arXiv subjects

Vicente Muñoz

Publications and source records attributed to Vicente Muñoz.

At least 55 records · Page 3Linked to original sources

arXiv2016

Homotopic properties of Kähler orbifolds

We prove the formality and the evenness of odd-degree Betti numbers for compact Kähler orbifolds, by adapting the classical proofs for Kähler manifolds. As a consequence, we obtain examples of symplectic orbifolds not admitting any Kähler orbifold structure. We also review the known examples of non-formal simply connected Sasakian manifolds, and produce an example of a non-formal quasi-regular Sasakian manifold with Betti numbers $b_1=0$ and $b_2\,> 1$.

math.DG↗

arXiv2015

Formality of 7-dimensional 3-Sasakian manifolds

We prove that any simply connected compact 3-Sasakian manifold, of dimension seven, is formal if and only if its second Betti number is $b_2<2$. In the opposite, we show an example of a 7-dimensional Sasaki-Einstein manifold, with second Betti number $b_2\geq 2$, which is formal. Therefore, such an example does not admit any 3-Sasakian structure. Examples of 7-dimensional simply connected compact formal Sasakian manifolds, with $b_2\geq 2$, are also given.

math.DG↗

arXiv2015

On formality of Sasakian manifolds

We investigate some topological properties, in particular formality, of compact Sasakian manifolds. Answering some questions raised by Boyer and Galicki, we prove that all higher (than three) Massey products on any compact Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using this we produce a method of constructing simply connected K-contact non-Sasakian manifolds. On the other hand, for every $n \geq 3$, we exhibit the first examples of simply connected compact Sasakian manifolds of dimension $2n + 1$ which are non-formal. They are non-formal because they have a non-zero triple Massey product. We also prove that arithmetic lattices in some simple Lie groups cannot be the fundamental group of a compact Sasakian manifold.

math.DG↗

arXiv2015

E-polynomial of the SL(3,C)-character variety of free groups

We compute the E-polynomial of the character variety of representations of a rank r free group in SL(3, C). Expanding upon existing techniques, we stratify the space of representations and compute the E-polynomial of each geometrically described stratum using fibrations. Consequently, we also determine the E-polynomial of its smooth, singular, and abelian loci and the corresponding Euler characteristic in each case. Along the way, we give new proofs of the SL(2,C), PGL(2,C), and PGL(3,C) cases.

math.AG↗

arXiv2015

Formality and the Lefschetz property in symplectic and cosymplectic geometry

We review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the $b_1=1$ case (in the coKähler/cosymplectic situation).

math.SG↗

arXiv2014

A $6$-dimensional simply connected complex and symplectic manifold with no Kähler metric

We construct a simply connected compact manifold which has complex and symplectic structures but does not admit Kähler metrics, in the lowest possible dimension where this can happen, that is, dimension 6. Such a manifold is automatically formal and has even odd-degree Betti numbers but it does not satisfy the Lefschetz property for any symplectic form.

math.SG↗

arXiv2014

Unified treatment of Explicit and Trace Formulas via Poisson-Newton formula

We prove that a Poisson-Newton formula, in a broad sense, is associated to each Dirichlet series with a meromorphic extension to the whole complex plane of finite order. These formulas simultaneously generalize the classical Poisson formula and Newton formulas for Newton sums. Classical Poisson formulas in Fourier analysis, explicit formulas in number theory and Selberg trace formulas in Riemannian geometry appear as special cases of our general Poisson-Newton formula.

math.NT↗

arXiv2014

E-polynomial of SL(2,C)-character varieties of complex curves of genus 3

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex curve of genus g=3 into $ SL(2,C), and also of the moduli space of twisted representations. The case of genus g=1,2 has already been done in [http://arxiv.org/abs/1106.6011]. We follow the geometric technique introduced in [http://arxiv.org/abs/1106.6011], based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations.

math.AG↗

arXiv2014

Manifolds which are complex and symplectic but not Kähler

The first example of a compact manifold admitting both complex and symplectic structures but not admitting a Kähler structure is the renowned Kodaira-Thurston manifold. We review its construction and show that this paradigm is very general and is not related to the fundamental group. More specifically, we prove that the simply-connected $8$-dimensional compact manifold of [\textsc{M. Fernández and V. Muñoz}, \emph{An 8-dimensional non-formal simply connected symplectic manifold}, Ann. of Math. (2) \textbf{167}, no. 3, 1045--1054, 2008.] admits both symplectic and complex structures but does not carry Kähler metrics.

math.SG↗

arXiv2014

On rotation of complex structures

We put in a general framework the situations in which a Riemannian manifold admits a family of compatible complex structures, including hyperkahler metrics and the Spin-rotations of arxiv:1302.2846. We determine the (polystable) holomorphic bundles which are rotable, i.e., they remain holomorphic when we change a complex structure by a different one in the family.

