0-Shifted Cosymplectic Geometry

• 0-shifted cosymplectic structure is a stack-theoretic extension of classical cosymplectic geometry that couples a closed 2-form and a nowhere-zero 1-form with a refined nondegeneracy condition.
• It employs closed basic forms on Lie groupoids and differentiable stacks to ensure Morita invariance while integrating aspects of precosymplectic and Poisson geometry.
• Its multiplicative formulation and Hamiltonian reduction framework offer practical insights into global geometric analysis in corank-1 Poisson settings.

A 0-shifted cosymplectic structure is a stack-theoretic and groupoid-theoretic extension of cosymplectic geometry in which the classical pair of a closed $2$-form and a closed $1$-form is required to satisfy a nondegeneracy condition compatible with Lie groupoid or differentiable stack data. In the stack formulation, the structure is given by closed basic forms $(\omega,\eta)$ on a Lie groupoid presentation, with $\eta$ nowhere vanishing and with the modified Lichnerowicz map $v\mapsto \iota_v\omega+\eta(v)\eta$ a quasi-isomorphism. In the multiplicative formulation, the relevant object is a cosymplectic groupoid, namely a Lie groupoid equipped with multiplicative closed forms $(\omega,\alpha)$ whose top wedge is nowhere zero. Together, these two viewpoints place 0-shifted cosymplectic geometry at the intersection of precosymplectic foliation theory, Poisson and symplectic groupoid theory, and Hamiltonian reduction on differentiable stacks (Garcia et al., 24 Aug 2025, Fernandes et al., 2023).

1. Classical antecedents and the defining stacky condition

Classical cosymplectic geometry starts from a triple

$(M,\omega,\eta),$

where $\omega\in \Omega^2(M)$ is closed, $\eta\in \Omega^1(M)$ is closed and nowhere zero, and

$TM=\ker(\omega)\oplus \ker(\eta).$

If $1$0, this is equivalent to requiring that

$1$1

be a volume form. The associated Lichnerowicz map is the bundle isomorphism

$1$2

and the Reeb vector field $1$3 is characterized by

$1$4

The 0-shifted theory is motivated not only by cosymplectic manifolds but also by precosymplectic manifolds. In the precosymplectic setting, $1$5 and $1$6 are still closed, but the condition is weakened to require that

$1$7

be a regular distribution strictly contained in $1$8. The corresponding foliation is denoted $1$9, and a basic lemma identifies the kernel of the Lichnerowicz map with this distribution: $(\omega,\eta)$0 Because the quotient by $(\omega,\eta)$1 is typically singular, the global theory is formulated in terms of Lie groupoids and differentiable stacks rather than ordinary manifolds (Garcia et al., 24 Aug 2025).

2. 0-shifted cosymplectic structures on Lie groupoids and differentiable stacks

Let

$(\omega,\eta)$2

be a Lie groupoid with source and target maps $(\omega,\eta)$3, Lie algebroid

$(\omega,\eta)$4

A differential form on $(\omega,\eta)$5 is basic if

$(\omega,\eta)$6

For source-connected groupoids this is equivalent to horizontality and invariance: $(\omega,\eta)$7

A 0-shifted cosymplectic structure on $(\omega,\eta)$8 is a pair of closed basic forms

$(\omega,\eta)$9

with $\eta$0 nowhere vanishing, such that for every $\eta$1 the Lichnerowicz map

$\eta$2

fits into the algebroid-to-cotangent cochain diagram as a quasi-isomorphism. In practical terms, this means that $\eta$3 is a chain map and

$\eta$4

The same condition is stated equivalently by the injectivity of the anchor $\eta$5, so any groupoid carrying such a structure is a foliation groupoid.

This formulation is Morita invariant: if

$\eta$6

is a Morita map, then $\eta$7 is 0-shifted cosymplectic on $\eta$8 if and only if $\eta$9 is 0-shifted cosymplectic on $v\mapsto \iota_v\omega+\eta(v)\eta$0. The structure therefore depends only on the differentiable stack $v\mapsto \iota_v\omega+\eta(v)\eta$1. In this sense, the theory defines a cosymplectic structure on a differentiable stack rather than merely on a chosen Lie groupoid presentation (Garcia et al., 24 Aug 2025).

