arXiv subjects

Ignasi Mundet i Riera

Publications and source records attributed to Ignasi Mundet i Riera.

27 records · Page 2Linked to original sources

arXiv2012

Higgs bundles and surface group representations in the real symplectic group

In this paper we study the moduli space of representations of a surface group (i.e., the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n,R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor-Wood type inequality. Our main result is a count of the number of connected components of the moduli space of maximal representations, i.e. representations with maximal Toledo invariant. Our approach uses the non-abelian Hodge theory correspondence proved in a companion paper arXiv:0909.4487 [math.DG] to identify the space of representations with the moduli space of polystable Sp(2n,R)-Higgs bundles. A key step is provided by the discovery of new discrete invariants of maximal representations. These new invariants arise from an identification, in the maximal case, of the moduli space of Sp(2n,R)-Higgs bundles with a moduli space of twisted Higgs bundles for the group GL(n,R).

math.AG↗

arXiv2004

A compactification of the moduli space of twisted holomorphic maps

We construct a compactification of the moduli space of twisted holomorphic maps with varying complex structure and bounded energy. For a given compact symplectic manifold $X$ with a compatible complex structure and a Hamiltonian action of $S^1$ with moment map $μ:X\to\imag\RR$, the moduli space which we compactify consists of equivalence classes of tuples $(C,P,A,ϕ)$, where $C$ is a smooth compact complex curve of fixed genus, $P$ is a principal $S^1$ bundle over $C$, $A$ is a connection on $P$ and $ϕ$ is a section of $P\times_{S^1}X$ satisfying $$\ov{\partial}_Aϕ=0,\qquad ι_{v}F_A+μ(ϕ)=c,$$ where $F_A$ is the curvature of $A$, $v$ is the restriction on $C$ of a volume form on the universal curve over $\oM_g$ and $c$ is a fixed constant. Two tuples $(C,P,A,ϕ)$ and $(C',P',A',ϕ')$ are equivalent if there is a morphism of bundles $ρ:P\to P'$ lifting a biholomorphism $C\to C'$ such that $ρ^*A'=A$ and $ρ^*ϕ'=ϕ$. The energy of $(C,P,A,ϕ)$ is $\|F_A\|_{L^2}^2+\|d_Aϕ\|_{L^2}^2 +\|μ(ϕ)-c\|_{L^2}^2$, and the topology of the moduli space is the natural one. We also incorporate marked points in the picture. There are two sources of non compactness. First, bubbling off phenomena, analogous to the one in Gromov--Witten theory. Second, degeneration of $C$ to nodal curves. In this case, there appears a phenomenon which is not present in Gromov--Witten: near the nodes, the section $ϕ$ may degenerate to a chain of gradient flow lines of $-\imagμ$.

math.SG↗

arXiv2003

Representations of the fundamental group of a closed oriented surface in Sp(4,R)

We classify the connected components of the space of representations of the fundamental group of a closed oriented surface of genus $\geq 2$ in $Sp(4,{\mathbf R})$. We prove that this is equivalent to classifying the connected components of the moduli space of representations (which consists of conjugacy classes of reductive representations). We then continue the analysis initiated by Gothen in math/9904114, relying on: (1) the relation between our moduli space of representations and the moduli space of certain (polystable) analogues of Higgs bundles (which follows essentially from work of Hitchin, Simpson and Corlette) and (2) the existence of a proper function on the moduli space whose set of local minima can be identified with the moduli space $N$ of (polystable) quadratic bundles of rank 2; we finally deduce our result from the classification of the connected components of $N$.

math.GT↗

arXiv2002

The symplectic vortex equations and invariants of Hamiltonian group actions

In this paper we define invariants of Hamiltonian group actions for central regular values of the moment map. The key hypotheses are that the moment map is proper and that the ambient manifold is symplectically aspherical. The invariants are based on the symplectic vortex equations. Applications include an existence theorem for relative periodic orbits, a computation for circle actions on a complex vector space, and a theorem about the relation between the invariants introduced here and the Seiberg--Witten invariants of a product of a Riemann surface with a two-sphere.

math.SG↗

arXiv2001

Parabolic vector bundles and equivariant vector bundles

Given a complex manifold $X$, a normal crossing divisor $D\subset X$ whose irreducible components $D_1,...,D_s$ are smooth, and a choice of natural numbers $r=(r_1,...,r_s)$, we construct a manifold $X(D,\ur)$ with an action of a torus $Γ$ and we prove that some full subcategory of the category of $Γ$-equivariant vector bundles on $X(D,r)$ is equivalent to the category of parabolic vector bundles on $(X,D)$ in which the lengths of the filtrations over each irreducible component of $X$ are given by $r$. When $X$ is Kaehler, we study the Kaehler cone of $X(D,r)$ and the relation between the corresponding notions of slope-stability.

