arXiv subjects

Vicente Munoz

Publications and source records attributed to Vicente Munoz.

18 recordsLinked to original sources

arXiv2022

Spin-harmonic structures and nilmanifolds

We introduce spin-harmonic structures, a class of geometric structures on Riemannian manifolds of low dimension which are defined by a harmonic unitary spinor. Such structures are related to SU(2) (dim=4,5), SU(3) (dim=6) and G_2 (dim=7) structures; in dimension 8, a spin-harmonic structure is equivalent to a balanced Spin(7) structure. As an application, we obtain examples of compact 8-manifolds endowed with non-integrable Spin(7) structures of balanced type.

math.DG↗

arXiv2020

E-polynomials of the SL(2,C)-character varieties of surface groups

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a once-punctured surface of any genus into SL(2,C), for any possible holonomy around the puncture. We follow the geometric technique introduced in arXiv:1106.6011, based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations.

math.AG↗

arXiv2020

Rationally elliptic toric varieties

We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described.

math.AG↗

arXiv2015

The SL(3,C)-character variety of the figure eight knot

We give explicit equations that describe the character variety of the figure eight knot for the groups SL(3,C), GL(3,C) and PGL(3,C). This has five components of dimension 2, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irreducible representations. Of these, one is distinguished as it contains the curve of irreducible representations coming from $Sym^2:SL(2,C) \to SL(3,C)$. The other two components are induced by exceptional Dehn fillings of the figure eight knot. We also describe the action of the symmetry group of the figure eight knot on the character varieties.

math.GT↗

arXiv2014

Simply-connected K-contact and Sasakian manifolds of dimension 7

We construct a compact simply-connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure. We also study rational homotopy properties of such manifolds, proving in particular that a simply-connected 7-dimensional Sasakian manifold has vanishing cup-product on the second cohomology and that it is formal if and only if all its triple Massey products vanish.

math.DG↗

arXiv2010

Geometric structures on loop and path spaces

Is is known that the loop space associated to a Riemannian manifold admits a quasi-symplectic structure. This article shows that this structure is not likely to recover the underlying Riemannian metric by proving a result that is a strong indication of the "almost" independence of the quasi-symplectic structure with respect to the metric. Finally conditions to have contact structures on these spaces are studied.

math.SG↗

arXiv2010

The most inaccessible point of a convex domain

The inaccessibility of a point p in a bounded domain D \subset R^n is the minimum of the lengths of segments through p with boundary at \bd D. The points of maximum inaccessibility I_D are those where the inaccessibility achieves its maximum. We prove that for strictly convex domains, I_D is either a point or a segment, and that for a planar polygon I_D is in general a point. We study the case of a triangle, showing that this point is not any of the classical notable points.

math.MG↗

arXiv2010

Hodge theory for Riemannian solenoids

A measured solenoid is a compact laminated space endowed with a transversal measure. The De Rham $L^2$-cohomology of the solenoid is defined by using differential forms which are smooth in the leafwise directions and $L^2$ in the transversal direction. We develop the theory of harmonic forms for Riemannian measured solenoids, and prove that this computes the De Rham $L^2$-cohomology of the solenoid. This implies in particular a Poincare duality result.

math.DG↗

arXiv2008

Torelli theorem for the moduli spaces of pairs

Let X be a smooth projective curve of genus at least two over the complex numbers. A pair (E,ϕ) over X consists of an algebraic vector bundle E over X and a holomorphic section ϕof E. There is a concept of stability for pairs which depends on a real parameter τ. Here we prove that the third cohomology groups of the moduli spaces of τ-stable pairs with fixed determinant and rank at least two are polarised pure Hodge structures, and they are isomorphic to H^1(X) with its natural polarisation (except in very few exceptional cases). This implies a Torelli theorem for such moduli spaces. We recover that the third cohomology group of the moduli space of stable bundles of rank at least two and fixed determinant is a polarised pure Hodge structure, which is isomorphic to H^1(X). We also prove Torelli theorems for the corresponding moduli spaces of pairs and bundles with non-fixed determinant.

math.AG↗

arXiv2008

On non-formality of a simply-connected symplectic 8-manifold

We show an alternative construction of the first example of a simply-connected compact symplectic non-formal 8-manifold given in arXiv:math/0506449. We also give an alternative proof of its non-formality using higher order Massey products.

