AI 中文总结
研究高皮卡数有理齐性空间中舒伯特簇的同调刚性与舒尔刚性问题，证明长根情形下光滑舒伯特簇有同调刚性，给出子图型舒伯特簇同调刚性列表，还表明长根情形下子图型舒伯特簇除非有纤维丛结构否则有舒尔刚性。
AI 中文摘要
在有理齐性空间\(X = G/P\)上的舒伯特簇\(X_0\)，若\(X\)上任何与\(X_0\)代表相同同调类的子簇\(Z\)，对某个\(g\in{\rm Aut_0}(X)\)必有\(Z = g\cdot X_0\)，则称\(X_0\)是同调刚性的。若进一步，\(X\)上同调类是\(X_0\)同调类倍数\(r\)的任何子簇\(Z\)，对某些\(g_1,\cdots,g_r\in{\rm Aut_0}(X)\)必有\(Z = g_1\cdot X_0+\cdots+g_r\cdot X_0\)，则称\(X_0\)是舒尔刚性的。皮卡数为一的有理齐性空间中舒伯特簇的同调刚性和舒尔刚性已有大量研究。本文研究高皮卡数有理齐性空间中舒伯特簇的这两种刚性问题。证明在长根情形（包括\(G\)为\(ADE\)型的所有情形）下，光滑舒伯特簇具有同调刚性。给出了具有/不具有同调刚性的子图型舒伯特簇的完整列表。此外，对于子图型的舒伯特簇\(X_0\)，证明在长根情形下它具有舒尔刚性，除非\(X_0\)在射影空间上有纤维丛结构。
英文摘要
A Schubert variety $X_0$ on a rational homogenous space $X=G/P$ is said to be homologically rigid, if any subvariety $Z$ on $X$ representing the same homology class with $X_0$ must satisfy $Z=g\cdot X_0$ for some $g\in{\rm Aut_0}(X)$. We say $X_0$ is Schur rigid, if furthermore any subvariety $Z$ on $X$ whose homology class is a multiple $r$ of that of $X_0$ must satisfy $Z=g_1\cdot X_0+\cdots+g_r\cdot X_0$ for some $g_1,\cdots ,g_r\in{\rm Aut_0}(X)$. Homological rigidity and Schur rigidity of Schubert varieties in rational homogeneous spaces of Picard number one have been well studied in extensive literature. In this paper, we study both rigidity problems of Schubert varieties in rational homogeneous spaces of higher Picard numbers. We show that in the long root cases, including all cases when $G$ is of type $ADE$, smooth Schubert varieties have homological rigidity. Besides, we give the complete list of Schubert varieties of subdiagram type with/without homological rigidity. Furthermore, for a Schubert variety $X_0$ of subdiagram type, we show that it has Schur rigidity in long root cases unless $X_0$ admits a fiber bundle structure over the projective space.
