论文标题
傅立叶 - 木粉转换用于精细压缩Prym品种
Fourier-Mukai transform for fine compactified Prym varieties
论文作者
论文摘要
考虑一个有限的覆盖$β：c \ to x $的光滑投射曲线$ x $的x $，通过减少，投影，平面曲线$ c $。与$ c $，$ q $和$ q'$上的两个一般极化相关，一个人可以构造相应的压缩Prym品种$ \ OVERLINE {\ MATHRM {p}}_β（q）$和$ \ overline {\ Mathrm {p}}_β（q'q''）$。考虑$γ$是一组线束，其扭力与$β$相吻合。在本文中，我们在$ \ Overline {\ Mathrm {p}}_β（q）$的衍生类别之间构建了一个傅立叶 - 木叶转换，而$ \ overline {\ mathrm {p}}}_β（q'β（q''）$的$γ$ equivariant派生类别的类别。因此，我们获得了$ \ mathrm {sl}（n，\ mathbb {c}）$ - Hitchin Fiber及其关联的$ \ Mathrm {pgl}（N，\ Mathbb {c}）$ - Hitchin fiber的密集型奇数光谱图。然后，我们的工作提供了由Arinkin和Melo-Rapagnetta-Viviani构建的傅立叶 - 木叶变换的扩展，这对应于$ \ mathrm {gl}（n，\ mathbb {c}）$ - 在这类单频谱曲线中的Hitchin纤维。
Consider a finite covering $β: C \to X$ of a smooth projective curve $X$ by a reduced, projective, planar curve $C$. Associated to two general polarizations on $C$, $q$ and $q'$, one can construct the corresponding compactified Prym varieties $\overline{\mathrm{P}}_β(q)$ and $\overline{\mathrm{P}}_β(q')$. Consider $Γ$ to be the group of line bundles whose torsion coincides with the order of $β$. In this article we construct a Fourier-Mukai transform between the derived categories of $\overline{\mathrm{P}}_β(q)$ and the $Γ$-equivariant derived category of $\overline{\mathrm{P}}_β(q')$. Hence, we obtain a derived equivalence between the $\mathrm{SL}(n,\mathbb{C})$-Hitchin fibre and its associated $\mathrm{PGL}(n,\mathbb{C})$-Hitchin fibre for a dense class of singular spectral curves. Our work then provides the extension of the Fourier-Mukai transform constructed by Arinkin and Melo-Rapagnetta-Viviani, which corresponds to autoduality of $\mathrm{GL}(n,\mathbb{C})$-Hitchin fibres in this class of singular spectral curves.
