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Crystalline and infinitesimal Poisson stacks

Primarily human-written textHuman understanding: all partsmath.AG — Algebraic Geometrymath.AT — Algebraic Topologymath.SG — Symplectic Geometry

Contributed by Aleksandr Karapetyan ↗

Version 1 / Oct 04, 2026 / CC BY 4.0

Abstract

A systematic way to construct Poisson brackets on moduli stacks in algebraic geometry is through shifted Poisson structures on derived stacks. The standard construction, due to Calaque--Pantev--To{\"e}n--Vaqui{\'e}--Vezzosi, imports Gelfand--Kazhdan formal geometry to derived stacks in order to establish formal descent for polyvector fields. The result is a graded Lie algebra which we call the ``crystalline'' polyvector fields on a derived stack. Using instead the formal derived geometry machinery of Gaitsgory--Rozenblyum, we construct a graded Lie algebra of ``infinitesimal'' polyvector fields which has a direct geometric interpretation: it governs deformations of the formal shifted cotangent stack, together with its canonical exact symplectic form. Our main result is an equivalence between crystalline and infinitesimal polyvector fields on a derived stack as graded Lie algebra objects, providing a deformation-theoretic perspective on the graded Lie algebra constructed by Calaque--Pantev--To{\"e}n--Vaqui{\'e}--Vezzosi.

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Version history

  1. v1Submitted by Aleksandr KarapetyanInitial depositCurrentOct 04, 2026