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Vicente Muñoz

Publications and source records attributed to Vicente Muñoz.

At least 37 records · Page 2Linked to original sources
arXiv2022

Asymptotically holomorphic theory for symplectic orbifolds

We extend Donaldson's asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a symplectic orbifold such that the symplectic form defines an integer cohomology class, we prove that there exist sections of large tensor powers of the prequantizable line bundle such that their zero sets are symplectic suborbifolds. We then derive a Lefschetz hyperplane theorem for these suborbifolds, that computes their real cohomology up to middle dimension. We also get the hard Lefschetz and formality properties for them, when the ambient manifold satisfies those properties.

math.SG↗
arXiv2022

On strongly inflexible manifolds

An oriented closed connected N-manifold M is inflexible if it does not admit self-maps of unbounded degree. In addition, if all the maps from any other oriented closed connected N-manifold have bounded degree, then M is said to be strongly inflexible. The existence of simply-connected inflexible manifolds was established by Arkowitz and Lupton. However, the existence of simply-connected strongly inflexible manifolds is still an open question. We provide an algorithm relying on Sullivan models that allow us to prove that all, but one, of the known examples of simply-connected inflexible manifolds are not strongly inflexible.

math.GT↗
arXiv2021

Purely coclosed $G_2$-structures on nilmanifolds

We classify 7-dimensional nilpotent Lie groups, decomposable or of nilpotency step at most 4, endowed with left-invariant purely coclosed $G_2$-structures. This is done by going through the list of all 7-dimensional nilpotent Lie algebras given by Gong, providing an example of a left-invariant 3-form $φ$ which is a pure coclosed $G_2$-structure (that is, it satisfies $d*φ=0$, $φ\wedge dφ=0$) for those nilpotent Lie algebras that admit them; and by showing the impossibility of having a purely coclosed $G_2$-structure for the rest of them.

math.DG↗
arXiv2021

Hodge-Deligne polynomials of SL(2,C)-character varieties for curves of small genus

We compute the Hodge-Deligne polynomials of the moduli spaces of representations of the fundamental group of a complex surface into SL(2,C), for the case of small genus g, and allowing the holonomy around a fixed point to be any matrix of SL(2,C), that is Id, -Id, diagonalisable, or of either of the two Jordan types. For this, we introduce a new geometric technique, based on stratifying the space of representations, and on the analysis of the behaviour of the Hodge-Deligne polynomial under fibrations.

math.AG↗
arXiv2021

The point counting problem in representation varieties of torus knots

We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1$ and $AGL_2$ for an arbitrary field $k$. In the case that $k = F_q$ is a finite field this gives rise to the count of the number of points of the representation variety, while for $k = C$ this calculation returns the E-polynomial of the representation variety. We discuss the interplay between these two results in sight of Katz theorem that relates the point count polynomial with the E-polynomial. In particular, we shall show that several point count polynomials exist for these representation varieties, depending on the arithmetic between m,n and the characteristic of the field, whereas only one of them agrees with the actual E-polynomial.

math.AG↗
arXiv2020

A K-contact simply connected 5-manifold with no semi-regular Sasakian structure

We construct the first example of a 5-dimensional simply connected compact manifold that admits a K-contact structure but does not admit a semi-regular Sasakian structure. For this, we need two ingredients: (a) to construct a suitable simply connected symplectic 4-manifold with disjoint symplectic surfaces spanning the homology, all of them but one of genus 1 and the other of genus g>1, (b) to prove a bound on the second Betti number $b_2$ of an algebraic surface with $b_1=0$ and having disjoint complex curves spanning the homology when all of them but one are of genus 1 and the other of genus g>1.

math.DG↗
arXiv2020

Representation variety for the rank one affine group

The aim of this paper is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the representation variety into simpler pieces; the arithmetic method, focused on counting their number of points over finite fields; and the quantum method, which performs the computation by means of a Topological Quantum Field Theory. We also discuss the corresponding moduli spaces of representations and character varieties, which turn out to be non-equivalent due to the non-reductiveness of the underlying group.