math.AG↗

arXiv2000

Hamiltonian Gromov-Witten invariants

In this paper we introduce invariants of semi-free Hamiltonian actions of $S\sp 1$ on compact symplectic manifolds (which satisfy some technical conditions related to positivity) using the space of solutions to certain gauge theoretical equations. These equations generalize at the same time the vortex equations and the holomorphicity equation used in Gromov-Witten theory. In the definition of the invariants we combine ideas coming from gauge theory and the ideas underlying the construction of Gromov-Witten invariants. This paper is based on a part of my PhD Thesis (see math/9912150).

math.SG↗

arXiv2000

Lifts of smooth group actions to line bundles

Let X be a compact manifold with a smooth action of a compact connected Lie group G. Let $L\to X$ be a complex line bundle. Using the Cartan complex for equivariant cohomology, we give a new proof of a theorem of Hattori and Yoshida which says that the action of G lifts to L if and only if the first Chern class $c\sb 1(L)$ of L can be lifted to an integral equivariant cohomology class in $H\sp 2\sb G(X;\ZZ)$, and that the different lifts of the action are classified by the lifts of $c\sb 1(L)$ to $H\sp 2\sb G(X;\ZZ)$. As a corollary of our method of proof, we prove that, if the action is Hamiltonian and $\nabla$ is a connection on L which is unitary for some metric on L and whose curvature is G-invariant, then there is a lift of the action to a certain power $L\sp d$ (where d is independent of L) which leaves fixed the induced metric on $L^d$ and the connection $\nabla\sp{\otimes d}$. This generalises to symplectic geometry a well known result in Geometric Invariant Theory.

math.DG↗

arXiv1999

Yang-Mills-Higgs theory for symplectic fibrations

Our aim in this work is to study a system of equations which generalises at the same time the vortex equations of Yang-Mills-Higgs theory and the holomorphicity equation in Gromov theory of pseudoholomorphic curves. We extend some results and definitions from both theories to a common setting. We introduce a functional generalising Yang-Mills-Higgs functional, whose minima coincide with the solutions to our equations. We prove a Hitchin-Kobayashi correspondence allowing to study the solutions of the equations in the Kaehler case. We give a structure of smooth manifold to the set of (gauge equivalence classes of) solutions to (a perturbation of) the equations (the so-called moduli space). We give a compactification of the moduli space, generalising Gromov's compactification of the moduli of holomorphic curves. Finally, we use the moduli space to define (under certain conditions) invariants of compact symplectic manifolds with a Hamiltonian almost free action of S^1. These invariants generalise Gromov-Witten invariants. This is the author's Ph.D. Thesis. A chapter of it is contained in the paper math.DG/9901076. After submitting his thesis in April 1999, the author knew that K. Cieliebak, A. R. Gaio and D. Salamon had also arrived (from a different point of view) at the same equations, and had developed a very similar programme (see math.SG/9909122).

math.SG↗

arXiv1999

A Hitchin-Kobayashi correspondence for Kaehler fibrations

Let $X$ be a compact Kaehler manifold and $E\to X$ a principal $K$ bundle, where $K$ is a compact connected Lie group. Let ${\cal A}^{1,1}$ be the set of connections on $E$ whose curvature lies in $Ω^{1,1}(E\times_{Ad} {\frak k})$, where ${\frak k}$ is the Lie algebra of $K$. Endow $\frak k$ with a nondegenerate biinvariant bilinear pairing. This allows to identify $\{\frak k}\simeq{\frak k}^*$. Let $F$ be a Kaehler left $K$-manifold and suppose that there exists a moment map $μ$ for the action of $K$ on $F$. Let ${\cal S}=Γ(E\times_K F)$. In this paper we study the equation $$ΛF_A+μ(Φ)=c$$ for $A\in {\cal A}^{1,1}$ and a section $Φ\in {\cal S}$, where $c\in{\frak k}$ is a fixed central element. We study which orbits of the action of the complex gauge group on ${cal A}^{1,1}\times{\cal S}$ contain solutions of the equation, and we define a positive functional on ${cal A}^{1,1}\times{\cal S}$ which generalises the Yang-Mills-Higgs functional and whose local minima coincide with the solutions of the equation.

math.DG↗