4.3 Non degenerate coisotropic structures
4.3.1 Definition of non-degeneracy
We start by first looking at the affine case. Recall the following notion, which is taken from [CPTVV, Definition 1.4.18].
Definition 4.3.1. Given an algebra A ∈ Pn+1 −algM, we say that A is non-degenerate if the morphism
DRint(A)−→Polint(A, n) induced by the Poisson bracket is an equivalence inMgr.
Let f:A→B a map insideCAlgM. Using the results of [MS, Section 2.7] we know that there is a natural gradedP[n+2,n+1]-structure on the couple
(Pol(A, n+ 1),Pol(B/A, n)),
such that the underlying morphism of graded commutative algebras is the map Pol(A, n+ 1)−→Pol(B/A, n)
induced byLA→LA⊗AB →LB.
Moreover, the fiber of the above morphism is Pol(f, n+ 1). Then we know from Section3.4.1 thatPol(f, n+ 1) is in fact a gradedPn+2-algebra. By definition, ann-shifted coisotropic structure on f is a morphism in the∞-category of graded dg Lie algebras
k(2)[−1]−→Pol(f, n+ 1)[n+ 1].
Our next goal is to show that one can use such a dg Lie morphism to endow the algebra (Pol(A, n+ 1),Pol(B/A, n))
with a mixed structure, in the sense of Section 3.4.2. In other words, we will show that there is a morphism of spaces
Cois(f, n)−→MixP[n+2,n+1](Pol(A, n+ 1),Pol(B/A, n)).
This will immediately follow from the following slightly more general statement.
Proposition 4.3.1. Let (R, S) be a graded P[n+2,n+1]-algebra in M. As in Section 3.4.1, let Ugr(R, S) be the fiber of the underlying morphism of graded commutative algebras R → S. Then there is a canonical morphism of spaces
MapLiegr
M(1M[−1](2), Ugr(R, S)[n+ 1])−→MixP[n+2,n+1](R, S).
Proof. By Definition 3.4.8, the space of mixed structures on (R, S) fits in a Cartesian square of spaces
where we considered R → Zgr(S) as a graded Pn+2-algebra inside the category Mor(M) of mor-phisms ofM.
We start by noticing that there is a natural map MapLiegr where1Mor(M) is the monoidal unit ofMor(M), that is to say the identity map of1M. Composing this with the map of Proposition3.4.7, we immediately get our desired morphism toMixPn+2(R→ Zgr(S)).
On the other hand, since Ugr(R, S) →Defgr(S)[−n−1]is a Pn+2-map, by definition of mixed structures we get that there is a morphism of spaces
MapLiegr
M(1M[−1](2), Ugr(R, S)[n+ 1])−→MixPn+1(S).
The results of Section 3.4.1 assure that the square Ugr(R, S) //
R
Defgr(S)[−n−1] //Zgr(S)
is a pullback of gradedPn+2-algebras, and thus the mixed structures onR →Zgr(S)and onSinduce the same mixed structure on Zgr(S). But now using the above presentation of MixP[n+2,n+1](R, S) as a limit, we do get our desired morphism
MapLiegr
M(1M[−1](2), U(R, S)[n+ 1])−→MixP[n+2,n+1](R, S), which concludes the proof.
Corollary 4.3.2. Let A→B be a morphism of commutative algebras inM. There is a canonical map of spaces
Cois(f, n)−→MixP[n+2,n+1](Pol(A, n+ 1),Pol(B/A, n)),
where (Pol(A, n+ 1),Pol(B/A, n))is considered with its canonical graded P[n+2,n+1]-algebra struc-ture.
Proof. This is simply follows from the definition of coisotropic structures and from the previous proposition.
Recall that there is a natural forgetful ∞-functor
P[n+2,n+1]−alggrM −→Mor(CAlggrM), which induces a map
Cois(f, n)−→MixComm(Pol(A, n+ 1)→Pol(B/A, n)).
In particular, given aP[n+1,n]-algebra(A, B), the map
Pol(A, n+ 1)−→Pol(B/A, n)
becomes a morphism of graded mixed commutative algebras. By definition, the weight zero compo-nent of this map is the underlying map f :A→B inCAlgM, so that by the universal property of the de Rham algebra (see [CPTVV, Section 1.4]) one gets a commutative square of graded mixed commutative algebras
Pol(A, n+ 1) //Pol(B/A, n)
DR(A)
OO //DR(B)
OO
where we haveDR(A)'SymA(LA[−1]) as graded commutative algebras.
Definition 4.3.2. We say that aP[n+1,n]-algebra(A, B)isnon-degenerate if the two vertical arrows in the diagram above are equivalences.
LetDR(f)denote the fiber of the mapDR(A)→DR(B). From the above diagram, we see that there is a natural arrow of graded mixed algebrasDR(f)→Pol(f, n+ 1). Notice that the space of morphisms in the∞-category of graded modules
k(2)−→DR(f)[n+ 1]
is by definition the space of closed 2-forms of degreen−1 onA, having restriction toB homotopic to 0. Equivalently, this can be described as the space of isotropic structures on the map A → B (whereAis only a pre-symplectic algebra). The moduleDR(f)fits into a diagram of graded mixed modules
DR(f)[n+ 1]−→Pol(f, n+ 1)[n+ 1]←−k(2)
Definition 4.3.3. Let again (A, B) be a P[n+1,n]-algebra. The space of isotropic structures com-patible with the given P[n+1,n]-structureis the space of dotted arrows making the following diagram commute
DR(f)[n+ 1]
k(2)
77
//Pol(f, n+ 1)[n+ 1]
in the ∞-category of graded mixed modules.
