Ten Compatible Poisson Brackets on $\mathbb P^5$

We give explicit formulas for ten compatible Poisson brackets on $\mathbb P^5$ found in arXiv:2007.12351.


Introduction
The goal of this paper is to present explicit formulas for certain algebraic Poisson brackets on P 5 .
Recall that two Poisson brackets { still a Poisson bracket (i.e., satisfies the Jacobi identity).Pairs of compatible Poisson brackets play an important role in the theory of integrable systems.
With every normal elliptic curve C in P n one can associate naturally a Poisson bracket on P n , called a Feigin-Odesskii bracket of type q n+1,1 .The corresponding quadratic Poisson brackets on A n+1 arise as quasi-classical limits of Feigin-Odesskii elliptic algebras.On the other hand, they can be constructed using the geometry of vector bundles on C (see [2,8]).
It was discovered by Odesskii-Wolf [6] that for every n there exists a family of 9 linearly independent mutually compatible Poisson brackets on P n , such that their generic linear combinations are Feigin-Odesskii brackets of type q n+1,1 .In [3], this construction was explained and extended in terms of anticanonical line bundles on del Pezzo surfaces.It was observed in [3,Example 4.6] that in this framework one also obtains 10 linearly independent mutually compatible Poisson brackets on P 5 .In this paper, we will produce explicit formulas for these 10 brackets (see Theorem 3.2).
2 Homological perturbation for P n 2.1 Formula for the homotopy Let H = p≥0, q∈Z H p (P n , O(q)) be the cohomology algebra of line bundles on P n , and A = p≥0, q∈Z C p (P n , O(q)), d the Čech complex with respect to the standard open covering U i = (x i ̸ = 0) of P n .There is a natural dg-algebra structure on A, such that the corresponding cohomology algebra is H.The multiplication on A is defined as follows.For α ∈ C p (P n , O(q)) and β ∈ C p ′ (P n , O(q ′ )), we define αβ ∈ C p+p ′ (P n , O(q + q ′ )) by where on the right hand side we use the multiplication map O(q) ⊗ O(q ′ ) → O(q + q ′ ).
The homological perturbation lemma equips H with a minimal A ∞ -structure (m n ), where m 2 is the usual product on H.We will use the form of this lemma due to Kontsevich-Soibelman [5], which gives formulas for m n as sums over trees.To apply homological perturbation, we need the following data: a projection π : A → H, an inclusion ι : H → A, and a homotopy Q such that πι = id H and id A −ιπ = dQ + Qd.
and H i = 0 for i ̸ = 0, n.We define ι in degree zero by ι(f ) k = f for k = 0, 1, . . ., n.We define ι in degree n by ι(g) 0...n = g.We define the projection in degree zero to be To define π in degree n, we observe that and we let π be the natural projection to H n .To define the homotopy, we use that A decomposes as a direct sum of chain complexes where A(⃗ e) consists of all elements in A whose components are scalar multiples of x ⃗ e := x e 0 0 x e 1 1 • • • x en n .In other words, A(⃗ e) is the ⃗ e-isotypical summand with respect to the action of the group (which is equal to −∞ if all e i are negative).There is then a standard homotopy Q defined on an element γ ∈ A(⃗ e) p by Q(γ , if all e i are negative).
Note that the condition I ⊃ {0 ≤ i ≤ n | e i < 0} guarantees that x ⃗ e is a regular section of the appropriate line bundle over U i 0 ...ip .Clearly, these elements form a basis for A and our homotopy operator Q is given by With these data one can in principle calculate all the higher products on the cohomology algebra H. Below, we will get explicit formulas in the case we need.

