HIGHER DIRECT IMAGES OF DUALIZING SHEAVES OF LAGRANGIAN FIBRATIONS

DAISUKE MATSUSHITA

Abstract. For a Lagrangian fibration X → S between smooth projective manifolds, we prove that R^{i}f_{∗}O_{X} ≅ Ω_{S}^{i}. It follows that if X is an irreducible symplectic manifold, Hodge numbers h^{p,q}(S) = h^{p,q}(ℙ^{n}), where n = dim S.

1. Introduction

We begin with the definition of Lagrangian fibrations.

Definition 1.1. Let X be a Kähler manifold with a holomorphic symplectic form ω and S a normal variety. A proper surjective morphism f: X → S is said to be a Lagrangian fibration if a general fibre F of f is a Lagrangian submanifold with respect to ω, that is, the restriction of the symplectic 2-form ω|_{F} is identically zero and dim F = (1/2) dim X.

In title, we investigate higher direct images of the dualizing sheaf of a Lagrangian fibration. Our result is the following.

Theorem 1.2. Let f : X → S be a Lagrangian fibration between smooth projective manifolds. Then

R^{i}f_{∗}O_{X} ≅ Ω_{S}^{i}.

In particular, the sheaves R^{i}f_{∗}O_{X} are locally free.

Remark 1. Let f^{∘} : X^{∘} → S^{∘} be the smooth part of f and H := R^{i}f_{∗}^{∘}ℂ. It is known that the higher direct images of the dualizing sheaf is isomorphic to the natural extension of H if the discriminant locus is a normal crossing divisor [11, Theorem 2.6], [15, Theorem 2.6] and [16, Theorem 1]. However, there is a projective Lagrangian fibration whose discriminant locus is not a normal crossing divisor. In that case, there is no natural extension of H. The above theorem asserts that H can be extended as a locally free sheaf even if the discriminant locus is not necessarily normal crossing. Example. Let X := Hilb^{2}T, where T is a projective K3 surface which has an elliptic fibration T → ℙ^{1}. Then X admits a Lagrangian fibration over ℙ^{2} whose discriminant locus is not a normal crossing divisor.

1991 Mathematics Subject Classification. Primary 14E40, Secondary 14D05.

*Research Fellow of the Japan Society for the Promotion of Science.

2 DAISUKE MATSUSHITA

Remark 2. The assumption projectivity is crucial in our present proof. The author suspect, however, that the the most part of our results holds for Lagrangian fibrations. See Remark 4 in Section 3.

Combining Theorem 1.2 with [12, Theorem 2] and [13, Theorem 1], we obtain the following corollary.

Corollary 1.3. Let f : X → S be a fibre space of a projective irreducible symplectic manifold over a smooth projective base S. Then S is Fano and Hodge numbers of S are the following:

h^{q}(S, Ω_{S}^{p}) = \{ \begin{matrix}1 & if p = q, \\ 0 & if p ≠ q. \end{matrix}

Every Hodge number of S is the same as that of ℙ^{n}, where n = dim S.

Remark 3. Every Hodge number of a quadric hypersurface Q in ℙ^{2n} is equal to that of ℙ^{2n−1} and Q is not isomorphic to ℙ^{2n−1} (n ≥ 2). The author does not know whether there exists an example such that a fibre space of an irreducible symplectic manifold f : X → S whose base space S is not isomorphic to a projective space. Recently, Miyaoka [2] announced that S is isomorphic to a projective space if f admits a global section. Note that if f admits a global section, then S is smooth. The author would like to state one question.

Question 1.4. Under the assumptions of Theorem 1.2, we have

R^{n−1}f_{∗}ω_{X/S} ≅ T_{S},

which is the dual of R^{1}f_{∗}O ≅ Ω_{S}^{1}. Note that ω_{X/S} is the relative dualizing sheaf and n = dim S. Then is R^{n−1}f_{∗}ω_{X/S} nef? In particular, is R^{n−1}f_{∗}ω_{X/S} ample if X is an irreducible symplectic manifold?

Note that it is known that R^{i}f_{∗}ω_{X/S} is weakly positive.

Corollary 1.5. If dim X = 4, then T_{S} is nef. In particular, T_{S} is ample if X is an irreducible symplectic manifold.

This paper is organised as follows. In section 2, we prepare for the proof of Theorem 1.2. Theorem 1.2, Corollary 1.3 and Corollary 1.5 are proved in section 3. Section 4 is devoted to the classification of singular fibres over codimension one points which needs for the proof of Theorem 1.2.

