SNC Log Symplectic Structures on Fano Products

Abstract This paper classifies Poisson structures with the reduced simple normal crossing divisor on a product of Fano varieties of Picard number 1. The characterization of even-dimensional projective spaces from the viewpoint of Poisson structures is given by Lima and Pereira. In this paper, we generalize the characterization of projective spaces to any dimension.


Introduction
Geometry of log symplectic form is well investigated, and the term log symplectic is slightly abused. In the eld of di erential geometry, it mostly refers to a generically symplectic Poisson structure with the reduced and smooth divisor [ , ]. Such a structure is o en called topologically stable Poisson [ ], b-Poisson [ ], or b-log symplectic [ ] in the eld of topological geometry. In Goto's de nition [ ], log symplectic indicates the generically symplectic Poisson strucuture with the reduced and simple normal crossing degeneracy divisor. In this paper, we use this terminology following Pym's de nition [ ]. at is, we do not suppose that the reduced degeneracy divisor is smooth or simple normal crossing. is is a reasonable de nition in terms of holomorphic Poisson structures and the eld of algebraic geometry, as the degeneracy divisors usually have singularities in the higher-dimensional case [ , eorem ]. In fact, some of the Feigin and Odesskii's examples have elliptic singular points.
One of the main bene ts of holomorphic Poisson structure is that holomorphic Poisson manifolds also have the almost complex structure. Interestingly, in terms of the symplectic geometry, the known examples of projective irreducible holomorphic symplectic manifolds are only of four types [ ].
erefore, the analogous problem is worth investigating. Since Poisson structures can be regarded as a generalization of symplectic forms, there should be many more types of such Poisson manifolds. Indeed, one can easily nd such examples on the projective space. However, the following theorem suggests that the Poisson structure still imposes severe constraints.

eorem . ([ ])
Let (X, Π) be a log symplectic structure with the simple normal crossing degeneracy divisor (say snc log symplectic structure) on a complex Fano variety X with cyclic Picard group of even dimension n ≥ . en X is a projective space, and Π is a diagonal Poisson structure. e diagonal Poisson structure on a projective space (or an a ne space) is de ned as a generically symplectic Poisson structure whose degeneracy divisor is the union of all coordinate hyperplanes. A local form of a diagonal Poisson structure can be given by Π = ∑ i< j c i j x i x j ∂ ∂xi ∧ ∂ ∂x j for a general system of coe cients c i j and homogeneous coordinates x i of X.
e degeneracy divisor must have singularities when we treat the higherdimensional variety, so we usually assume a xed kind of the singularity of the degeneracy divisor.
is principle is justi ed, because the Zariski closure of set of log symplectic structures or the set of SNC log symplectic structures forms a union of connected components in the moduli space of all Poisson bivector elds [ ].
ere are two cases used to classify holomorphic log symplectic manifolds. One is when the singularity is not a simple normal crossing singularity, and the other is when the Picard group of a variety is not cyclic. Pym [ ] researches the case where all the singular points are elliptic singular points. So one of the main unsolved situations is that the variety is not of Picard rank . e author [ ] classi es the case of the blowing up of projective spaces along a linear subspace. e focus of this paper is a product of varieties.
In this paper, we prove the following theorem.

eorem .
Let X i be a complex Fano variety of Picard number and of dimension n i ≥ , let X = ∏ m i= X i be a product of even dimension n = ∑ m i= n i , and let Π be a SNC log symplectic structure on X. en we have X i = P ni , and Π is a diagonal Poisson structure.
A diagonal Poisson structure on the product of projective spaces indicates that the degeneracy divisor is the union of all coordinate hyperplanes. Let x i , . . . , x ini denote a homogeneous coordinate system of X i and c i jk l ∈ C.
en we can express the diagonal Poisson structure on the product of the projective spaces in the following form: is theorem gives the characterization of projective spaces of any dimensions.

