论文标题
小g-varieties
Small G-varieties
论文作者
论文摘要
如果每个非平凡的$ g $ -Orbit in $ x $中的每个非平地$ g $ -orbit对最高重量矢量的轨道都是同构的，则具有半圣经组$ g $的仿射品种称为“小”。这样的品种$ x $带有乘法组$ \ mathbb {k}^*$通勤的典型行动。我们表明，$ x $由$ \ mathbb {k}^*$ - 多种$ x^u $固定点下的固定点下的$ x^u $确定。此外，如果$ x $平滑，则$ x $是$ g $ - 向量捆绑包，而不是商$ x // g $。如果$ g $是类型的$ a_n $（$ n> 1 $），$ c_n $，$ e_6 $，$ e_7 $或$ e_8 $，我们表明所有仿射$ g $ - varieties to to a the One varieties都很小。结果，我们有以下结果。如果$ n> 4 $，则每个光滑的仿射$ sl_n $ - 尺寸的变化$ <2n $是$ \ mathrm {sl} _n $ - vector束，上面平稳的$ x // x // \ x // \ mathrm {sl} _n $，带有光纤与自然代表或其二重奏相同。
An affine varieties with an action of a semisimple group $G$ is called "small" if every non-trivial $G$-orbit in $X$ is isomorphic to the orbit of a highest weight vector. Such a variety $X$ carries a canonical action of the multiplicative group $\mathbb{K}^*$ commuting with the $G$-action. We show that $X$ is determined by the $\mathbb{K}^*$-variety $X^U$ of fixed points under a maximal unipotent subgroups $U$ of $G$. Moreover, if $X$ is smooth, then $X$ is a $G$-vector bundle over the quotient $X// G$. If $G$ is of type $A_n$ ($n>1$), $C_n$, $E_6$, $E_7$ or $E_8$, we show that all affine $G$-varieties up to a certain dimension are small. As a consequence we have the following result. If $n>4$, every smooth affine $SL_n$-variety of dimension $<2n$ is an $\mathrm{SL}_n$-vector bundle over the smooth quotient $X//\mathrm{SL}_n$, with fiber isomorphic to the natural representation or its dual.
