundle is a framed principal $G$--bundle $(E_G,\, \phi)$ together with a section $\theta\, \in\, H^0(X,\, \text{ad}(E_G)\otimes K_X\otimes{\mathcal O}_X(D))$ such that $\theta(x)$ is compatible with the framing $\phi$ for every $x\, \in\, D$. We construct a holomorphic symplectic structure on the moduli space $\mathcal{M}_{FH}(G)$ of stable framed $G$--Higgs bundles. Moreover, we prove that the natural morphism from $\mathcal{M}_{FH}(G)$ to the moduli space $\mathcal{M}_{H}(G)$ of $D$-twisted $G$--Higgs bundles $(E_G,\, \theta)$ that forgets the framing, is Poisson. These results generalize \cite{BLP} where $(G,\, \{H_x\}_{x\in D})$ is taken to be $(\text{GL}(r,{\mathbb C}),\, \{\text{I}_{r\times r}\}_{x\in D})$. We also investigate the Hitchin system for $\mathcal{M}_{FH}(G)$ and its relationship with that for $\mathcal{M}_{H}(G)$.Comment: Final versio

On moduli spaces of Hitchin pairs

Let $X$ be a compact Riemann surface $X$ of genus at--least two. Fix a holomorphic line bundle $L$ over $X$. Let $\mathcal M$ be the moduli space of Hitchin pairs $(E ,\phi\in H^0(End(E)\otimes L))$ over $X$ of rank $r$ and fixed determinant of degree $d$. We prove that, for some numerical conditions, $\mathcal M$ is irreducible, and that the isomorphism class of the variety $\mathcal M$ uniquely determines the isomorphism class of the Riemann surface $X$.Comment: 18 pages; final version, accepted in Math. Proc. Cambridge Phil. So

Hodge polynomials of the SL(2, C)-character variety of an elliptic curve with two marked points

We compute the Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2, C). When we fix the conjugacy classes of the representations around the marked points to be diagonal and of modulus one, the character variety is diffeomorphic to the moduli space of strongly parabolic Higgs bundles, whose Betti numbers are known. In that case we can recover some of the Hodge numbers of the character variety. We extend this result to the moduli space of doubly periodic instantons

Agua, salud y análisis costo/beneficio social

In this paper, it is shown the relationship between coverage in water and sanitation, and hydric disease’s incidence. There are synthesized the situations of the more affected regions and there are presented the Millennium Development Goals on the subject. Briefly, there are summarized the social cost/benefit analysis applicable to public projects, and it is studied the particular case of its application to the Millennium Development Goals in water and sanitation

Water, health and social cost/benefit analysis

In this paper, it is shown the relationship between coverage in water and sanitation, and hydric disease’s incidence. There are synthesized the situations of the more affected regions and there are presented the Millennium Development Goals on the subject. Briefly, there are summarized the social cost/benefit analysis applicable to public projects, and it is studied the particular case of its application to the Millennium Development Goals in water and sanitation.water; health; Millennium Development Goals