In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety of Picard number 1 to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support it by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types $\mathrm {A}_n$ and $\mathrm {D}_n$, that is, flag varieties $\operatorname {Fl}(1,n;n+1)$ and isotropic orthogonal Grassmannians $\operatorname {OG}(2,2n)$; in particular, we construct on each of those an exceptional collection invariant with respect to the entire automorphism group. For $\operatorname {OG}(2,2n)$ this is the first exceptional collection proved to be full.

在我们之前的论文中，我们提出了一个猜想，该猜想将光滑的Fano Picard 1号变种的小量子同调环的结构与其派生相干滑轮的结构联系起来。在这里，我们对这个猜想进行概括，使其更加精确，并通过Dynkin类型$ \ mathrm {A} _n $和$ \ mathrm {D} _n $的简单代数组的（共）伴随齐次变体示例来证明这一点，即是，标志品种$ \ operatorname {Fl}（1，n; n + 1）$和各向同性正交Grassmannians $ \ operatorname {OG}（2,2n）$；尤其是，我们针对每个自构群构造了一个例外集合不变性。对于$ \ operatorname {OG}（2,2n）$ 这是第一个被证明是完整的非凡收藏。