Calculation of m 4 for P 2
We now specialize to the case of the projective plane P 2 .We have no higher products of odd degree because H and H ⊗n only live in even degrees.Also, for degree reasons the product m 4 will only be non-zero on elements e ⊗ f ⊗ g ⊗ h ∈ H ⊗4 where one or two of the arguments lie in H 2 and the rest in H 0 .Below, we will explicitly compute the product m 4 involving one argument in H 2 .Thus, the following special case of the multiplication in A will be relevant: for a monomial x ⃗ e and a Laurent monomial x ⃗ e ′ , we have We use the formula where the sum runs over all rooted binary trees with 4 leaves labeled e, f , g and h (from left to right).For each such tree T the expression m T (e, f, g, h) is computed by moving the inputs through that tree, applying ι at the leaves, applying the homotopy Q on each interior edge, multiplying elements of A at each inner vertex and finally applying the projection π at the bottom.
We have to sum over the following five trees, which we denote T 1 , . . ., T 5 , respectively, Let us first consider the case e ∈ H 2 and f, g, h ∈ H 0 and let's take them all to be basis elements of H 2 and H 0 : where α 0 , α 1 , α 2 < 0 and a i , b i , c i ≥ 0 for i = 0, 1, 2. In this case only one of the trees above can be non-zero in the expression for m 4 (e, f, g, h), namely T 5 , because in all other trees at some point the homotopy Q will be applied to an element of A 0 .Below is a picture of the different summands in A • and the possible ways the homotopy Q can map a monomial element in each summand: (2) (4) When computing m T 5 (e, f, g, h) we should move e through this diagram; at every node it gets multiplied by one of the other arguments and then it moves downwards along one of the arrows.
We see that we get non-zero result if we move either along (1) followed by (2) or along (3) followed by ( 4) (so that we land in • 2 ).We claim that only the second route is possible.The reason is that at each node we multiply e by a monomial, so the exponents of x 0 , x 1 , x 2 will not decrease at any time.By the definition of Q, if e gets moved along (1) then after the multiplication at • 0,1,2 the exponent of x 1 is non-negative while the exponent of x 2 is negative.Hence, after performing the multiplication at • 0,2 the exponent of x 1 is still non-negative.It follows then from the definition of Q that e cannot move along (2) after moving along (1).Now comes the computation of m T 5 (e, f, g, h).Below, we denote by µ the multiplication in A. Then m T 5 (e, f, g, h) = πµ(Qµ(Qµ(e, f ), g), h) where the symbols ( * ), ( * * ) are ( * * * ) mean that we get zero unless the following conditions hold: ( * ) In the end, we have Similarly, we compute m 4 applied to e, f , g, h in any given order.We have 3 Feigin-Odesskii brackets

Bivectors on projective spaces
It is well known that every G m -invariant bivector on a vector space V leads to a bivector on the projective space PV .A bivector on V can be thought of as a skew-symmetric bracket {•, •} on the polynomial algebra S(V * ), which is a biderivation.Such a bracket is G m -invariant if and only if the bracket of two linear forms is a quadratic form.In other words, such a bracket can be viewed as a skew-symmetric pairing b : The corresponding bivector Π on the projective space PV is determined by the skew-symmetric forms Π v on T * v PV for each point ⟨v⟩ ∈ PV .We have an identification It is easy to see that under this identification we have where s 1 , s 2 ∈ ⟨v⟩ ∨ .Here we take the value of the quadratic form b(s 1 ∧ s 2 ) at v. We can use the above formula in reverse.Namely, suppose for some bivector Π on PV we found a skew-symmetric pairing b such that (3.1) holds.Then the G m -invariant bracket {•, •} on S(V ) given by b induces the bivector Π on PV .Note that if Π is a Poisson bivector on PV , it is not guaranteed that the G m -invariant bracket {•, •} on S(V ) is also Poisson, i.e., satisfies the Jacobi identity (but it is known that {•, •} can be chosen to be Poisson, see [1,7]).

Recollections from [3]
Below, we will denote simply by L 1 L 2 the tensor product of line bundles L 1 and L 2 .
Let ξ be a line bundle of degree n on an elliptic curve C. We fix a trivialization ω C ≃ O C .Then the associated Feigin-Odesskii Poisson structure Π (to which we will refer as FO bracket) on PH 1 ξ −1 ≃ PH 0 (ξ) * is given by the formula (see [3, where ⟨ϕ⟩ ∈ P Ext 1 (ξ, O), and s 1 , s 2 ∈ ⟨ϕ⟩ ⊥ .Here we use the Serre duality pairing ⟨•, •⟩ between H 0 (ξ) and H 1 ξ −1 and the triple Massey product that also agrees with the triple product m 3 obtained by homological perturbation from the natural dg enhancement of the derived category of coherent sheaves on C.There is some ambiguity in a choice of m 3 but for s 1 , s 2 ∈ ⟨ϕ⟩ ⊥ , the expression in the right-hand side of (3.2) is well defined.Next, assume that S is a smooth projective surface, L is a line bundle on S such that H * (S, LK S ) = 0, and let C ⊂ S be a smooth connected anticanonical divisor (which is an elliptic curve), so we have an exact sequence of coherent sheaves on S, We have a natural restriction map The exact sequence shows that under our assumptions this restriction map is an isomorphism.Thus, the FO bracket on PH 0 (L| C ) * associated with (C, L| C ) (defined up to rescaling) can be viewed as a Poisson structure on a fixed projective space PV * , where By [3,Theorem 4.4], the Poisson brackets on PV * associated with different anticanonical divisors are compatible.More precisely, we get a linear map from H 0 S, K −1

S
to the space of bivectors on PV * , whose image lies in the space of Poisson brackets.