Acknowledgement. The author would like to express his gratitude to Professors A. Fujiki, Y. Miyaoka, S. Mori and N. Nakayama for their encouragement and suggestions. The author also thanks to Dr. O. Fujino who gave him useful comments. The main idea of this note was obtained during author’s stay at École Normale Supérieure, Paris. The author wishes to thank to Professor A. Beauville who gave him an opportunity to stay at E. N. S.

2. Preliminary

2.1. We collect definitions and fundamental material which are needed in the proof of Theorem 1.2. First we consider a relative compactification of an Abelian fibration.

Proposition 2.1. Let f^{∘} : X^{∘} → Δ^{∗}(t_{1})×Δ^{n−1}(t_{2}, ⋯, t_{n}) be a smooth projective Abelian fibration. Assume that f^{∘} has the following properties:

(1) f^{∘} has a projective relative compactification f′ : Y → Δ^{n}.

(2) For a general point p of the discriminant locus of f′, f^{′−1}(p) has a reduced component.

(3) The monodromy matrix T of R^{1}f_{∗}^{∘}ℂ around Δ^{∗}(t_{1}) is unipotent.

(4) The rank of T − I is at most one, where I is the identity matrix. Then there exists a projective relative compactification f : X → Δ^{n} which has the following properties:

(1) Every singular fibre of f is an Abelian variety or a cycle of several copies of a ℙ^{1}-bundle over an Abelian variety. In the latter case, E_{i} ∩ E_{j} forms sections of the rulings of E_{i} and E_{j}, where E_{i} and E_{j} are irreducible components of a singular fibre.

(2) The morphism f^{−1}(D) → D is a topologically locally trivial deformation, where D is the discriminant locus of f.

(3) K_{X} is f-trivial.

(4) Every fibre of f is reduced.

Proof. Let τ(s) be the period matrix of the fibre of f at s and t a point of D of f. Since D is smooth, lim_{s→t} τ(s) exists in a rational component of degree g′. From the assumption (4), 0 ≤ g′ ≤ 1. From the assumption (2), f′ admits a meromorphic section which is a holomorphic section of f^{∘}. Hence assertions of Proposition 2.1 are deduced from [17, Theorem 5.3] except projectivity of the relative compactification f : X → Δ^{n}.

Lemma 2.2. The relative compactification f is projective.

Proof. By changing birational model, we may assume that there is a morphism Y → X. Let H be an f′-ample divisor on Y and H′ its proper transform on X. Then H′ is an f-big divisor on each component of a fibre of f. If g′ = 0, every fibre of f is an Abelian variety. Thus every big divisor on each component of a fibre of f is ample. Therefore H′ is f-ample. If g′ = 1, we need the following claim:

4 DAISUKE MATSUSHITA

Claim 2.3. Let s be a point of D and E_{i} an irreducible component of the fibre of f at s. Then there is a section e of the ruling of E_{i} such that −K_{E_{i}} ≡ 2e and e is nef.

Proof. Let C be a smooth curve on Δ^{n} such that C ∩ D = {s}. We consider the restriction f_{C} : X_{C} := X ×_{Δ^{n}} C → C. We denote by X_{s} the fibre of f at s and by E_{i} each irreducible component of X_{s}. Let E_{i+1} and E_{i−1} be next components of E_{i} and F_{ij} := E_{i} ∩ E_{j}. Since K_{X_{C}} is f_{C}-trivial and the dual graph of X_{s} is S^{1}, K_{E_{i}} ∼ −F_{ii+1} − F_{ii−1} by adjunction formula. Because each E_{i} is a ℙ^{1}-bundle over an Abelian variety, −K_{E_{i}} is nef. In order to prove Claim 2.3, it is enough to show that F_{ii+1} and F_{ii−1} are numerically equivalent on E_{i}. Let X_{t} be a general fibre of f_{C}. Since X_{s} is a deformation retract of X_{C}, there is a collapsing morphism c_{t} : X_{t} → X_{s}. Let E′_{i} := c_{t}^{−1}(E_{i}) and F′_{ij} := c_{t}^{−1}(F_{ij}). Note that E′_{i} → E_{i} is a real blow up along F_{ij} and F′_{ij} is a S^{1}-bundle over F_{ij}. We consider the following Mayer-Vietoris sequences:

0 → ℂ_{X_{t}} → ⊕_{E′_{i}}ℂ_{E′_{i}} → ⊕_{F′_{ij}}ℂ_{F′_{ij}} → 0
0 → ℂ_{X_{s}} → ⊕_{E_{i}}ℂ_{E_{i}} → ⊕_{F_{ij}}ℂ_{F_{ij}} → 0