3. Multiplicative cosymplectic groupoids and the groupoid-level model

A complementary formulation is provided by cosymplectic groupoids. A cosymplectic structure on an odd-dimensional manifold $v\mapsto \iota_v\omega+\eta(v)\eta$2 is a pair $v\mapsto \iota_v\omega+\eta(v)\eta$3 such that

$v\mapsto \iota_v\omega+\eta(v)\eta$4

is a volume form. In the groupoid context, the notation $v\mapsto \iota_v\omega+\eta(v)\eta$5 is used for the $v\mapsto \iota_v\omega+\eta(v)\eta$6-form and $v\mapsto \iota_v\omega+\eta(v)\eta$7-form. A cosymplectic groupoid is a Lie groupoid

$v\mapsto \iota_v\omega+\eta(v)\eta$8

equipped with a cosymplectic structure $v\mapsto \iota_v\omega+\eta(v)\eta$9 for which both forms are multiplicative: $(\omega,\alpha)$0 Here $(\omega,\alpha)$1 is multiplication on the space $(\omega,\alpha)$2 of composable pairs.

This multiplicative structure is the groupoid-level analogue of an ordinary cosymplectic manifold and is the natural setting for a “0-shifted cosymplectic” object in the multiplicative language. Several structural consequences follow. The Reeb vector field $(\omega,\alpha)$3, defined by

$(\omega,\alpha)$4

is bi-invariant: $(\omega,\alpha)$5 and it is both left- and right-invariant. It is also complete. Since $(\omega,\alpha)$6 is closed, $(\omega,\alpha)$7 is integrable; since $(\omega,\alpha)$8 is multiplicative, $(\omega,\alpha)$9 is moreover a multiplicative distribution. The restriction of $(M,\omega,\eta),$0 to the leaves of $(M,\omega,\eta),$1 is symplectic, so $(M,\omega,\eta),$2 defines a regular Poisson structure $(M,\omega,\eta),$3 of corank $(M,\omega,\eta),$4, whose symplectic foliation is exactly $(M,\omega,\eta),$5. The induced Poisson structure on the base $(M,\omega,\eta),$6 is denoted $(M,\omega,\eta),$7, and the Poisson structure on the base obtained from $(M,\omega,\eta),$8 coincides with the one obtained from $(M,\omega,\eta),$9 (Fernandes et al., 2023).

A basic proposition gives the numerical and geometric constraints: $\omega\in \Omega^2(M)$0 $\omega\in \Omega^2(M)$1 is multiplicative, $\omega\in \Omega^2(M)$2 is multiplicative, and the Reeb vector field is a bi-invariant Poisson vector field. Another key result shows that the identity section $\omega\in \Omega^2(M)$3 lies in a single symplectic leaf $\omega\in \Omega^2(M)$4 of $\omega\in \Omega^2(M)$5, and that leaf is a Lie subgroupoid. The restricted structure

$\omega\in \Omega^2(M)$6

is then a symplectic groupoid integrating $\omega\in \Omega^2(M)$7. Thus every cosymplectic groupoid contains a canonical symplectic subgroupoid integrating its base Poisson manifold (Fernandes et al., 2023).

4. Structural relations with Poisson, oversymplectic, and infinitesimal geometry

Cosymplectic groupoids sit precisely at the interface of two corank-$\omega\in \Omega^2(M)$8 theories. A Poisson groupoid $\omega\in \Omega^2(M)$9 is of corank $\eta\in \Omega^1(M)$0 when $\eta\in \Omega^1(M)$1 has constant rank $\eta\in \Omega^1(M)$2. Such a Poisson groupoid is cosymplectic exactly when there exists a vector field $\eta\in \Omega^1(M)$3 that is a Poisson vector field, is transverse to the symplectic foliation, and is bi-invariant. This characterizes the additional “Reeb direction” needed to pass from a regular corank-$\eta\in \Omega^1(M)$4 Poisson groupoid to a cosymplectic one.