Corollary .
Let X be a complex Fano variety of Picard number . Suppose that X × X admits a SNC log symplectic structure, then X is a projective space.

Poisson Structures
Only in this chapter, we assume that the base eld is an algebraivally closed and of characteristic . Let X be a smooth projective variety. A Poisson structure on X is a bivector eld Π ∈ Γ(X, ∧ T X ) such that the Schouten bracket [Π, Π] ∈ Γ(X, ∧ T X ) vanishes identically. When we de ne a bilinear map by { f , g} = Π(d f , d g) for f , g ∈ O X , the bracket satis es the following properties:

SNC Log Symplectic Structures on Fano Products
In general, (a) and (c) hold for every bracket de ned by a bivector eld. Jacobi identity holds if and only if the bivector elds vanishes the Schouten bracket.
We say a Poisson structure Π has rank k at a point x ∈ X if Π k (x) ≠ and Π k+ (x) = . For the largest number k that satis es Π k ≠ , we say Π is of rank k. If dim X = n and rank Π = n, then we call the Poisson structure Π generically symplectic. We set D k− (Π) ∶= {x ∈ X rank x Π < k}. We call this set k-th degeneracy locus. If Π has rank k, then the divisorial part of D k− (Π) is called the degeneracy divisor of Π, and we denote it by D(Π). If Π is a generically symplectic Poisson structure, then the degeneracy loci forms a divisor, that is, is an e ective anti-canonical divisor.

Outline of Pym's Proof
In this section, we review an outline of the proof of eorem . given by Pym. For the product case, the proof will also be given along this outline. erea er, we assume that a variety X is over a complex number eld C. Pym's proof is consist of three steps. First claim is that Fano index of X has inequality is claim can be proved by an inductive argument on the dimension of X. e key ideas of the induction are generalized in Lemmas . and . in this paper. e classication of Fano varieties with high-index are well known.
eorem . ([ , , ]) Let X be a n-dimensional Fano variety of Picard number . Suppose i X ≥ n − . en X is in one of the following cases: (i) i X = n + and X = P n ; (ii) i X = n and X = Q n ⊂ P n+ , a smooth quadric hypersurface; (iii) i X = n − and X is one of the following varieties: (a) a degree-six hypersurface in the (n+ )-dimensional weighted projective space P( , , , . . . , ); (b) a double cover of P n branched over a smooth quartic hypersurface; (c) a smooth cubic hypersurface in P n+ ; (d) an intersection of two smooth quadric hypersurfaces in P n+ ; (e) a linear section of the Grassmannian Gr( , ) in its plucker embedding.
As a next step, Pym developed the key lemma. We can calculate each variety of high index one by one; we then obtain the conclusion that the variety must be a projective space.
eorem . ([ , Prop . ]) Let D = D(Π) = ∑ k j= D j be a irreducible decomposition of the degeneracy divisor. We write [D j ] = c (O X (D j )). en the following relation holds in the cohomology rings of X:

Example .
First case is X = P n , n ≥ . Let H be an ample generator of the Picard group, D(Π) = ∑ k j= D j a irreducible decomposition. en we write D j = d j H. e Euler sequence leads the equation ch(T X ) = ( n + )e H − . erefore, the le -hand side of ( . ) becomes ( n + ) sinh [H]. We compare the intersection number of both sides of ( . ) by intersecting the degree i − cycle H i− . en we obtain where the top equation is obtained when i = n − . is straightforward calculation gives a unique solution of integers d j : Since we suppose that the degeneracy divisor is a simple normal crossing one, it is composed of coordinate hyperplanes.
If the variety is not a projective space, then we can con rm that it is unsuitable in terms of the above lemma. We show this in one case, but it can be con rmed in the remaining cases as well.

