Abstract:We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus $2$ curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a degeneration of the Hilbert–Chow morphism associated with the moduli space of certain torsion sheaves on an abelian surface, where the symmetries induced by Markman’s monodromy operators play a crucial role.

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中文翻译：

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希钦纤维化、阿贝尔曲面和 P=W 猜想

摘要：我们通过阿贝尔曲面研究了希钦纤维的拓扑结构。我们建立了 $2$ 曲线和任意秩的 P=W 猜想。在更高的属和任意秩中，我们证明 P=W 适用于由甚至重言类生成的上同调子代数。此外，我们表明，所有重言式生成器都位于 P = W 猜想所预测的反常过滤的正确部分。结合 Mellit 最近的工作，这降低了对反常过滤的乘法性的完全猜想。我们的主要技术是将 Hitchin 纤维化研究为与阿贝尔表面上某些扭轮模空间相关的 Hilbert-Chow 态射的简并，其中由 Markman 单极算子引起的对称性起着至关重要的作用。

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