The second relation concerns oversymplectic groupoids of corank $\eta\in \Omega^1(M)$5, namely Lie groupoids with a closed multiplicative $\eta\in \Omega^1(M)$6-form $\eta\in \Omega^1(M)$7 of constant corank $\eta\in \Omega^1(M)$8. In that setting, $\eta\in \Omega^1(M)$9 is a line distribution, and when it is simple the quotient inherits a symplectic groupoid structure. For a proper orientable corank-$TM=\ker(\omega)\oplus \ker(\eta).$0 oversymplectic groupoid with simple $TM=\ker(\omega)\oplus \ker(\eta).$1, there exists a multiplicative $TM=\ker(\omega)\oplus \ker(\eta).$2-form $TM=\ker(\omega)\oplus \ker(\eta).$3 making $TM=\ker(\omega)\oplus \ker(\eta).$4 cosymplectic if and only if the associated multiplicative Chern class vanishes. More precisely, if

$TM=\ker(\omega)\oplus \ker(\eta).$5

is a proper $TM=\ker(\omega)\oplus \ker(\eta).$6-central extension, the obstruction is

$TM=\ker(\omega)\oplus \ker(\eta).$7

and

$TM=\ker(\omega)\oplus \ker(\eta).$8

This provides an objective clarification of a possible misconception: a multiplicative closed $TM=\ker(\omega)\oplus \ker(\eta).$9-form of corank $1$00 does not automatically extend to a cosymplectic structure; the obstruction is cohomological (Fernandes et al., 2023).

Infinitesimally, the structure is rigid. A multiplicative $1$01-form $1$02 induces an IM $1$03-form

$1$04

and a multiplicative $1$05-form $1$06 induces an IM $1$07-form

$1$08

There is also a distinguished central section $1$09 coming from the Reeb field. The Lie algebroid $1$10 fits into

$1$11

with kernel spanned by $1$12, and the splitting is given by

$1$13

Hence

$1$14

The bracket and anchor are the standard ones induced by the base Poisson structure $1$15: $1$16

$1$17

The associated Lie bialgebroid is correspondingly

$1$18

In the source $1$19-connected case, the classification becomes explicit: $1$20 with multiplicative forms

$1$21

where $1$22 is the source $1$23-connected symplectic groupoid integrating $1$24. This shows that, infinitesimally and in the universal integration, the extra direction is a trivial central $1$25-extension (Fernandes et al., 2023).

5. Hamiltonian actions, moment maps, and reduction

The Hamiltonian theory begins in the precosymplectic setting. Let $1$26 be a Lie group with Lie algebra $1$27. An action on $1$28 is precosymplectic if

$1$29

It is Hamiltonian if there exists a moment map

$1$30

such that for each $1$31,

$1$32

and $1$33 is $1$34-equivariant.

A distinguished ideal is defined by

$1$35

with connected normal subgroup $1$36. Then $1$37 acts trivially on the leaf spaces $1$38 and $1$39, and the moment map descends to

$1$40

If $1$41 is connected, the image lies in an affine translate $1$42 with $1$43 coadjoint-fixed; after shifting, one may assume

$1$44

The action is called clean when

$1$45

or equivalently, using $1$46,

$1$47

The stacky form of this theory uses a foliation Lie $1$48-group

$1$49

associated to a crossed module $1$50. A Hamiltonian action on a 0-shifted cosymplectic groupoid $1$51 is encoded by a moment map morphism

$1$52

subject to three conditions: the $1$53-action on $1$54 is precosymplectic; for each $1$55,

$1$56

and $1$57 is $1$58-equivariant in the $1$59-group sense (Garcia et al., 24 Aug 2025).

Reduction is a 0-shifted cosymplectic analogue of Marsden–Weinstein–Meyer reduction. Suppose $1$60 carries such a Hamiltonian action and

$1$61

is a regular value of

$1$62

Then

$1$63

is a Lie subgroupoid. Using the foliation $1$64, one constructs a foliation groupoid

$1$65

with a regular Lie $1$66-group action and a $1$67-equivariant Morita map

$1$68

Here regularity means: the action is locally leafwise transitive; the action of $1$69 on $1$70 is free; and if $1$71 fixes a point of $1$72, then $1$73.

If $1$74 acts freely on $1$75, then $1$76 is a manifold and $1$77 is a principal $1$78-bundle. The reduced 0-shifted cosymplectic groupoid is

$1$79

with reduced forms $1$80 determined by

$1$81

where

$1$82

Its groupoid structure is given by

$1$83

$1$84

$1$85

For an ordinary Lie group action on a cosymplectic manifold, if $1$86 is a regular value of $1$87, then

$1$88

is a cosymplectic stack; it is an orbifold if the action is proper and a manifold if free and proper (Garcia et al., 24 Aug 2025).

