FUJIKI RELATION ON SYMPLECTIC VARIETIES

DAISUKE MATSUSHITA

Abstract. We generalize Fujiki relation of Beauville-Bogomolov quadratic form on a symplectic variety. As an application, we study a fibre space structure of a projective symplectic variety.

1. Introduction

We start with the definition of a symplectic variety.

Definition 1.1. A compact Kähler variety Z is said to be a symplectic variety if Z satisfies the following conditions: For a smooth locus U on Z, there exists a nondegenerate holomorphic 2-form ω on U. This form is extended to a regular form ω̃ on Z̃, where ν : Z̃ → Z is a resolution of singularities of Z.

In [6, Theorem 8 (2)], Namikawa induced a quadratic form q_{Z} on H^{2}(Z, ℂ), which is a natural extension of Beauville-Bogomolov quadratic form defined in [1, Théorème 5] on singular varieties. In [3, Theorem 4.7], Fujiki proved that Beauville-Bogomolov quadratic form has special relation with cup products. We prove that q_{Z} has same properties.

Theorem 1.2. Let Z be a 2n-dimensional symplectic variety and ν : Z̃ → Z a resolution of singularities of Z. Assume that

(1) The codimension of the singular locus of Z is greater than four.

(2) For the smooth locus U of Z, the natural restriction map

H^{2}(Z, ℚ) ≅ H^{2}(U, ℚ)

is isomorphism.

(3) dim H^{1}(Z, O_{Z}) = 0 and dim H^{2}(Z, O_{Z}) = 1. According to [6, Theorem 8 (2)], we define the quadratic form q_{Z} on H^{2}(Z, ℂ) by

q_{Z}(α) := n ∫Z̃(ω̃¯ω̃)^{n−1}α̃^{2} + (1 − 2n) (∫Z̃ ω̃^{n} ¯ω̃^{n−1}α̃ ∫Z̃ ω̃^{n−1} ¯ω̃^{n}α̃) ,

where α̃ := ν^{∗}α and α ∈ H^{2}(Z, ℂ). Then q_{Z}(α) satisfies the following equation:

C_{Z}q_{Z}(α)^{n} = α^{2n}.

1991 Mathematics Subject Classification. Primary 14F40, Secondary 14D06.

Note that a constant C_{Z} depends on only Z. Moreover the index of q_{Z} is (3, h^{2}(Z, ℂ)− 3).

Remark 1. By [7, Theorem], when Z is symplectic varieties with only terminal singularities, the condition of codimension of singular locus is always satisfied. Remark 2. Namikawa obtained the index of q_{Z} in [8, Corollary 8] by different method.

Theorem 1.3. Let Z be a symplectic variety which satisfies the conditions of Theorem 1.2 and D a Cartier divisor on Z. Then Riemann-Roch formula of D is expressed as follows:

χ(D) = ∑_{k=0}^{n} a_{k}q_{Z}(D)^{k},

where a_{k} (0 ≤ k ≤ n) are constants which depend on only Z.

As an application of Theorem 1.2, we obtain the following result:

Corollary 1.4. Let Z be a 2n-dimensional projective symplectic variety which satisfies the conditions of Theorem 1.2. Assume that there exists a surjective morphism f : Z → B over a projective normal variety B and a general fibre F of f has positive dimension. Then F and B has the following properties:

(1) dim B = n and Picard number of B is one.

(2) F is an n-dimensional abelian variety.

(3) For the singular locus Z_{sing} of Z, f(Z_{sing}) forms a proper closed subset of B and the restriction ω to F is identically zero.

(4) For every effective divisor D of Z, dim f(D) ≥ dim B − 1.

Remark 3. Comparing the above corollary with [9, Theorem 2], there exist two difference: One is ampleness of −K_{B} and another one is ℚ-factoriality of B. In section 3, we construct an example such that −K_{B} is not ample. We prove Theorem 1.2 and 1.3 in Section 2. A proof of Corollary 1.4 and an example are given in Section 3. Acknowledgement. The author express his thanks to Professors A. Fujiki and Yo. Namikawa for their advice and encouragement.

2. Proof of Theorem 1.2

(2.1) We start with the investigation of the rank of q_{Z}.

Lemma 2.1. The rank of q_{Z} is greater than three.

