arXiv subjects

Ralph R. Gomez

Publications and source records attributed to Ralph R. Gomez.

8 recordsLinked to original sources

arXiv2025

Non-existence of extremal Sasaki metrics via the Berglund-Hübsch transpose

We use the Berglund-Hübsch transpose rule from classical mirror symmetry in the context of Sasakian geometry and results on relative K-stability in the Sasaki setting developed by Boyer and van Coevering to exhibit examples of Sasaki manifolds of big Sasaki cones that do not admit any extremal Sasaki metrics at all. Previously, examples with this feature were produced by Boyer and van Coevering for Brieskorn-Pham polynomials or their deformations. Our examples are based on the more general framework of invertible polynomials. In particular, we construct families of links that preserve the emptiness of the extremal Sasaki-Reeb cone via the Berglund-Hübsch rule: if the link does not admit extremal Sasaki metrics then its Berglund-Hübsch dual preserves this property and moreover this dual admits a representative in its local moduli with a larger Sasaki-Reeb cone which remains obstructed to admitting extremal Sasaki metrics. Some of the examples exhibited here have the homotopy type of a sphere or are rational homology spheres.

math.DG↗

arXiv2024

Beglund-Hübsch transpose and Sasaki-Einstein rational homology 7-spheres

We show that links of invertible polynomials coming from the Johnson and Kollár list of Kähler-Einstein 3-folds that are rational homology 7-spheres remain rational homology 7-spheres under the so-called Berglund-Hübsch transpose rule coming from classical mirror symmetry constructions. Actually, this rule produces twins, that is, links with same degree, Milnor number and homology H_3, with the exception of iterated Thom-Sebastiani sums of singularities of chain and cycle type, where the torsion and the Milnor number may vary. The Berglund-Hübsch transpose rule not only gives a framework to better understand the existence of SasakiEinstein twins but also gives a mechanism for producing new examples of Sasaki-Einstein twins in the rational homology 7 -sphere setting. We also give reasonable conditions for a Sasaki-Einstein rational homology 7-sphere to remain Sasaki-Einstein under the BH-transpose rule. In particular, we found 75 new examples of Sasaki-Einstein rational homology 7-spheres arising as links of not well-formed hypersurface singularities.

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arXiv2022

Berglund-Hübsch Transpose Rule and Sasakian Geometry

We apply the Berglund-Hübsch transpose rule from BHK mirror symmetry to show that to an $n-1$-dimensional Calabi-Yau orbifold in weighted projective space defined by an invertible polynomial, we can associate four (possibly) distinct Sasaki manifolds of dimension $2n+1$ which are $n-1$-connected and admit a metric of positive Ricci curvature. We apply this theorem to show that for a given K3 orbifold, there exists four seven dimensional Sasakian manifolds of positive Ricci curvature, two of which are actually Sasaki-Einstein.

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arXiv2016

Sasaki-Einstein 7-Manifolds, Orlik Polynomials and Homology

Let $L_f$ be a link of an isolated hypersurface singularity defined by a weighted homogenous polynomial $f.$ In this article, we give ten examples of $2$-connected seven dimensional Sasaki-Einstein manifolds $L_f$ for which $H_{3}(L_f, \mathbb{Z})$ is completely determined. Using the Boyer-Galicki construction of links $L_f$ over particular Kähler-Einstein orbifolds, we apply a valid case of Orlik's conjecture to the links $L_f $ so that one is able to explicitly determine $H_{3}(L_f,\mathbb{Z}).$ We give ten such new examples, all of which have the third Betti number satisfy $10\leq b_{3}(L_{f})\leq 20$.

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arXiv2015

Generalized CoKähler Geometry and an Application to Generalized Kähler Structures

In this paper we define the notion of a generalized coKähler structure and prove that the product $M_{1}\times M_{2}$ of generalized contact metric manifolds $(M_i, Φ_i,E_{\pm,i}, G_i)$, $ i=1, 2$, where $M_{1}\times M_{2}$ is endowed with the product generalized complex structure induced from $Φ_1$ and $Φ_2$, is generalized Kähler if and only if $(M_i, Φ_i, E_{\pm,i}, G_i) ,\ \ i=1,2$ are generalized coKähler structures. We also prove that products of generalized coKähler and generalized Kähler manifolds admit a generalized coKähler structure. We use these product constructions to give nontrivial examples of generalized coKähler structures. Finally, we show the analogs of these theorems hold in the setting of twisted generalized geometries. We use these theorems to construct new examples of twisted generalized Kähler structures on manifolds that do not admit a classical Kähler structure and we give examples of twisted generalized coKähler structures on manifolds which do not admit a classical coKähler structure.

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arXiv2015

On Products of Generalized Geometries

In this paper we address what generalized geometric structures are possible on products of spaces that each admit generalized geometries. In particular we consider, first, the product of two odd dimensional spaces that each admit a generalized almost contact structure, and then subsequently, the product of an odd dimensional space that admits a generalized almost contact structure and an even dimensional space that admits a generalized almost complex structure. We also draw attention to the relationship of the Courant bracket to the classical notion of normality for almost contact structures.

math.DG↗

arXiv2013

A Note on Smale Manifolds and Lorentzian Sasaki-Einstein Geometry

In this note, we construct new examples of Lorentzian Sasaki-Einstein (LSE) metrics on Smale manifolds $M.$ It has already been established in \cite{Gmz2} that such metrics exist on the so-called torsion free Smale manifolds, i.e. the $k$-fold connected sum of $S^{2}\times S^{3}.$ Now, we show that LSE metrics exist on Smale manifolds for which $H_{2}(M,\mathbb{Z})_{tor}$ is nontrivial. In particular, we show that most simply-connected positive Sasakian rational homology 5-spheres are also negative Sasakian (hence Lorentzian Sasaki-Einstein). Moreover, we show that for each pair of positive integers $(n,s)$ with $n,s >1$, there exists a Lorentzian Sasaki-Einstein Smale manifold $M$ such that $H_{2}(M,{\mathbb{Z}})_{tors}=(\mathbb{Z}/n)^{2s}$. Finally, we are able to construct so-called mixed Smale manifolds (connect sum of torsion free Smale manifolds with rational homology spheres) which admit LSE metrics and have arbitrary second Betti number. This gives infinitely many examples which do not admit positive Sasakian structures. These results partially address the open problems

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arXiv2009

Lorentzian Sasaki-Einstein Metrics on Connected Sums of $S^{2}\times S^{3}$

Negative Sasakian manifolds, where the first Chern class of the contact subbundle is a torsion class, can be viewed as Seifert-$S^1$ bundles where the base orbifold has an ample orbifold canonical class. We use this framework to settle completely an open problem formulated by C.Boyer and K.Galicki which asks whether or not arbitrary connected sums of $S^2\times S^3$ are negative Sasakian manifolds. As a consequence of the affirmative answer to this problem, there exists so-called Sasaki eta-Einstein and Lorentzian Sasaki-Einstein metrics on all of these five-manifolds and moreover all of these can be realized as links of isolated hypersurface singularities defined by weighted homogenous polynomials. The key step is to construct infinitely many hypersurfaces with branch divisors in weighted projective that contain only rational curves.

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