From the above sequences, we obtain

0 → H^{0}(X_{t}) → ⊕_{E′_{i}}H^{0}(E′_{i}) → ⊕_{F′_{ij}}H^{0}(F′_{ij}) →^{α_{1}} H^{1}(X_{t}),

and

0  0
↓  ↓
→ H^{1}(X_{s}) → ⊕_{E_{i}}H^{1}(E_{i}) →^{α_{4}} ⊕_{F_{ij}}H^{1}(F_{ij}) →
↓  ↓
→^{α_{1}} H^{1}(X_{t}) →^{α_{2}} ⊕_{E′_{i}}H^{1}(E′_{i}) →^{α_{3}} ⊕_{F′_{ij}}H^{1}(F′_{ij}) → H^{2}(X_{t}) →
↓  ↓
0 → ⊕_{F_{ij}}H^{0}(F_{ij}) → ⊕_{E_{i}⊃F_{ij}}H^{0}(F_{ij}) → ⊕_{F_{ij}}H^{0}(F_{ij}) → 0
↓ β  ↓
⊕_{E_{i}}H^{2}(E_{i}) → ⊕_{F_{ij}}H^{2}(F_{ij})

Note that each column is Gysin sequence. Let m be the dimension of a general fibre of f. Then dim Im(α_{1}) = 1 and dim Ker(α_{4}) = 2m − 2, because X_{s} is a cycle of several copies of a ℙ^{1}-bundle over an Abelian variety. Since dim Im(α_{1}) + dim Im(α_{2}) = 2m, dim Ker(α_{3}) = 2m − 1. Hence β is not isomorphism. The morphism β is defined by

H^{0}(F_{ii−1}) ⊕ H^{0}(F_{ii+1}) ∋ (λ_{1}, λ_{2}) ↦ (λ_{1}[F_{ii−1}] + λ_{2}[F_{ii+1}]) ∈ H^{2}(E_{i}).

If F_{ii−1} and F_{ii+1} is not numerically equivalent, then β is injection. That is a contradiction. □

HIGHER DIRECT IMAGES 5

We go back to the proof of Lemma 2.2. From Claim 2.3, the nef cone of each irreducible component of a singular fibre is generated by a section and divisors on an Abelian variety. Therefore every big divisor on each irreducible component of a singular fibre is ample and projectivity of f is deduced similarly as in the case that g′ = 0. □ We complete the proof of Proposition 2.1. □

2.2. For the later use, we prove the following Lemma.

Lemma 2.4. Let f : X → Δ^{n}(t_{1}, ⋯, t_{n}) be a projective Abelian fibration such that (1) The discriminant locus D of f is defined by t_{1} = 0.

(2) The morphism f^{−1}(D) → D is a topologically locally trivial fibration.

(3) Every singular fibre of f is a normal crossing variety whose every irreducible component is a ℙ^{1}-bundle over an Abelian variety.

(4) K_{X} is f-nef. Then every bimeromorphic map Φ : X −−→ X which commutes with f is an isomorphism.

Proof. We derive a contradiction by assuming that Φ is not isomorphism. Since X is a relative minimal model and Φ commutes with f, Φ is an isomorphism over codimension one point of X. Let H be an f-ample divisor on X and H′ the proper transform by Φ. If Φ is not isomorphism, then H′ is not f-nef. Hence by [6, Theorem 3-1-1], there exists an extremal contraction ν : X → X̄. Let E be an irreducible component of f^{−1}(D). From assumption (2) and (3), each irreducible component of a fibre of E → D is a ℙ^{1}-bundle over an Abelian variety. Since D is simply connected, every fibre of E → D is irreducible. Thus E → D factors E → A → D, where E → A is a ℙ^{1}-bundle and A → D is a smooth Abelian fibration. Let ℓ be a rational curve which is contacted by ν. Then ℓ must be a fibre of the ruling of E. Hence ν contracts every fibre of the ruling of E and the exceptional locus of ν has codimension one. That derives a contradiction. □

Definition 2.5. Let f : X → Δ^{n} be a projective Abelian fibration which satisfies the assertions (1), (2) (3) and (4) of Proposition 2.1. We call X as a toroidal model of type I or II according to the singular fibre of f is an Abelian variety or a cycle of several copies of a ℙ^{1}-bundle over an Abelian variety.