Proof. Let ν : Z̃ → Z be a resolution of singularities of Z and ω̃ a extension of ω on Z̃. By the definition of q_{Z}, q_{Z}(ω̃ + ¯ ω̃) > 0 and q_{Z}(ω̃ − i¯ω̃) > 0. Let α be an ample divisor on Z. If we choose a suitable exceptional divisor E, ν^{∗}α − E becomes an ample divisor. From [6, Remark (1) (p24)], ω̃^{n−1}|_{E} = 0. Hence

q_{Z}(α) = n ∫Z̃(ω̃¯ω̃)^{n−1}(ν^{∗}α − E)^{2}.

Thus q_{Z}(α) > 0 by Hodge-Riemann bilinear relation. □ (2.2) Let D be the Kuranishi space of Z, Z the Kuranishi family and Z_{s} the fibre at s ∈ D. By [6, Theorem 8], for every point of s ∈ D, there exists an isomorphism ϕ_{s} : H^{2}(Z, ℂ) → H^{2}(Z_{s}, ℂ). The period map p : D → ℙ(H^{2}(Z, ℂ)) defined by p(s) := ϕ_{s}^{−1}(ω_{s}), where ω_{s} is a symplectic form of Z_{s}.

Lemma 2.2. Let ν_{s} : Z̃_{s} → Z_{s} be a resolution of Z_{s}. We define a quadric form on H^{2}(Z_{s}, ℂ) by

q_{Z_{s}}(α) := n ∫Z̃_{s}(ω̃_{s} ¯ω̃_{s})^{n−1}α̃^{2} + (1 − 2n) (∫Z̃_{s} ω̃_{s}^{n} ¯ω̃_{s}^{n−1}α̃ ∫Z̃_{s} ω̃_{s}^{n−1} ¯ω̃_{s}^{n}α̃) ,

where α̃ := ν_{s}^{∗}α and α ∈ H^{2}(Z_{s}, ℂ). Then q_{Z_{s}} is not depend on the choice of Z̃_{s} and

q_{Z}(α) = q_{Z}(ϕ_{s}^{−1}(ω_{s} + ω̄_{s}))q_{Z_{s}}(ϕ_{s}(α)).

Proof. We prove that q_{Z_{s}} is independent of the choice of resolution by similar argument in the proof of [6, Theorem 8 (2)] and obtain the rank of q_{Z_{s}} is greater than three by similar argument of Lemma 2.1. The period map p_{s} : D → ℙ(H^{2}(Z_{s}, ℂ)) is defined by p_{s}(t) := ϕ_{s}(p(t)).

Claim 2.3. Let Q_{s} := {q_{Z_{s}}(α) = 0|α ∈ H^{2}(Z_{s}, ℂ)}. Then p_{s}(D) ⊂ Q_{s}.

Proof. Let α ∈ H^{2}(Z_{s}, ℂ). For the smooth locus of U_{s} of Z_{s}, there exists Hodge decomposition of H^{2}(U_{s}, ℂ) and an isomorphism H^{2}(U_{s}, ℂ) ≅ H^{2}(Z, ℂ) by [6, Theorem 8]. Hence we write α as α = aω_{s} + w + bω̄_{s}. By direct calculation

∫Z̃_{s} α̃^{n+1} ¯ω̃_{s}^{n−1} = (n + 1)q_{Z_{s}}(α) (∫Z̃_{s} α̃ω̃_{s}^{n−1} ¯ω̃_{s}^{n})^{n−1} .

Let E be an irreducible component of the exceptional locus of Z̃_{s} → Z_{s}. Assume α := ϕ_{s}ϕ_{t}^{−1}(ω_{t}). Then α̃^{n+1}|_{U_{s}} = 0 because α|_{U_{s}} is a holomorphic two form on the smooth locus U_{t} of Z_{t} and U_{t} is deformation equivalent to U_{s} by [6, Theorem 8]. By [6, Remark (1) (p24)], ω̃_{s}^{n−1}|_{E} = 0. Hence the left hand side equals zero. If t is very near to s, we obtain ∫Z̃_{s} α̃ω̃_{s}^{n−1} ¯ω̃_{s}^{n} ≠ 0. Thus we obtain q_{Z_{s}}(α) = 0. □ We continue the proof of Lemma 2.2. By the above Claim, ϕ_{s} maps an open set of Q_{0} to an open set of Q_{s} isomorphically. Since both quadric surfaces are irreducible, we obtain that ϕ_{s}(Q_{0}) = Q_{s}. Since q_{Z_{s}}(ω_{s} + ω̄_{s}) = 1, we are done. □ (2.3) The following Lemma is the key of the proof of Theorem 1.2 and 1.3 which is based on arguments in [2, Lemma 1.9] and [4, Theorem 5.12].

