arXiv subjects

Lucía Martín-Merchán

Publications and source records attributed to Lucía Martín-Merchán.

11 recordsLinked to original sources

arXiv2026

Hodge theoretic results for nearly Kähler manifolds in all dimensions

We generalize to nearly Kähler manifolds of arbitrary dimensions most of the Hodge-theoretic results for nearly Kähler $6$-manifolds that were established by Verbitsky. In particular, for a compact nearly Kähler manifold of any dimension, the (appropriately defined) Hodge numbers are related to the Betti numbers in the same way as on a compact Kähler manifold. In the $6$-dimensional case, Verbitsky was able to say slightly more using the induced $\mathrm{SU}(3)$ structure. We discuss potential extensions of this to twistor spaces over positive scalar curvature quaternionic-Kähler manifolds, which are a particular class of $(4n+2)$-dimensional nearly Kähler manifolds equipped with a special $\mathrm{SU}(n) \! \cdot \! \mathrm{U}(1)$ structure.

math.DG↗

arXiv2025

Holomorphic linking numbers, ABC Massey products, and Calabi-Yau 3-folds

On compact Kähler manifolds, we relate ABC Massey products arising from complex analytic cycles to holomorphic linking numbers. This enables us to construct a family of simply connected projective 3-folds with trivial canonical bundle, equipped with a non-vanishing ABC Massey product.

math.DG↗

arXiv2025

Closed $\mathrm{G}_2$ structures on compact manifolds satisfying the known topological obstructions to holonomy $\mathrm{G}_2$ metrics

We construct new compact manifolds endowed with closed $\mathrm{G}_2$ structures that satisfy the topological properties found by Joyce and Baraglia for the existence of a torsion-free $\mathrm{G}_2$ structure in the same cohomology class. Those manifolds arise as resolutions of orbifolds of the form $M/\mathbb{Z}_2$, where $M$ itself does not admit any torsion-free $\mathrm{G}_2$ structure. We develop an equivariant resolution procedure for closed $\mathrm{G}_2$ orbifolds with $\mathbb{Z}_2$ isotropy, and analyze some topological properties of the resolved manifolds, including their first Pontryagin class and certain triple Massey products.

math.DG↗

arXiv2024

Compact holonomy $\mathrm{G}_2$ manifolds need not be formal

We construct a compact, simply connected manifold with holonomy $\mathrm{G}_2$ that is non-formal. We use the construction method of compact torsion-free $\mathrm{G}_2$ manifolds developed by D.D. Joyce and S. Karigiannis. A non-vanishing triple Massey product is obtained by arranging the singular locus in a particular configuration.

math.DG↗

arXiv2024

Extrinsic geometry of calibrated submanifolds

Given a calibration $α$ whose stabilizer acts transitively on the Grassmanian of calibrated planes, we introduce a nontrivial Lie-theoretic condition on $α$, which we call compliancy, and show that this condition holds for many interesting geometric calibrations, including Kähler, special Lagrangian, associative, coassociative, and Cayley. We determine a sufficient condition that ensures compliancy of $α$, we completely characterize compliancy in terms of properties of a natural involution determined by a calibrated plane, and we relate compliancy to the geometry of the calibrated Grassmanian. The condition that a Riemannian immersion $ι\colon L \to M$ be calibrated is a first order condition. By contrast, its extrinsic geometry, given by the second fundamental form $A$ and the induced tangent and normal connections $\nabla$ on $TL$ and $D$ on $NL$, respectively, is second order information. We characterize the conditions imposed on the extrinsic geometric data $(A, \nabla, D)$ when the Riemannian immersion $ι\colon L \to M$ is calibrated with respect to a calibration $α$ on $M$ which is both parallel and compliant. This motivate the definition of an infinitesimally calibrated Riemannian immersion, generalizing the classical notion of a superminimal surface in $\mathbb R^4$.

math.DG↗

arXiv2024

The twisted G$_2$ equation for strong G$_2$-structures with torsion

We discuss general properties of strong G$_2$-structures with torsion and we investigate the twisted G$_2$ equation, which represents the G$_2$-analogue of the twisted Calabi-Yau equation for SU$(n)$-structures introduced by Garcia-Fernández - Rubio - Shahbazi - Tipler. In particular, we show that invariant strong G$_2$-structures with torsion do not occur on compact non-flat solvmanifolds. This implies the non-existence of non-trivial solutions to the twisted Calabi-Yau equation on compact solvmanifolds of dimensions $4$ and $6$. More generally, we prove that a compact, connected homogeneous space admitting invariant strong G$_2$-structures with torsion is diffeomorphic either to $S^3 \times T^4$ or to $S^3 \times S^3 \times S^1$, up to a covering, and that in both cases solutions to the twisted G$_2$ equation exist. Finally, we discuss the behavior of the homogeneous Laplacian coflow for strong G$_2$-structures with torsion on these spaces.

math.DG↗

arXiv2022

Exact G$_2$-structures on compact quotients of Lie groups

We show that the compact quotient $Γ\backslash\mathrm{G}$ of a seven-dimensional simply connected Lie group $\mathrm{G}$ by a co-compact discrete subgroup $Γ\subset\mathrm{G}$ does not admit any exact $\mathrm{G}_2$-structure which is induced by a left-invariant one on $\mathrm{G}$.

math.DG↗

arXiv2021

A compact non-formal closed $G_2$ manifold with $b_1=1$

We construct a compact manifold with a closed $G_2$ structure not admitting any torsion-free $G_2$ structure, which is non-formal and has first Betti number $b_1=1$. We develop a method of resolution for orbifolds that arise as a quotient $M/Z_2$ with $M$ a closed $G_2$ manifold under the assumption that the singular locus carries a nowhere-vanishing closed 1-form.

math.DG↗

arXiv2020

Resolution of $4$-dimensional symplectic orbifolds

We give a method to resolve 4-dimensional symplectic orbifolds making use of techniques from complex geometry and gluing of symplectic forms. We provide some examples to which the resolution method applies.

math.SG↗

arXiv2018

Spinorial classification of Spin(7) structures

We describe the different classes of $\mathrm{Spin(7)}$ structures in terms of spinorial equations. We relate them to the spinorial description of $\mathrm{G}_2$ structures in some geometrical situations. Our approach enables us to analyze invariant $\mathrm{Spin(7)}$ structures on quasi abelian Lie algebras.

math.DG↗

arXiv2018

Tight neighborhoods of contact submanifolds

We prove that any small enough neighborhood of a closed contact submanifold is always tight under a mild assumption on its normal bundle. The non-existence of $C^0$--small positive loops of contactomorphisms in general overtwisted manifolds is shown as a corollary.

math.SG↗