Now suppose that(A, B)is a non-degenerateP[n+1,n]-algebra. Then the mapDR(f)→Pol(f, n+
1) is an equivalence, and therefore the space of compatible isotropic structures on f : A → B is contractible. Notice that in this case the 2-form is automatically non-degenerate: in fact, we have an induced equivalence
DR(A)'Pol(A, n+ 1)
which by Theorem 3.2.5 of [CPTVV] can be used to turn the given non-degenerate shifted Poisson structure onAto a non-degenerate closed 2-form, so thatAis actually a shifted symplectic algebra.
Moreover, the fact that also DR(B) → Pol(B/A, n) is an equivalence implies that the isotropic structure is actually Lagrangian.
As a consequence, for every morphism of cdga f :A→B, there is a well-defined map of spaces Pnd[n+1,n](f)−→Lagr(f)
wherePnd[n+1,n](f)is the space of non-degenerateP[n+1,n]-structure on(A, B)such that the underlying cdga map isf.
Now let us deal with the general case. Let X be a n-shifted Poisson stack, and let f :L→ X be a morphisms of derived Artin stacks, locally of finite presentation. Suppose we are given a point γ ∈Cois(f, n). This means in particular that we have a map of graded dg Lie algebras
k[−1](2)−→Pol(X, n+ 1)[n+ 1]
such that the induced map
k[−1](2)−→Γ(LDR,Polt(PL/f∗PX, n)[n+ 1]
is homotopic to zero. Looking at weight 2 components, the shifted Poisson structure onX induces by adjunction a morphism of perfect complexes on X
π#: LX →TX[−n],
and the coisotropic condition implies that the induced mapLf →Tf[−n+ 1]is homotopic to zero.
This in turn gives the existence of dotted arrows in the following diagram Lf[−1] //
f∗LX //
f∗π#
LL
TL[−n] //f∗TX[−n] //Tf[−n+ 1]
where both horizontal rows are fiber sequences of perfect complexes on L.
Definition 4.3.4. With notations as above, the coisotropic structureγ onf is called non-degenerate if the n-Poisson structure on X is non-degenerate, and the previous diagram is an equivalence of fiber sequences. Equivalently, γ is non-degenerate if π# and the dotted arrows are equivalences of perfect complexes.
The space of non-degenerate coisotropic structures on f will be denoted Coisnd(f).
By Theorem4.1.5, the datum of a coisotropic structure onf :L→Xis equivalent to the datum of a compatible P[n+1,n]-structure on the couple (f∗PX(∞),PL(∞)) in the category of DLDR (∞)-modules.
Corollary 4.3.5. Let f : L → X be a map between derived Artin stacks, locally of finite pre-sentation, and suppose X is equipped with a n-shifted Poisson structure. A coisotropic structure γ ∈ Cois(f, n) is non-degenerate in the sense of Definition 4.3.4 if and only if the corresponding P[n+1,n]-algebra
(f∗PX(∞),PL(∞))
in the category of DLDR(∞)-modules is non-degenerate in the sense on Definition 4.3.2.
This is an immediate consequence of the general correspondence between geometric differential calculus on derived stacks and algebraic differential calculus on the associated prestacks of Tate principal parts, as exposed in [CPTVV].
We also have a similar result for the symplectic case.
Corollary 4.3.6. Let f : L → X again be a map between derived Artin stacks, locally of finite presentation, and supposeωX is an-shifted closed 2-form onX, such that f∗ω∼0 inside the space of closed 2-forms on L. The formω canonically induces a n-shifted closed 2-form ω0 on the algebra f∗PX(∞), relative to DLDR(∞), such that its restriction to PL(∞) is homotopic to zero.
Then ω is a Lagrangian structure on f if and only if ω0 is a Lagrangian on the couple (f∗PX(∞),PL(∞))
in the category of DLDR(∞)-modules.
Notice that in the classical case (i.e. if X is a smooth Poisson scheme and f : L → X is a coisotropic sub-scheme), then f is non-degenerate if and only if the Poisson structure on X comes from a symplectic structure, andLis a Lagrangian sub-scheme. Our next goal is to prove a derived extension of this result.
Theorem 4.3.7. Let X be a non-degenerate n-shifted Poisson stack, and let f : L → X be a morphism of derived Artin stacks. Let Coisnd(f, n) be the subspace ofCois(f, n) of connected com-ponents of non-degenerate coisotropic structures on f. Then X is canonically n-shifted symplectic, and there exists an equivalence of spaces
Coisnd(f, n)→Lagr(f, n) .
The proof of this theorem will closely follow the proof of Theorem 3.2.5 in [CPTVV]. The result will follow from a slightly more general statement. We will need some general constructions on stacks of Lie algebras and of mixed modules; this is already contained in [CPTVV], and we refer to that paper for more details.