4 DAISUKE MATSUSHITA

Lemma 2.4. Let ν : Z̃ → Z be a resolution of Z and D^{∘} the open set set of D such that the morphism Z̃ → D is smooth over D^{∘}. We fix one point s of D^{∘}. Let α ∈ H^{2}(Z_{s}, ℂ) and τ a (k, k)-form on Z̃_{s}. If τ is a (k, k)-form on every fibre Z̃_{t} near s,

(1)  τ · ν^{∗}α^{2(n−k)} = C_{k}q_{Z_{s}}(α)^{n−k},

where C_{k} is a constant depending on only Z_{s}.

Proof. Let S_{s} be the hypersurface in H^{2}(Z_{s}, ℂ) defined by

{τ · ν^{∗}α^{2(n−k)} = 0|α ∈ H^{2}(Z_{s}, ℂ)}.

If we prove that Q_{s} = S_{s}, we obtain equation (1) because both hand sides of (1) have same degree and same zero locus. We choose an open set V of Q_{s} which is contained in p_{s}(D^{∘}). For every point α ∈ V, there exists the point t of D^{∘} such that α defines a symplectic form on the smooth part of Z_{t}. Thus τ · (ν^{∗}α)^{n−k+1} = 0 in H^{2n+2}(Z̃_{t}, ℂ) and hence τ · (ν^{∗}α)^{n−k+1} = 0 in H^{2n+2}(Z̃_{s}, ℂ). By analytic continuation, we obtain τ · (ν^{∗}α)^{n−k+1} = 0 for every point of Q_{s}. Therefore we obtain Q_{s} ⊂ S_{s}. We prove the opposite inclusion. Assume the contrary. Then there exists β ∈ H^{2}(Z_{s}, ℂ) such that τ · ν^{∗}β^{2(n−k)} = 0 and β ∉ Q_{s}. We choose a general element γ of H^{2}(Z_{s}, ℂ) such that τ · γ^{2(n−k)} ≠ 0. Then the line which pass β and γ intersect Q_{s} with two points. Let δ_{0} and δ_{1} be these points. We write

β = λ_{11}δ_{0} + λ_{12}δ_{1}, γ = λ_{21}δ_{0} + λ_{22}δ_{1}.

We remark that λ_{∗∗} ≠ 0 because β, γ and δ_{∗} are mutually distinct. Since τ · (ν^{∗}δ_{0})^{n−k+1} = τ · (ν^{∗}δ_{1})^{n−k+1} = 0,

τ · (ν^{∗}β)^{2(n−k)} = _{2(n−k)}C_{n−k}λ_{11}^{n−k}λ_{12}^{n−k}δ_{0}^{n−k}δ_{1}^{n−k} = 0
τ · (ν^{∗}γ)^{2(n−k)} = _{2(n−k)}C_{n−k}λ_{21}^{n−k}λ_{22}^{n−k}δ_{0}^{n−k}δ_{1}^{n−k} ≠ 0.

That derives a contradiction. □ (2.4)Proof of Theorem 1.2. From Lemma 2.4, we obtain

C_{Z}q_{Z_{s}}(ϕ_{s}(α))^{n} = ϕ_{s}(α)^{2n}.

Since ϕ_{s}(α)^{2n} = α^{2n}, we obtain that C_{Z}q_{Z}(α)^{n} = α^{2n} by Lemma 2.2. We investigate the index of q_{Z}. From [6, Proposition 9], there exists Hodge decomposition of H^{2}(Z, ℂ). Let A be an ample divisor on Z and H an element of H^{1,1}(Z, ℂ)_{ℝ} such that q_{Z}(H, A) = 0. We consider the following equation:

(2)  Cq_{Z}(λH + A)^{n} = (λH + A)^{2n}.

If we compare λ term of both hand sides, we obtain H.A^{2n−1} = 0 from the assumption q_{Z}(A, H) = 0. By Hodge-Riemann bilinear relation,

H^{2}.A^{2n−2} · A^{2n} ≤ (H.A^{2n−1})^{2} = 0.

