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We use the Berglund-H\"ubsch 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 complexity 3 or complexity 4 that do not admit any extremal Sasaki metrics in its whole Sasaki-Reeb cone which is of Gorenstein type. 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 examples of links with the following property: if the link satisfies the Lichnerowicz obstruction of Gauntlett, Martelli, Sparks and Yau, then its Berglund-H\"ubsch dual preserves this obstruction and moreover this dual admits a perturbation in its local moduli which is obstructed to admitting extremal Sasaki metrics in its whole Sasaki-Reeb cone. In particular, we exhibit examples that have the homotopy type of a sphere or are rational homology spheres., Comment: The previous version of this article has been corrected (minor inaccuracies and typos) and improved. We also have added subsection 4.3 with more examples for more general invertible polynomials

We show that links of invertible polynomials coming from the Johnson and Koll\'ar list of K\"ahler-Einstein 3-folds that are rational homology 7-spheres remain rational homology 7-spheres under the so-called Berglund-H\"ubsch 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-H\"ubsch 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., Comment: We have changed the title, corrected minor typos and added Remark 4.2

We apply the Berglund-H\"ubsch 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., Comment: Fixed some typos and added references

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 K\"ahler-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 $10\leq b_{3}(L_{f})\leq 20$.

In this paper we define the notion of a generalized coK\"ahler structure and prove that the product $M_{1}\times M_{2}$ of generalized contact metric manifolds $(M_i, \Phi_i,E_{\pm,i}, G_i)$, $ i=1, 2$, where $M_{1}\times M_{2}$ is endowed with the product generalized complex structure induced from $\Phi_1$ and $\Phi_2$, is generalized K\"ahler if and only if $(M_i, \Phi_i, E_{\pm,i}, G_i) ,\ \ i=1,2$ are generalized coK\"ahler structures. We also prove that products of generalized coK\"ahler and generalized K\"ahler manifolds admit a generalized coK\"ahler structure. We use these product constructions to give nontrivial examples of generalized coK\"ahler 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 K\"ahler structures on manifolds that do not admit a classical K\"ahler structure and we give examples of twisted generalized coK\"ahler structures on manifolds which do not admit a classical coK\"ahler structure., Comment: Final print version. To appear in Journal of Geometry and Physics

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., Comment: Final print version; To appear in Geometriae Dedicata

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

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., Comment: 12 pages. Corrected Corollary 4.4 and changed a few typos preceding this corollary