Feigin-Odesskii bracket for an anticanonical divisor
We keep the data (S, L) of the previous subsection.Let i : C → S be an anticanonical divisor in S, with the equation F ∈ H 0 S, K −1 S .We want to write a formula for the FO bracket Π = Π F on PV * in terms of higher products on the surface S and the equation F .For this we rewrite the right-hand side of formula (3.2).Let us write the triple product in this formula as MP C to remember that it is defined for the derived category of C. Proposition 3.1.
(i) In the above situation, given e ∈ V * and s 1 , s 2 ∈ ⟨e⟩ ⊥ , one has where we use the identification V * ≃ H 2 S, L −1 K S given by Serre duality and consider the A ∞ -products on S, obtained by the homological perturbation.
(ii) Assume that a generic anticanonical divisor is smooth (and connected).Then gives a collection of compatible Poisson brackets on PV depending linearly on F .
Proof .(i) By Serre duality, H * S, L −1 = 0, so the map induced by the exact sequence is an isomorphism.It is a standard fact that this isomorphism is the dual to the isomorphism H 0 (S, L) → H 0 (C, L| C ) given by the restriction, via Serre dualities on S and C. Let us denote by e C ∈ H 1 C, L −1 C the element corresponding to e ∈ H 2 S, L −1 K S under the above isomorphism.
We claim that the triple Massey product MP C (s (where the middle arrow has degree 1) agrees with the corresponding triple Massey product on S, Indeed, the relevant spaces are identified via the restriction maps.Let be the natural maps.Then we have to check that for s 1 , s 2 ∈ ⟨e⟩ ⊥ ⊂ H 0 (S, L), one has where we view this as equality of cosets in Hom(O S , L| C ).The A ∞ -identities imply that where s 2 | C r = r L s 2 , and Combining these two identities, we deduce our claim.Thus, it is enough to calculate the Massey product MP(s 1 | C , e C r L , s 2 ).Using the exact sequences (3.3) and (3.4), we can represent O C (resp.L C ) by the twisted complex [K S [1] In terms of these resolutions, the elements of Ext ), while the element of Hom(O C , L| C ) corresponding to s ∈ H 0 (S, L) ≃ H 0 (C, L| C ) is given by the natural map of twisted complexes induced by the multiplication by s.The elements of Hom(O S , L| C ) are identified with Hom(O S , L) ≃ hom 0 (O S , [LK S [1] → L]).Thus, the m 3 product we are interested is given by the following triple product in the category of twisted complexes over S: where we view e as a morphism of degree 1 from L to K S [1].Now the formula for m 3 on twisted complexes (see [4,Section 7.6]) gives m 4 (F, s 1 , e, s 2 ) − m 4 (s 1 , F, e, s 2 ) (here the insertions of F correspond to insertions of the differentials in the twisted complexes).
(ii) It is clear that Π F gives a linear map from H 0 S, ω −1

S
to the space of bivectors on PV .By (i), for generic F we get a Poisson bracket.Hence, this is true for all F .  ).The fact that these 10 brackets are linearly independent follows from the compatibility of this construction with the GL 3 -action and is explained in [3,Proposition 4.7].Now we will derive formulas for the brackets { , } F on the algebra of polynomials in 6 variables which induce the above Poisson brackets on PV ≃ P 5 , where * .
They depend linearly on F , so we will just give formulas for { , } x ⃗ c , where x ⃗ c runs through all 10 monomials of degree 3 in (x 0 , x 1 , x 2 ).Let us set Note that {x ⃗ e | ⃗ e ∈ ∆(n)} forms a basis for H 0 P 2 , O(n) when n ≥ 0, while x ⃗ e {0,1,2} | ⃗ e ∈ ∆(n) is a basis for H 2 P 2 , O(n) when n < 0. In particular, we use x ⃗ a | ⃗ a ∈ ∆(2) as a basis in V * = H 0 P 2 , O(2) .Our brackets should associate to a pair of elements of this basis a quadratic form in the same variables.
where the second sum is over the symmetric group on the letters {a, b, c} and 2. It is not true that formulas (3.5) define compatible Poisson brackets on the algebra of polynomials in 6 variables: this is true only for the induced brackets on P 5 (in other words, the relevant identities hold only for the ratios of coordinates x i /x j ). ■

3. 4
The case leading to 10 compatible brackets on P5   We can apply Proposition 3.1 to the case S = P 2 and L = O(2).Note that the assumptions are satisfied in this case since LK S = O(−1) has vanishing cohomology.Thus, for each F ∈ H 0 P 2 , O(3) giving a smooth cubic, we get a formula for the FO-bracket Π F onPH 0 P 2 , O(2)* = P 5 .Hence, we get a family of 10 (the dimension of H 0 P 2 , O(3) compatible brackets on P 5 (we also know this from[3, Proposition 4.7]