Nested Event Extraction (NEE) aims to extract complex event structures where an event contains other events as its arguments recursively. Nested events involve a kind of Pivot Elements (PEs) that simultaneously act as arguments of outer-nest events and as triggers of inner-nest events, and thus connect them into nested structures. This special characteristic of PEs brings challenges to existing NEE methods, as they cannot well cope with the dual identities of PEs. Therefore, this paper proposes a new model, called PerNee, which extracts nested events mainly based on recognizing PEs. Specifically, PerNee first recognizes the triggers of both inner-nest and outer-nest events and further recognizes the PEs via classifying the relation type between trigger pairs. The model uses prompt learning to incorporate information from both event types and argument roles for better trigger and argument representations to improve NEE performance. Since existing NEE datasets (e.g., Genia11) are limited to specific domains and contain a narrow range of event types with nested structures, we systematically categorize nested events in the generic domain and construct a new NEE dataset, called ACE2005-Nest. Experimental results demonstrate that PerNee consistently achieves state-of-the-art performance on ACE2005-Nest, Genia11, and Genia13. The ACE2005-Nest dataset and the code of the PerNee model are available at https://github.com/waysonren/PerNee.