math.AG↗
arXiv2020

Gompf connected sum for orbifolds and K-contact Smale-Barden manifolds

We develop the Gompf fiber connected sum operation for symplectic orbifolds. We use it to construct a symplectic 4-orbifold with $b_1=0$ and containing symplectic surfaces of genus 1 and 2 that are disjoint and span the rational homology. This is used in turn to construct a K-contact Smale-Barden manifold with specified 2-homology that satisfies the known topological constraints with sharper estimates than the examples constructed previously. The manifold can be chosen spin or non-spin.

math.DG↗
arXiv2020

Geometry of the SL(3,C)-character variety of torus knots

Let G be the fundamental group of the complement of the torus knot of type (m,n). This has a presentation G= . We find the geometric description of the character variety X(G) of characters of representations of G into SL(3,C), GL(3,C) and PGL(3,C).

math.GT↗
arXiv2020

Homology Smale-Barden manifolds with K-contact and Sasakian structures

Kollár has found subtle obstructions to the existence of Sasakian structures on 5-dimensional manifolds. In the present article we develop methods of using these obstructions to distinguish K-contact manifolds from Sasakian ones. In particular, we find the first example of a closed 5-manifold M with $H_1(M,Z)=0$ which is K-contact but which carries no semi-regular Sasakian structures.

math.SG↗
arXiv2020

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.

math.AG↗
arXiv2020

On the classification of Smale-Barden manifolds with Sasakian structures

Smale-Barden manifolds $M$ are classified by their second homology $H_2(M,{\mathbb Z})$ and the Barden invariant $i(M)$. It is an important and dificult question to decide when $M$ admits a Sasakian structure in terms of these data. In this work we show methods of doing this. In particular we realize all $M$ with $H_2(M)={\mathbb Z}^k\oplus(\oplus_{i=1}^r{\mathbb Z}_{m_i}^{2g_i})$ and $i=0,\infty$, provided that $k\geq 1$, $m_i\geq 2$, $g_i\geq 1$, $m_i$ are pairwise coprime. Using our methods we also contribute to the problem of the existence of definite Sasakian structures on rational homology spheres. Also, we give a complete solution to the problem of the existence of Sasakian structures on rational homology spheres in the class of semi-regular Sasakian structures.

math.DG↗
arXiv2018

A lax monoidal Topological Quantum Field Theory for representation varieties

We construct a lax monoidal Topological Quantum Field Theory that computes Deligne-Hodge polynomials of representation varieties of the fundamental group of any closed manifold into any complex algebraic group $G$. As byproduct, we obtain formulas for these polynomials in terms of homomorphisms between the space of mixed Hodge modules on $G$. The construction is developed in a categorical-theoretic framework allowing its application to other situations.

math.AG↗
arXiv2018

Transversality for the moduli space of Spin(7)-instantons

We construct the moduli space of Spin(7)-instantons on a hermitian complex vector bundle over a closed 8-dimensional manifold endowed with a (possibly non-integrable) Spin(7)-structure. We find suitable perturbations that achieve regularity of the moduli space, so that it is smooth and of the expected dimension over the irreducible locus.

math.DG↗
arXiv2017

On SO(3)-bundles over the Wolf spaces

We study the formality of the total space of principal SU(2) and SO(3)-bundles over a Wolf space, that is a symmetric positive quaternionic Kähker manifold. We apply this to conclude that all the 3-Sasakian homogeneous spaces are formal. We also determine the principal SU(2) and SO(3)-bundles over the Wolf spaces whose total space is non-formal.

math.DG↗
arXiv2017

Orientability of the moduli space of Spin(7)-instantons

Let $(M,Ω)$ be a closed $8$-dimensional manifold equipped with a generically non-integrable $\mathrm{Spin}(7)$-structure $Ω$. We prove that if $\mathrm{Hom}(H^{3}(M,\mathbb{Z}), \mathbb{Z}_{2}) = 0$ then the moduli space of irreducible $\mathrm{Spin}(7)$-instantons on $(M,Ω)$ with gauge group $\mathrm{SU}(r)$, $r\geq 2$, is orientable.

math.DG↗