Example .
Let X ⊂ P n+ be a smooth quadric hypersurface. Let H be an ample generator of the Picard group and let D(Π) = ∑ k j= D j be an irreducible decomposition. en we write D j = d j H. We have the relative Euler sequence: erefore, we obtain the chern character In the similar way, we compare the coe cients

SNC Log Symplectic Structures on Fano Products
If n ≥ , there are no solutions that satisfy the above system of equations.
Finally, we specify the form of the Poisson structure; r-matrix construction plays an important role.

De nition .
Let X be a variety equipped with an action of Lie group G, let g be its Lie algebra, and let r be a classical triangular r-matrix for G, that is, [r, r] = ∈ ∧ g.
en we obtain the Poisson structure on X by pushing r forward along the action map g → Γ(X, T X ). We call this construction of Poisson structures r-matrix construction.
When X is a n-dimensional projective space, we have already known that the degree of each irreducible component of the degeneracy divisor is by Example . . en one can show that such a Poisson structure must be obtained from r-matrix construction for the Lie group G = (C * ) n .

Numerical Properties of Fano Products
Let X = ∏ m i= X i be a variety of dimension n, let X i be a Fano variety of dimension n i ≥ and of Picard number with the projection p i ∶ X → X i , H i an ample generator of Pic(X i ), and let H i = p * i H i be a pull-back of H i . Recall that the intersection number is obtained by the following formula: where a system of integers d i satis es ∑ m i= d i = n and d is a degree of a projective variety X.

Proposition .
Settings are the same as the above. We write the irreducible decomposition of the degeneracy divisor D = D(Π) = ∑ k j= D j . We set D j = ∑ m i= a i j H i for some non-negative integer a i j ∈ N. en for every j, there uniquely exists ≤ i ≤ m such that a i j ≠ and a l j = for all l ≠ i. is indicates that we obtain Proof Since X is a product of manifolds X i , we have ch(T X ) = ∑

K. Okumura
We consider terms of degree of ( . ) and intersection numbers with respect to the is detects the coe cient of H i H l that is not contained in the le side of ( . ). As we calculated the intersection number, we have = d k j= a i j a l j .
is implies that a i j a l j = for every j. erefore, for some xed j, if there exists a non-zero integer a i j , then we nd that the other integers a l j = for all l ≠ i. Moreover, since H j is an e ective divisor on X, such an integer a i j indeed exists. Now, we have ch(T X ) = ∑ m i= ch(T Xi ); thus, the conclusion on the coe cients of D j suggests that the formula drops down to each variety X i and each divisor D j on X i such that is proposition and the discussion by Pym immediately lead the following corollary.

Corollary .
Settings are the same as in the above proposition. en X i = P ni or i Xi < n i − , where i Xi is the Fano index of X i .

Proof of the Main Theorem
In this section, we complete the proof of the main theorem. e rst goal is to show that X i = P ni for every i. For this aim, it is enough to con rm that i Xi ≥ n i − . In order to prove this, we will construct the triplet (X, Π, D(Π)) inductively. At rst, we take any irreducible component of D(Π) and name it D . en it is enough to show that we can choose another component D such that is again an SNC log symplectic structure on some Fano product.

Lemmas for the Induction
e settings are the same as in the above section; namely, let X i be a Fano manifold of Picard number and of dimension n i ≥ over a complex number eld C, let H i be an ample generator of Pic(X i ), let H i = p * i H i be a pull-back of H i , let X = ∏ m i= X i be a product of even dimension n = ∑ m i= n i , let p i ∶ X → X i be the i-th projection, let Π be a SNC log symplectic structure on X, let D = D(Π) = ∑ k j= D j be the degeneracy divisor and its irreducible decomposition, and let D j be a divisor on X i for some i such that p * i D j = D j .