FUJIKI RELATION 5

Hence H^{2}.A^{2n−2} < 0 if H ≢ 0. Comparing λ^{2} term of the both hand side of the equation (2), we obtain q_{Z}(H) < 0 if H ≢ 0. Combining Lemma 2.1, the index of q_{Z} is (3, h^{2}(Z, ℂ) − 3). □ (2.5) Proof of Theorem 1.3. From the proof of (1) of [6, Theorem 8],

ϕ_{s} : H^{2}(Z, ℂ) ≅ H^{2}(Z, ℂ) → H^{2}(Z_{s}, ℂ)

is isomorphism. Hence χ_{Z}(D) = χ_{Z_{s}}(ϕ_{s}(D)). By Lemma 2.2, it is enough to prove that there exists constants C_{k} (0 ≤ k ≤ n) and they satisfy

χ_{Z_{s}}(D) = ∑_{i=0}^{n} C_{k}q_{Z_{s}}(D)^{k},

for a Cartier divisor D of Z_{s}. Let us consider χZ̃_{s}(ν^{∗}D). By [6, Theorem 6], Z_{s} has only rational singularities. Thus χ_{Z_{s}}(D) = χZ̃_{s}(ν^{∗}D). By Serre duality

χ_{Z_{s}}(D) = χ_{Z_{s}}(−D).

Hence each term of Riemann-Roch formula of χZ̃_{s}(ν^{∗}D) is expressed

(Polynomial of Chern classes of Z̃_{s}) · (ν^{∗}D)^{2k}.

Since polynomials of Chern classes remain (k, k)-form under small deformation, we obtain each term of Riemann-Roch formula is expressed as C_{k}q_{Z_{s}}(D)^{k} by Lemma 2.4. □

3. Fibre space structure

(3.1) We prove Corollary 1.4 in three steps:

Step 1: dim B = n.

Step 2: For a general fibre F of f, the restriction of a symplectic form on

the smooth locus of F is identically zero. Step 3: F is an Abelian variety and f(Z_{sing}) forms a proper closed subset of

B. Step 4: ρ(B) = 1 Step 5: For every effective divisor D on Z, dim f(D) ≥ n − 1.

We fix some notations. Let A be an ample divisor on Z and H an ample divisor on B. (3.2) Step 1. From Theorem 1.2,

C_{Z}q_{Z}(A + λf^{∗}H)^{n} = (A + λf^{∗}H)^{2n}.

Since C_{Z}q_{Z}(f^{∗}H) = (f^{∗}H)^{2n} = 0, we obtain

(3)  C_{Z}(q_{Z}(A) + 2λq_{Z}(A, f^{∗}H))^{n} = (A + λf^{∗}H)^{2n}.

If we compare both hand sides the above equation, we obtain

A^{k}.(f^{∗}H)^{2n−k} = 0, (k < n)
2nA^{2n−1}.f^{∗}H = C_{Z}q_{Z}(A, f^{∗}H) · q_{Z}(A)^{n−1}.

By Lemma 2.1, q_{Z}(A) > 0. Hence q_{Z}(A, f^{∗}H) > 0 and

A^{k}.(f^{∗}H)^{2n−k} > 0, (k ≥ n).

Hence dim B = n. (3.3) Step 2. Let F be a general fibre of f and U_{F} the smooth part of F. In order to prove ω|_{U_{F}} ≡ 0, we prove

∫_{F} ω ∧ ω̄A^{n−2} = 0.

We have

∫_{F} ω ∧ ω̄A^{n−2} = c(ωω̄A^{n−2}(f^{∗}H)^{n}),

where c is a nonzero constant. Hence we will show ωω̄A^{n−2}(f^{∗}H)^{n} = 0. By Theorem 1.2,

C_{Z}q_{Z}(ω + ω̄ + sA + tf^{∗}H, ω + ω̄ + sA + tf^{∗}H)^{n} = (ω + ω̄ + sA + tf^{∗}H)^{2n}.

Calculating the left hand side, we obtain

C_{Z}(q_{Z}(ω + ω̄) + s^{2}q_{Z}(A) + 2sq_{Z}(ω + ω̄, A) + 2tq_{Z}(ω + ω̄, f^{∗}H) + 2stq_{Z}(A, f^{∗}H))^{n}.

From the definition of q_{Z} in Theorem 1.2,

q_{Z}(ω + ω̄, A) = q_{Z}(ω + ω̄, f^{∗}H) = 0.