6. Convexity, Morse–Bott theory, examples, and interpretive significance

Under compactness assumptions, the theory admits a Kirwan-type convexity theorem. Assume

$1$89

is of compact type, $1$90 is compact Lie, and $1$91 is compact. Choose a maximal $1$92-torus

$1$93

and a closed Weyl chamber $1$94. The moment image satisfies

$1$95

and the moment body is

$1$96

If the $1$97-action on $1$98 is clean, then $1$99 is a closed convex polyhedron. This is the cosymplectic-stack analogue of Kirwan convexity (Garcia et al., 24 Aug 2025).

The same clean-action hypothesis yields a Morse–Bott theorem. For $(\omega,\eta)$00, the component

$(\omega,\eta)$01

defines a Lie groupoid morphism

$(\omega,\eta)$02

of Morse–Bott type when the induced basic function on $(\omega,\eta)$03 is Morse–Bott. Under the clean $(\omega,\eta)$04-action hypothesis, each $(\omega,\eta)$05 is Morse–Bott, and every non-degenerate critical submanifold has even index (Garcia et al., 24 Aug 2025).

The examples in the literature organize the geometry. If $(\omega,\eta)$06 is a symplectic groupoid, then

$(\omega,\eta)$07

with

$(\omega,\eta)$08

is a cosymplectic groupoid; replacing $(\omega,\eta)$09 by $(\omega,\eta)$10 gives a similar example. Given a symplectic groupoid automorphism $(\omega,\eta)$11, the mapping torus

$(\omega,\eta)$12

inherits a multiplicative cosymplectic structure. In the proper, source-connected case, with the symplectic leaf $(\omega,\eta)$13 through the identity embedded, the Reeb flow returns $(\omega,\eta)$14 to itself at some time $(\omega,\eta)$15, defines a symplectomorphism

$(\omega,\eta)$16

and yields the normal form

$(\omega,\eta)$17

There is also a proper example on $(\omega,\eta)$18 with a specific multiplicative symplectic form and a transverse multiplicative vector field for which the Reeb orbit space need not be smooth, showing that the extension picture need not be central in the smooth sense. Another example shows that even when the underlying groupoid splits as a product, the cosymplectic structure itself need not be equivalent to the obvious product structure, so the multiplicative geometry carries finer information than the groupoid decomposition alone (Fernandes et al., 2023).

Reduction examples make the stacky character explicit. Starting from a toric symplectic manifold $(\omega,\eta)$19 and its symplectic mapping torus $(\omega,\eta)$20, or the trivial product $(\omega,\eta)$21, if $(\omega,\eta)$22 and $(\omega,\eta)$23 is a regular value of $(\omega,\eta)$24, then

$(\omega,\eta)$25

inherits a precosymplectic structure

$(\omega,\eta)$26

with reduced moment map

$(\omega,\eta)$27

For $(\omega,\eta)$28, the explicit moment map is

$(\omega,\eta)$29

and after reduction by $(\omega,\eta)$30,

$(\omega,\eta)$31

The reduced image is

$(\omega,\eta)$32

a polyhedral cone or orthant-type region. A further family of examples shows how the foliations $(\omega,\eta)$33 and $(\omega,\eta)$34 vary with the symplectomorphism used in the mapping torus. For $(\omega,\eta)$35 with $(\omega,\eta)$36-action,

$(\omega,\eta)$37

If $(\omega,\eta)$38, the foliations are trivial or product-like; if

$(\omega,\eta)$39

the induced foliation can have nontrivial holonomy, for example a leaf with holonomy $(\omega,\eta)$40, and similarly one can obtain holonomy $(\omega,\eta)$41 or $(\omega,\eta)$42. This clarifies why the natural quotient object is often a stack or groupoid rather than a manifold (Garcia et al., 24 Aug 2025).

Taken together, these results show that 0-shifted cosymplectic geometry is a globalization of precosymplectic geometry to differentiable stacks, while its multiplicative model is controlled by corank-$(\omega,\eta)$43 Poisson geometry, symplectic subgroupoids, Reeb dynamics, and trivial central extensions of cotangent Lie algebroids. A plausible implication is that the notion occupies for cosymplectic geometry the same structural role that 0-shifted symplectic structures occupy for symplectic geometry: it packages a closed $(\omega,\eta)$44-form together with a distinguished closed $(\omega,\eta)$45-form into a Morita-invariant, reduction-compatible, and Poisson-generating theory on stacks (Garcia et al., 24 Aug 2025, Fernandes et al., 2023).