论文标题
在一类QuaternionicKähler歧管上的遗传学结构
Hermitian structures on a class of quaternionic Kähler manifolds
论文作者
论文摘要
任何QuaternionicKähler歧管$（\ bar n，g _ {\ bar n}，\ Mathcal Q）$配备了杀伤矢量字段$ x $，无处消失的Quaternionic Moment Map带有可集成的几乎复杂的复杂结构$ J_1 $ J_1，是Quaternionic结构结构$ \ MATHCAL $ \ MATHCAL Q $ \ MATHCAL Q $。使用HK/QK通信，我们研究了通过更改$ x $ x $和$ j_1x $的发行版本上更改$ j_1 $的符号，从而获得了几乎Hermitian结构$（g _ {\ bar n}，\ tilde j_1）$的属性。特别是，我们为其整合性提供了必要和足够的条件，并在合并中是Kähler。我们表明，对于包含单循环变形的C-MAP空间的大型QuaternionicKähler歧管，结构$ \ tilde J_1 $是可集成的。我们还表明，$ \ tilde j_1 $的集成性意味着$（g _ {\ bar n}，\ tilde j_1）$是在第四维度中的kähler，但不在更高的维度中。在QuaternionicKähler对称空间的一环变形的特殊情况下，两平面的复杂的grassmannians双重构造，我们构建了第三个规范的Hermitian结构$（g _ {\ bar n}，\ hat hat j_1）$。最后，我们对QuaternionicKähler进行了完整的本地分类四倍，其中$ \ tilde j_1 $是可以集成的，并表明这些是本地对称性的，或者携带同一性$ 1 $ 1 $等速度，由一个lie代数代数$ \ mathfrak {o}（2）\ ltimes \ ltimes \ ltimes \ nise \ niis \ rak { r）$，$ \ mathfrak {u}（2）$，或$ \ mathfrak {u}（1,1）$。
Any quaternionic Kähler manifold $(\bar N,g_{\bar N},\mathcal Q)$ equipped with a Killing vector field $X$ with nowhere vanishing quaternionic moment map carries an integrable almost complex structure $J_1$ that is a section of the quaternionic structure $\mathcal Q$. Using the HK/QK correspondence, we study properties of the almost Hermitian structure $(g_{\bar N},\tilde J_1)$ obtained by changing the sign of $J_1$ on the distribution spanned by $X$ and $J_1X$. In particular, we derive necessary and sufficient conditions for its integrability and for it being conformally Kähler. We show that for a large class of quaternionic Kähler manifolds containing the one-loop deformed c-map spaces, the structure $\tilde J_1$ is integrable. We do also show that the integrability of $\tilde J_1$ implies that $(g_{\bar N},\tilde J_1)$ is conformally Kähler in dimension four, but not in higher dimensions. In the special case of the one-loop deformation of the quaternionic Kähler symmetric spaces dual to the complex Grassmannians of two-planes we construct a third canonical Hermitian structure $(g_{\bar N},\hat J_1)$. Finally, we give a complete local classification of quaternionic Kähler four-folds for which $\tilde J_1$ is integrable and show that these are either locally symmetric or carry a cohomogeneity $1$ isometric action generated by one of the Lie algebras $\mathfrak{o}(2)\ltimes\mathfrak{heis}_3(\mathbb R)$, $\mathfrak{u}(2)$, or $\mathfrak{u}(1,1)$.