math.DG↗

arXiv2014

Non-formal co-symplectic manifolds

We study the formality of the mapping torus of an orientation-preserving diffeomorphism of a manifold. In particular, we give conditions under which a mapping torus has a non-zero Massey product. As an application we prove that there are non-formal compact co-symplectic manifolds of dimension $m$ and with first Betti number $b$ if and only if $m=3$ and $b \geq 2$, or $m \geq 5$ and $b \geq 1$. Explicit examples for each one of these cases are given.

math.AT↗

arXiv2013

Motives and the Hodge Conjecture for moduli spaces of pairs

Let $C$ be a smooth projective curve of genus $g\geq 2$ over $\mathbb C$. Fix $n\geq 1$, $d\in {\mathbb Z}$. A pair $(E,ϕ)$ over $C$ consists of an algebraic vector bundle $E$ of rank $n$ and degree $d$ over $C$ and a section $ϕ\in H^0(E)$. There is a concept of stability for pairs which depends on a real parameter $τ$. Let ${\mathfrak M}_τ(n,d)$ be the moduli space of $τ$-polystable pairs of rank $n$ and degree $d$ over $C$. Here we prove that for a generic curve $C$, the moduli space ${\mathfrak M}_τ(n,d)$ satisfies the Hodge Conjecture for $n \leq 4$. For obtaining this, we prove first that ${\mathfrak M}_τ(n,d)$ is motivated by $C$.

math.AG↗

arXiv2013

Spin(7)-instantons, stable bundles and the Bogomolov inequality for complex 4-tori

Using gauge theory for Spin(7)-manifolds of dimension 8, we develop a procedure, called Spin-rotation, which transforms a (stable) holomorphic structure on a vector bundle over a complex torus of dimension 4 into a new holomorphic structure over a different complex torus. We show non-trivial examples of this procedure by rotating a decomposable Weil abelian variety into a non-decomposable one. As a byproduct, we obtain a Bogomolov type inequality, which gives restrictions for the existence of stable bundles on an abelian variety of dimension 4, and show examples in which this is stronger than the usual Bogomolov inequality.

math.DG↗

arXiv2013

On the genus of meromorphic functions

We define the class of Left Located Divisor (LLD) meromorphic functions and their vertical order $m_0(f)$ and their convergence exponent $d(f)$. When $m_0(f)\leq d(f)$ we prove that their Weierstrass genus is minimal. This explains the phenomena that many classical functions have minimal Weierstrass genus, for example Dirichlet series, the $Γ$-function, and trigonometric functions.

math.CV↗

arXiv2013

Poisson-Newton formulas and Dirichlet series

We prove that a Poisson-Newton formula, in a broad sense, is associated to each Dirichlet series with a meromorphic extension to the whole complex plane. These formulas simultaneously generalize the classical Poisson formula and Newton formulas for Newton sums. Classical Poisson formulas in Fourier analysis, classical summation formulas as Euler-McLaurin or Abel-Plana formulas, explicit formulas in number theory and Selberg trace formulas in Riemannian geometry appear as special cases of our general Poisson-Newton formula. We also associate to finite order meromorphic functions general Poisson-Newton formulas that yield many classical integral formulas.

math.CV↗

arXiv2012

Hodge polynomials of the moduli spaces of rank 3 pairs

Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. A holomorphic triple $(E_1,E_2,ϕ)$ on $X$ consists of two holomorphic vector bundles $E_1$ and $E_2$ over $X$ and a holomorphic map $ϕ:E_2 \to E_1$. There is a concept of stability for triples which depends on a real parameter $σ$. In this paper, we determine the Hodge polynomials of the moduli spaces of $σ$-stable triples with $\rk(E_1)=3$, $\rk(E_2)=1$, using the theory of mixed Hodge structures. This gives in particular the Poincaré polynomials of these moduli spaces. As a byproduct, we recover the Hodge polynomial of the moduli space of odd degree rank 3 stable vector bundles.

math.AG↗

arXiv2012

The Hilali Conjecture for hyperelliptic spaces

The Hilali Conjecture predicts that for a simply-connected elliptic space, the total dimension of the rational homotopy does not exceed that of the rational homology. Here we give a proof of this conjecture for a class of elliptic spaces known as hyperelliptic.

math.AT↗

arXiv2012

The SU(2)-character varieties of torus knots

Let G be the fundamental group of the complement of the torus knot of type (m,n). We study the relationship between SU(2) and SL(2,C)-representations of this group, looking at their characters. Using the description of the SL(2,C)-character variety of G, X(G), we give a geometric description of Y(G)\subset X(G), the set of characters arising from SU(2)-representations.

math.AG↗