发表机构
Pontificia Universidad Católica del Perú（秘鲁天主教 Pontificia 大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在有理同调$(4n-1)$-球面上构造了新的Sasaki-Einstein度量，通过改进循环多项式的估计，并证明Berglund-Hübsch转置规则保持拓扑和度量存在性。
AI 中文摘要
我们在有理同调$(4n-1)$-球面（$n>1$）上找到了Sasaki-Einstein度量，这些球面由指数为1的循环多项式构造，且这些多项式切割出有理簇。这里找到的Einstein度量与Boyer和Galicki在arXiv:math/0311355中发现的度量不等价。我们的发现源于对由循环多项式定义的超曲面的估计改进，该估计由Johnson和Kollár提出，用于确定Kähler-Einstein轨道度量。我们还构造了包含上述有理簇作为余维二子簇的加权超曲面，并且由于对循环多项式的精细估计，我们找到了这些超曲面的权重和次数的条件，使得它们对应的光滑链环承认Sasaki-Einstein度量。最后，我们研究了Berglund-Hübsch转置规则对所研究链环的拓扑和Sasaki-Einstein度量存在性的影响，并将arXiv:2311.15998中关于有理同调7-球面的所有结果推广到有理同调$(4n-1)$-球面，即我们证明了在转置规则下这两个性质的不变性。
英文摘要
We find Sasaki-Einstein metrics on rational homology $(4n-1)$-spheres for $n>1$ built from cyclic polynomials of index 1 cutting out rational varieties. The Einstein metrics found here are inequivalent to the ones found by Boyer and Galicki in arXiv:math/0311355. Our findings are consequence of an improvement, for hypersurfaces defined by cycle polynomials, on the estimate given by Johnson and Kollár to determine Kähler-Einstein orbifold metrics. We also construct weighted hypersurfaces that contain the rational varieties described above as codimension two subvarieties and, due to the refined estimate for cyclic polynomials, we find conditions on the weights and degrees of these hypersurfaces so their corresponding smooth links admit Sasaki-Einstein metrics. Finally we study the effect of the Berglund-Hübsch transpose rule on the topology and on the existence of Sasaki-Einstein metrics on the links studied and generalize all the results given in arXiv:2311.15998 for rational homology 7-spheres to rational homology $(4n-1)$-spheres, that is, we show invariance of these two features under the transpose rule.
CommentsSlight changes were made, for instance, some typos or misprints were fixed, added some remarks