Thus we conclude that ωω̄A^{n−2}(f^{∗}H)^{n} = 0 by comparing the s^{n−2}t^{n} term of both hands sides. (3.4) Step 3. Let U be the smooth locus of Z. If we choose a point x of B generally, f^{∘} : U → B is smooth at x and U_{F} = F ∩ U. We consider the following diagram:

0 → f^{∗}T_{B,x} → T_{U} → T_{F_{U_{F}}} → 0
↓
0 → Ω_{U_{F}}^{1} → Ω_{U}^{1} → f^{∗}Ω_{B,x}^{1} → 0.

From the above diagram and Step 2, h^{0}(T_{U_{F}}) = n. Then F is an Abelian variety by the following Lemma.

Lemma 3.1. Let F be a normal variety such that K_{F} ∼ O_{F} and h^{0}(T_{U_{F}}) = dim F, where U_{F} is the smooth locus of F. Then F is an Abelian variety.

FUJIKI RELATION 7

Proof. Since F is normal, dim(F \ U_{F}) ≤ dim F − 2. Hence h^{0}(Θ_{F}) = dim F by analytic continuation. Let F̃ be a resolution of F. Then h^{0}(TF̃) = h^{0}(Θ_{F}) = dim F. Since K_{F} ∼ O_{F}, we have an injection TF̃ → Ωdim F −1 ; F̃. Hence h^{dim F −1}(OF̃) = h^{0}(Ωdim F −1 ; F̃) ≥ dim F. By Serre duality h^{dim F −1}(OF̃) = h^{1}(KF̃) From Grauert-Riemenschneider vanishing theorem, h^{1}(KF̃) = h^{1}(π_{∗}KF̃) = h^{1}(O_{F}). Hence h^{1}(F, O_{F}) ≥ dim F. By [5, Theorem 13], F is an abelian variety. □ Since F is a complete intersection in Z, Z is smooth in a neighbourhood of F and f(Z_{sing}) forms a proper closed subset of B. (3.5)Step 4. Let D be a Cartier divisor on B. We prove that f^{∗}(D + λH) ≡ 0 for a suitable number λ. From Theorem 1.2, q_{Z} is nondegenerate. Hence it is enough to prove that

q_{Z}(f^{∗}(D + λH)) = q_{Z}(f^{∗}(D + λH), A) = 0.

From Theorem 1.2, we obtain C_{Z}q_{Z}(f^{∗}(D+λH))^{n} = (f^{∗}(D+λH))^{2n} = 0 for every λ. Hence if we choose λ suitably, we obtain q_{Z}(f^{∗}(D + λH), A) = 0. □ (3.6) Step 5. Let D be an effective divisor on Z. We derive a contradiction assuming that dim f(D) < dim B − 1. Under this assumption, we obtain that D.f^{∗}H^{n−1} is numerically trivial. The following equation

C_{Z}q_{Z}(D + tf^{∗}H)^{n} = (D + tf^{∗}H)

tells us that q_{Z}(D, f^{∗}H) = 0 because D^{n}f^{∗}H^{n} = 0. We consider the following equation:

(sD + tf^{∗}H + A)^{2n} = C_{Z}q_{Z}(sD + tf^{∗}H + A)^{n}.

Comparing st^{n−1}-term of the both hand sides, we obtain that q_{Z}(D, A)q_{Z}(A, f^{∗}H) = 0. For an effective divisor E on Z, we consider the following equation:

C_{Z}q_{Z}(tE + A)^{n} = (tE + A)^{2n}.

Comparing t-term of the both hand sides, we obtain that q_{Z}(D, A) > 0 and q_{Z}(f^{∗}H, A) > 0. That derives a contradiction. □ (3.7) Under the conditions of Corollary 1.4, there exists an example such that −K_{B} is not ample. Example. Let E_{i} be an elliptic curve. We consider the abelian 6-fold Z̃ := E_{1} × ⋯ × E_{6}. Let G be the finite subgroup of GL(6, ℂ) generated by

( \begin{matrix}ζ_{6} & 0 & 0 & 0 & 0 & 0 \\ 0 & ζ_{6}^{5} & 0 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 & ζ_{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & ζ_{3}^{2} \end{matrix} ) and ( \begin{matrix}0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \end{matrix} ) ,

where ζ_{n} means a n-th root of unity. Then the quotient Z := Z̃/G satisfies the conditions Theorem 1.2 and it admits a fibration Z → (E_{1} × E_{3} × E_{5})/G. By direct calculation, K_{E^{3}/G} ≡ 0.

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Division of Mathematics, Graduate School of Science, Hokkaido University, Sapporo, 060-0810 Japan

 E-mail address: matusita@math.sci.hokudai.ac.jp