math.SG↗

arXiv2007

Symplectic resolutions, Lefschetz property and formality

We introduce a method to resolve a symplectic orbifold into a smooth symplectic manifold. Then we study how the formality and the Lefschetz property of the symplectic resolution are compared with that of the symplectic orbifold. We also study the formality of the symplectic blow-up of a symplectic orbifold along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture of Babenko and Taimanov.

math.SG↗

arXiv2007

Torelli theorem for the moduli spaces of connections on a Riemann surface

Let $(X,x_0)$ be any one--pointed compact connected Riemann surface of genus $g$, with $g\geq 3$. Fix two mutually coprime integers $r>1$ and $d$. Let ${\mathcal M}_X$ denote the moduli space parametrizing all logarithmic $\text{SL}(r,{\mathbb C})$--connections, singular over $x_0$, on vector bundles over $X$ of degree $d$. We prove that the isomorphism class of the variety ${\mathcal M}_X$ determines the Riemann surface $X$ uniquely up to an isomorphism, although the biholomorphism class of ${\mathcal M}_X$ is known to be independent of the complex structure of $X$. The isomorphism class of the variety ${\mathcal M}_X$ is independent of the point $x_0 \in X$. A similar result is proved for the moduli space parametrizing logarithmic $\text{GL}(r,{\mathbb C})$--connections, singular over $x_0$, on vector bundles over $X$ of degree $d$.

math.AG↗

arXiv2005

Weakly Lefschetz symplectic manifolds

The harmonic cohomology of a Donaldson symplectic submanifold and of an Auroux symplectic submanifold are compared with that of its ambient space. We also study symplectic manifolds satisfying a weakly Lefschetz property, that is, the $s$-Lefschetz propery. In particular, we consider the symplectic blow-ups of the complex projective space along weakly Lefschetz symplectic submanifolds. As an application we construct, for each even integer $s\geq 2$, compact symplectic manifolds which are $s$-Lefschetz but not $(s+1)$-Lefschetz.

math.SG↗

arXiv2004

The Geography of Non-formal Manifolds

We show that there exist non-formal compact oriented manifolds of dimension $n$ and with first Betti number $b_1=b\geq 0$ if and only if $n\geq 3$ and $b\geq 2$, or $n\geq (7-2b)$ and $0\leq b\leq 2$. Moreover, we present explicit examples for each one of these cases.

math.DG↗

arXiv2001

Codimension one symplectic foliations

We define the concept of symplectic foliation on a symplectic manifold and provide a method of constructing many examples, by using asymptotically holomorphic techniques.

math.SG↗

arXiv2001

Semipositive bundles and Brill-Noether theory

We prove a Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles $E$ and $F$ over a complex manifold under the condition that $E^*\ox F$ is Griffiths $k$-positive. We apply this result to find some homotopy groups of the Brill-Noether loci for a generic curve.

math.DG↗

arXiv1997

Donaldson invariants for connected sums along surfaces of genus 2

We relate the Donaldson invariants of two four-manifolds $X_i$ with embedded Riemann surfaces of genus 2 and self-intersection zero with the invariants of the manifold X which appears as a connected sum along the surfaces. When the original manifolds are of simple type with $b_1=0$ and $b^+>1$, X is of simple type with $b_1=0$ and $b^+>1$ as well, and the relationship between the invariants is expressed as constraints in the basic classes for X. Also we give some applications. For instance, if $X_i$ have both $b_1=0$ then X is of simple type with $b_1=0$, $b^+>1$, and has no basic classes evaluating zero on the Riemann surface. Finally, we prove that any four-manifold with $b^+>1$ and with an embedded surface of genus 2, self-intersection zero and representing an odd homology class, is of finite type of second order.

dg-ga↗

arXiv1997

Gluing formulae for Donaldson invariants for connected sums along surfaces

We solve a conjecture of Morgan and Szabo (Embedded genus 2 surfaces in four-manifolds, Preprint) about the relationship of the basic classes of two four-manifolds $X_i$ of simple type with $b_1=0$, $b^+>1$, such that there are embedded Riemann surfaces of genus $g \geq 2$ and self-intersection zero (and representing odd homology classes) with the basic classes of the manifold X which appears as a connected sum along the surfaces (supposing this latter one is of simple type). This is also expressed as constraints in the basic classes of X. The result is in accordance with the results on Seiberg-Witten invariants (Morgan, Szabo and Taubes, A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture).

dg-ga↗