preprint Barbara Bolognese author Alex Küronya author Martin Ulirsch author 114697 Let $X$ be a complex abelian variety. We prove an analogue of both the (cohomological) $P=W$ conjecture and the geometric $P=W$ conjecture connecting the finer topological structure of the Dolbeault moduli space of topologically trivial semistable Higgs bundles on $X$ and the Betti moduli space of characters of the fundamental group of $X$. The geometric heart of our approach is the spectral data morphism for Dolbeault moduli spaces on abelian varieties that naturally factors the Hitchin morphism and whose target is not an affine space of pluricanonical sections, but a suitable symmetric product. 2023 eng 2303.03734 B. Bolognese, A. Küronya, M. Ulirsch, ArXiv:2303.03734 (2023). Bolognese, Barbara, Alex Küronya, and Martin Ulirsch. “$P=W$ Phenomena on Abelian Varieties.” <i>ArXiv:2303.03734</i>, 2023. B. Bolognese, A. Küronya, and M. Ulirsch, “$P=W$ phenomena on abelian varieties,” <i>arXiv:2303.03734</i>. 2023. Bolognese, B., Küronya, A., &#38; Ulirsch, M. (2023). $P=W$ phenomena on abelian varieties. In <i>arXiv:2303.03734</i>. @article{Bolognese_Küronya_Ulirsch_2023, title={$P=W$ phenomena on abelian varieties}, journal={arXiv:2303.03734}, author={Bolognese, Barbara and Küronya, Alex and Ulirsch, Martin}, year={2023} } Bolognese B, Küronya A, Ulirsch M. $P=W$ phenomena on abelian varieties. <i>arXiv:230303734</i>. Published online 2023. Bolognese, Barbara, et al. “$P=W$ Phenomena on Abelian Varieties.” <i>ArXiv:2303.03734</i>, 2023. 663122026-07-08T06:32:00Z2026-07-08T06:32:25Z