AI 中文总结
本文研究格拉斯曼几何中爱因斯坦标度与相容度量的存在性，针对反自对偶共形结构的共形到爱因斯坦问题，改进相关条件并推广至(2n,2)-格拉斯曼结构，还得到了相容度量的代数障碍及次极大对称模型的相关性质。
AI 中文摘要
我们研究格拉斯曼几何中爱因斯坦标度与相容度量的存在性问题，其中包含反自对偶（ASD）共形结构的共形到爱因斯坦问题这一特殊情形。在ASD情形下，给定外尔曲率中代数形式的一般性条件，我们得到了局部爱因斯坦标度存在的充要条件，这些条件改进了文献中已有的结果。我们还证明，在黎曼符号下，非里奇平坦的ASD-凯勒度量局部共形为爱因斯坦度量当且仅当它具有一个全纯 Killing 向量场，其反自对偶导数与里奇旋量成比例。接下来，我们考虑(2n,2)-格拉斯曼结构的ASD共形到爱因斯坦问题的若干推广情形。当n>1时，给定类似的一般性条件，我们得到了局部爱因斯坦标度存在的充要条件。利用 tractor 微积分，我们得到了相容度量存在的代数障碍，并证明该几何的唯一次极大对称模型具有线性独立相容度量的次极大数目。
英文摘要
We study the existence of Einstein scales and compatible metrics in Grassmannian geometry. As a special case, this includes the conformal-to-Einstein problem for anti-self-dual (ASD) conformal structures. In the ASD setting, given a genericity condition algebraic in the Weyl curvature, we obtain necessary and sufficient conditions for the existence of a local Einstein scale that refine those previously appearing in the literature. We also prove that in Riemannian signature, a non-Ricci-flat ASD-Kähler metric is locally conformally Einstein if and only if it has a holomorphic Killing vector field with anti-self-dual derivative proportional to the Ricci spinor. Next, we consider some generalisations of the ASD conformal-to-Einstein problem for $(2n,2)$-Grassmannian structures. When $n > 1$, given an analogous genericity condition, we obtain necessary and sufficient conditions for the existence of a local Einstein scale. Using tractor calculus, we obtain algebraic obstructions to the existence of compatible metrics and show that the unique submaximally symmetric model of the geometry has the submaximal number of linearly independent compatible metrics.
Comments28 pages
