The quantization conjecture revisited
The quantization conjecture revisited
复制标题
重新审视量子化猜想
DOI：
--
复制
发表时间：
1998
期刊：
影响因子：
--
通讯作者：
C. Teleman
中科院分区：
文献类型：
--
作者：
C. Teleman
证明了 Guillemin 和 Sternberg 量子化猜想的强版本。对于平滑、紧致、极化簇 (X, L) 上的还原群作用，L 在 GIT 商 X//G 上的上同调等于 X 上同调的不变部分。这推广了全局截面上的 [GS] 定理，并加强了其随后对黎曼罗赫数的扩展 ([JK], [li])。值得注意的副产品是在定义偏振或移位去奇异化的微小变化下 X//G 上向量丛上同调的不变性，以及布托定理的新证明。还研究了 X 及其地层的等变全纯形式和等变 Hodgeto-de Rham 谱序列，显示了其崩溃。一个应用是曲线上 C 丛模栈的 Borel-Weil-Bott 定理 [Ti] 的新证明，以及具有抛物线结构的丛的模栈和空间的类似陈述。还显示了这些堆栈的 Hodge-to de Rham 序列的崩溃。
A strong version of the quantization conjecture of Guillemin and Sternberg is proved. For a reductive group action on a smooth, compact, polarized variety (X, L), the cohomologies of L over the GIT quotient X//G equal the invariant part of the cohomologies over X. This generalizes the theorem of [GS] on global sections, and strengthens its subsequent extensions ([JK], [li]) to RiemannRoch numbers. Remarkable by-products are the invariance of cohomology of vector bundles over X//G under a small change in the defining polarization or under shift desingularization, as well as a new proof of Boutot's theorem. Also studied are equivariant holomorphic forms and the equivariant Hodgeto-de Rham spectral sequences for X and its strata, whose collapse is shown. One application is a new proof of the Borel-Weil-Bott theorem of [Ti] for the moduli stack of C-bundles over a curve, and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures. Collapse of the Hodge-to-de Rham sequences for these stacks is also shown.