We study the local classification problem for differential Pfaffian forms on a supermanifold M that are homogeneous with respect to a given homogeneity structure on M. The most familiar examples of homogeneity structures are those associated with vector bundle structures. Our aim is to show that, for a homogeneous form of fixed degree, there exist homogeneous Darboux coordinates. As a consequence, we obtain Darboux-type normal forms for homogeneous Pfaffian forms, recovering as special cases the classical Darboux theorem together with its contact and presymplectic counterparts. To formulate an analogue of Darboux classification in the supergeometric setting, we associate to a differential form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> the characteristic distribution <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\chi (\alpha )=\ker (\alpha )\cap \ker (\textrm{d}\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>ker</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mo>ker</mo> <mo stretchy="false">(</mo> <mtext>d</mtext> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and define the class of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> as the rank of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\chi (\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We prove that, under suitable regularity and constant-rank assumptions, this distribution completely determines the local equivalence problem for homogeneous Pfaffian forms. Our results apply equally well to ordinary (purely even) manifolds.

A Darboux classification of homogeneous Pfaffian forms on graded manifolds

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2026年6月15日

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Janusz Grabowski,
Asier López-Gordón

摘要

We study the local classification problem for differential Pfaffian forms on a supermanifold M that are homogeneous with respect to a given homogeneity structure on M. The most familiar examples of homogeneity structures are those associated with vector bundle structures. Our aim is to show that, for a homogeneous form of fixed degree, there exist homogeneous Darboux coordinates. As a consequence, we obtain Darboux-type normal forms for homogeneous Pfaffian forms, recovering as special cases the classical Darboux theorem together with its contact and presymplectic counterparts. To formulate an analogue of Darboux classification in the supergeometric setting, we associate to a differential form $\alpha $ $α$ the characteristic distribution $\chi (\alpha )=\ker (\alpha )\cap \ker (\textrm{d}\alpha )$ $χ (α) = ker (α) \cap ker (d α)$ , and define the class of $\alpha $ $α$ as the rank of $\chi (\alpha )$ $χ (α)$ . We prove that, under suitable regularity and constant-rank assumptions, this distribution completely determines the local equivalence problem for homogeneous Pfaffian forms. Our results apply equally well to ordinary (purely even) manifolds.