AI 中文总结
本文构造了具有孤立Gorenstein典范奇点且不可光滑化的Calabi-Yau三维流形，利用Altmann形变理论与Corti-Filip-Petracci准则，并在Kreuzer-Skarke分类中识别出约8.27%的多胞形存在此阻碍。
AI 中文摘要
我们构造了具有孤立Gorenstein典范奇点的射影Calabi-Yau三维流形，这些奇点不允许光滑化，作为Gorenstein Fano环面四维流形中的反典范超曲面。一个例子具有唯一奇点，即第一Hirzebruch曲面上的反典范锥。另一个例子具有唯一奇点，其约化极小万有基是光滑曲线，但其形变均为奇异。该构造结合了Altmann的形变理论与Corti-Filip-Petracci的光滑化准则：环境扇多胞形的二维面若具有原始边，且没有分解为单位线段和幺模三角形的Minkowski分解，则阻碍光滑化。我们还分类了217个孤立Gorenstein环面三维奇点，其多边形边向量在某个格基中的坐标位于$[-2,2]$内，记录了它们的形变类型和光滑化分量。在所用数据库副本的完整性和无重复性前提下，对Kreuzer-Skarke分类的扫描发现，在其$473{,}800{,}776$个多胞形中有$39{,}175{,}536$个（约$8.27\%$）存在此阻碍，因此其一般反典范超曲面不允许光滑化。
英文摘要
We construct projective Calabi-Yau threefolds with isolated Gorenstein canonical singularities that admit no smoothing, as anticanonical hypersurfaces in Gorenstein Fano toric fourfolds. One example has a unique singularity, the anticanonical cone over the first Hirzebruch surface. Another has a unique singularity whose reduced miniversal base is a smooth curve but whose deformations are all singular. The construction combines Altmann's deformation theory with the smoothing criterion of Corti-Filip-Petracci: a two-dimensional face of the ambient fan polytope with primitive edges obstructs smoothing if it has no Minkowski decomposition into unit segments and unimodular triangles. We also classify the 217 isolated Gorenstein toric threefold singularities whose polygon edge vectors have coordinates in $[-2,2]$ in some lattice basis, recording their deformation types and smoothing components. Subject to completeness and absence of repetitions in the database copy used, a scan of the Kreuzer-Skarke classification finds this obstruction in $39{,}175{,}536$ of its $473{,}800{,}776$ polytopes (approximately $8.27\%$), whose generic anticanonical hypersurfaces therefore admit no smoothing.
Comments18 pages
