Published paper
Abstract
The paper develops Poisson brackets in algebraic geometry through Poisson schemes, modules, Lie algebroids, degeneration loci, and applications to projective Poisson varieties.
Role in dependence graphs
Proof-critical source
On Lagrangianity of -supports of holonomic -modules and --modules
This paper is included only for the following marked statement:
- Corollary 2.4 · Journal of Mathematical Sciences p. 1417 / scan p. 5The singular locus of a Poisson scheme is a Poisson subvariety.
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Exact reviewed source
Published journal scan · Journal of Mathematical Sciences 84 (1997), no. 5, 1413–1444
Alexander Polishchuk. Algebraic geometry of Poisson brackets. Journal of Mathematical Sciences 84 (1997), no. 5, 1413–1444.
Open audited source ↗01Statements4 reported findingsNo statement issues found
The foundational results on Poisson subschemes, modules, Batalin–Vilkovisky operators, degeneration loci, log-symplectic complexes, and projective Poisson structures are correct under their stated hypotheses.
Corollary 2.4No issue found
The singular locus is a Poisson subscheme
Journal page 1417 · Lemma 2.1 and Corollaries 2.3–2.4
The structure map is a morphism of Poisson modules. Lemma 2.1 makes the annihilator of each coherent cokernel a Poisson ideal, so its Fitting degeneration loci are Poisson subschemes. The singular subscheme is the appropriate degeneration locus of , giving exactly the cited conclusion used downstream in the cluster-algebra argument.
Theorems 3.6, 4.1, and 6.1No issue found
The BV and Poisson-module correspondences are correct
Journal pages 1420–1428 · Sections 3–6
A square-zero second-order generator on the exterior algebra determines the Lie-algebroid bracket and flat connection, with the inverse formulas satisfying the required signs. Specializing to the cotangent Lie algebroid gives the canonical Poisson differential and the stated duality formulas. The rank-two projectivization construction follows from the induced determinant-module connection and preserves Jacobi.
Theorems 9.1, 9.3, and 10.6No issue found
The degeneration and logarithmic-complex results are supported
Journal pages 1431–1437 · Sections 9–10
The Chern-class and stability argument forces the asserted rank degeneration for corank-one foliations and projective-space Poisson tensors. For a simple-normal-crossings Pfaffian divisor, the local residue sequences identify the Poisson complex with logarithmic forms, and the filtration argument yields the announced hypercohomology description and degeneration consequence.
Theorems 13.1, 14.3, and 15.3No issue found
The projective classification and Hamiltonian-vector-field conclusions are correct
Journal pages 1439–1444 · Sections 13–15
Adjunction and the induced conormal Poisson module restrict the possible smooth curve components. In , the cubic intersection calculations and the elliptic degree-five resolution give the stated classification alternatives. For an irreducible reduced Pfaffian divisor, a Hamiltonian vector field would yield an inadmissible regular unit or a nontrivial abelian quotient of the complement, and the Fano Picard-rank-one specialization closes the argument.
02Proofs2 reported findingsNo proof issues found
The local Poisson-module calculations, BV identities, degeneration-locus arguments, logarithmic residue sequences, and projective intersection computations are correct and complete.
Sections 1–8No issue found in the reviewed proof
The foundational algebraic proofs close
Journal pages 1413–1431 · Sections 1–8
The annihilator and Fitting-ideal arguments are local and stable under the Hamiltonian derivations. The derived-bracket formulas reproduce the Lie algebroid operations with consistent degrees and signs, while the projective-bundle and Poisson-divisor constructions follow by direct transition-function calculations. All external algebraic facts invoked here are used within their stated scopes.
Sections 9–15No issue found in the reviewed proof
The geometric proof chains are complete
Journal pages 1431–1444 · Sections 9–15
The Bott-vanishing, Chern-class, residue, and adjunction inputs apply to the bundles and divisors at hand. The simple-normal-crossings local model supports each filtered-complex identification, and the later curve and Pfaffian arguments track scheme structure and smoothness explicitly. No missing case or contradictory numerical intersection was found.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.