Lemma .
Let Π be a SNC log symplectic structure on X = ∏ m i= X i with n i ≥ and ω = Π − a two-form corresponding to Π, that is, Π¬ω = . We take an arbitrary irreducible component D j of the degeneracy divisorD(Π), and we set the one-form α = Res D j ω. en α is not identically zero and has poles along D j ∩ (D(Π) D j ).
Proof α is evidently a non-zero meromorphic one-form that may have poles along D j ∩(D(Π) D j ). It is enough to show that α is not holomorphic. We suppose that D j is a divisor on X i and set X = ∏ k≠i X k . en D j = D j × X holds. Applying Kunneth's

SNC Log Symplectic Structures on Fano Products
formula, the following equation holds: By the Kodaira vanishing theorem, the latter term is . We apply the Lefshetz hyperplane theorem for Hodge decomposition (see e.g., m i= X i be a product, let p i ∶ X → X i be an i-th projection, let D be a simple normal crossing divisor on X, and let α ∈ Γ(X, Ω X (log D)) with residues λ j , that is, Res D j α = λ j . We write the irreducible decomposition D = ∑ k j= D j and D j = ∑ i a i j H i . en the following equation holds: k j= a i j λ j = f or al l i.

Proof
ere exists a residue exact sequence e injection ι j ∶ D j → X induces ι j * ; H p (X, C) → H p (D j , C). Its dual is ι j * ∨ ∶ (H p (D j )) ∨ → (H p (X, C)) ∨ . e Poincaré duality suggests that there is a map (H p (D j )) ∨ = H n− −p (D j , C) → H n−p (X, C) = (H p (X, C)) ∨ . In particular, when we take p = n − , we obtain the morphism ι j * ∶ H (D j , C) → H , (X). Due to the construction of ι j * , this sends ↦ [c (D j )]. en the connecting morphism δ is a sum of ι j * . Recall that we assumed that H (X i , O Xi ) = , so δ factors through the inclusion Pic(X) ⊗ C → H (X, Ω X ) and Pic(X) = ∑ m i= Pic(X i ). As α is an element of H (X, Ω X (log D)), δ(φ(α)) = . e i-th component contained in Pic(X i ) of this equation is ∑ k j= λ j a i j = . ∎

Induction
Now, we focus on the rst component X and show that X = P n . If we can prove this claim, then we can show that X = ∏ m i= P ni by changing the index of the variety. We take an irreducible component D of D(Π) and suppose that D is a divisor on X . en, Lemma . ensures that there exists some irreducible component D such that Res D (Res D ω) ≠ .
We will nd two cases: (A) D is a divisor on X , (B) D is a divisor on X .
en [ , Lemma . ] ensures that Π Y is a generically symplectic Poisson structure. By construction, we can write Y = Y × ∏ m i= X i . Applying the residue theorem for α = Res D ω and the variety D = D × ∏ m i= X i , we obtain the third irreducible component D such that D is a divisor on X . en D ensures that D Y is non-empty, and so D Y is a simple normal crossing divisor. e adjunction formula indicates that D Y is an anti-canonical divisor; that is, Y i is a Fano variety. Owing to the construction of Y, we have D(Π Y ) = D Y . If n ≥ , by applying the Lefshetz hyperplane theorem twice, we see that us, Y i is also the variety of Picard number . erefore, we obtain a new SNC log symplectic triplet Next, we consider the case where n = , . en Y i does not satisfy the hypothesis of Lefshetz hyperplane theorem. As we have three divisors D , D , D on X i , the Fano index i X of X becomes greater than . Since we suppose that n ≤ , Corollary . indicates that X = P n . Moreover, as we computed in Example . , we see that the degree a i j ≠ of the divisor D j equals .
is means that we just obtain the projective space as a degree hyperplane of the projective space when we cut out some irreducible component D , D of D(Π) in the step of the induction. erefore, we can discuss this in the same way, because of the two facts, that is, H (P n , Ω P n ) = and Picard number of the projective space is . Finally, we can also replace X with Y even if n = , .
We can also nd that D(Π Y ) = (D(Π) − D − D ) Y . en the residue theorem ensures that there exist D such that D is a divisor on X in the similar way as above. By switching the role of D and D , we obtain D with D is a divisor on X . erefore, D and D is a Fano variety. If n ≥ and n ≥ , then we can apply the Lefshetz hyperplane theorem and obtain that ρ(D ) = ρ(D ) = .
Next, we suppose n i = , for i = or . e Fano index i Xi of X i is greater than , since D(Π) has at least components D i and D i+ that are derived from X i . erefore, Corollary . leads to X i = P .

Dimension ≤ 2 and Summary.
e outline of our inductive process is as follows. First, we take D = a p * H . Next, we take D whose existence is ensured by Lemma . . Although we distinguish two cases by the source of D , in any case, we obtain the SNC log symplectic triplet (Y , Π Y , D Y ). en we take D again and repeat steps until we nd that Y is a projective space, and this will be achieved when the dimension becomes smaller than . Finally, we go back through the induction steps one by one and count the Fano index; then we nd that X = P n .
Before the conclusion, we note the following. On the above construction, we treat the case of n i ≥ , that is, our initial condition. In both Case A and Case B, the step reduces the dimension of the component. In particular, when we repeat Case B, n = dim X may be equal to or smaller than , and this is out of our assumptions. But if we encounter a such case, then that component must pass through Case B of n = . is means that we can treat such a low-dimensional component as a degree

SNC Log Symplectic Structures on Fano Products
hyperplane cut of the projective space. us, our induction works for every stage of the inductive process.

Poisson Structures on the Product of Projective Spaces
Proposition .
[ , Exercise . ] Let (X = ∏ m i= P ni , Π) be a SNC log symplectic structure with n i ≥ .
en, Π must be induced by r-matrix construction for the standard action of the torus G = (C * ) n × (C * ) n × ⋅ ⋅ ⋅ × (C * ) nm .
Proof Let Y be an orbit of a general point x and let g be a Lie algebra of G. e action of G preserves the degeneracy divisor D(Π) = X Y. For the action map a∶ g → Γ(Y , T Y ) and the element ξ ∈ g, we construct an element ξ X ∈ Γ(X, T X ). ξ X satis es two conditions, ξ X Y = a(ξ) and ξ X U = for any open subset U contained in X Y. So we obtain the mapã∶ g → Γ(X, T X ).
Next, we consider the basis of Γ(X, T X (− log D)). For a standard a ne coordinate x , . . . , x n , the vector elds x ∂x , . . . , x n ∂x n form a basis on that a ne open set. As they can glue to that on a product of projective spaces, we see that they form a global basis for T X (− log D). erefore, T X (− log D) is a trivial bundle over X whose bers are canonically identi ed with g. Since holomorphic sections of a trivial bundle on a compact manifold are always constant, we obtain the isomorphism ∧ g ≃ Γ(X, ∧ T X (− log D)) as a vector space. ∎ When we take coordinates ([x ∶ x ∶ ⋅ ⋅ ⋅ ∶ x n ], [x ∶ ⋅ ⋅ ⋅ ∶ x n ], . . . , [x m ∶ ⋅ ⋅ ⋅ ∶ x mnm ]), this means that Poisson structure Π is an invariant against the following transformation: where j ≠ and c i j ∈ C * . We can write the Poisson structure by using the coordinate as follows: When we set y i j = c i j x i j , we have ∂ ∂ y i j = ∂x i j ∂ y i j ∂ ∂x i j = c i j ∂ ∂x i j . erefore, {x i j , x k l } must be divisible by x i j . For the same reason, x k l also divides by {x i j , x k l }. erefore, the following equation holds: {x i j , x k l } = c i jk l x i j x k l for some coe cient c i jk l ∈ C. For the general choice of the coe cients c i jk l , in order to make Π be a generically symplectic, the degeneracy divisor is a union of all coordinate hyperplanes. One can also con rm that bivector elds of this form vanishes the Schouten bracket.