2.3. We note variations of Hodge structures. Let f : X → S be a projective morphism between smooth manifold. Assume that there exists an open set U of S such that f is smooth over U and D := S \ U is a simple normal crossing divisor. Let f^{∘} : f^{−1}(U) → U be the smooth part of f. We denote the upper canonical extension of H := R^{i}f_{∗}^{∘}ℂ by ^{u}H and the lower canonical extension by ^{ℓ}H.

6 DAISUKE MATSUSHITA

Lemma 2.6. Let f : X → Δ^{n}(t_{1}, ⋯, t_{n}) be a projective morphism from a smooth manifold X to a polydisk Δ^{n}. Assume that f has the following properties:

(1) The discriminant locus D of f is defined by t_{1} = 0 and E := (f^{∗}D)_{red} is a simple normal crossing divisor.

(2) The morphism E → D is a locally trivial deformation. Let f^{∗}D = ∑ a_{i}E_{i}. Then

uF^{1}(H) ≅ f_{∗}(Ω_{X/Δ^{n}}^{1}(log E) ⊗ ∑(a_{i} − 1)E_{i}).

Proof. If dim S = 1, then this assertion is [5, VIII]. Thus 52^{u}F^{1}(H) and f_{∗}(Ω_{X/Δ^{n}}^{1}(log E) ⊗ ∑(a_{i} − 1)E_{i}) are isomorphic in codimension one points of S. By the assumptions of f, [18, Theorem 3.5] and the argument in the proof of [11, Lemma 2.11], f_{∗}(Ω_{X/Δ^{n}}^{1}(log E) ⊗ ∑(a_{i} − 1)E_{i}) is locally free. Therefore we obtain the assertion of Lemma 2.6. □

2.4. Finally, we note the mixed Hodge structure of a simple normal crossing variety.

Lemma 2.7. Let X = ∑ X_{i} be a simple normal crossing variety. We define

X^{[k]} := ∐_{i_{1}<···<i_{k}}(X_{i_{1}} ∩ ⋯ ∩ X_{i_{k}})^{∼},

where ∼ means the normalisation. For an index set I = {i_{1}, ⋯, i_{k}}, we define the inclusion δ_{j}^{I} as

δ_{j}^{I} : X_{i_{1}} ∩ ⋯ ∩ X_{i_{k}} → X_{i_{1}} ∩ ⋯ ∩ X_{i_{j−1}} ∩ X_{i_{j+1}} ∩ ⋯ ∩ X_{i_{k}}.

Then (1) There exists an exact sequence:

0 → Gr_{1}^{W} H^{1}(X, ℂ) →^{α} H^{1}(X^{[0]}, ℂ) →^{β} H^{1}(X^{[1]}, ℂ).

Note that α is induced from the normalization and

β := ⨁_{|I|=2} ∑_{j=1}^{2}(−1)^{j}(δ_{j}^{I})^{∗}.

(2) Assume that X is a cycle of several copies of a ℙ^{1}-bundle over an Abelian variety and every intersection of irreducible components forms a section of the ruling. Then the natural inclusion X_{i} → X induces an injection j^{∗} : Gr_{1}^{W} H^{1}(X, ℂ) → H^{1}(X_{i}, ℂ).

Proof. (1) By [4, Chapter 4], there is a spectral sequence

E_{1}^{p,q} = H^{q}(X^{[p]}, ℂ) ⟹ H^{p+q}(X, ℂ).

HIGHER DIRECT IMAGES 7

Since this spectral sequence degenerates at E_{2} level ([4, Chapter 4.8]), we obtain the desired exact sequence.

(2) Let p : H^{1}(X^{[0]}, ℂ) → H^{1}(X_{i}, ℂ) be the projection. Then the induced morphism is decomposed into j^{∗} = p ∘ α. From the definition of β, p^{−1}(0) ∩ Kerβ = 0 and we are done. □

3. Isomorphism between higher direct images and cotangent bundles

3.1. We prove Theorem 1.2, Corollary 1.3 and Corollary 1.5. From the following lemma, it is enough to consider R^{1}f_{∗}O_{X} for the proof of Theorem 1.2.

Lemma 3.1. Let f : X → S be a Lagrangian fibration between smooth projective manifolds. Assume that R^{1}f_{∗}O_{X} ≅ Ω_{S}^{1}. Then

R^{i}f_{∗}O_{X} ≅ Ω_{S}^{i}.

Proof. Since ω_{X} ≅ O_{X}, R^{i}f_{∗}O_{X} are torsion free by [10, Theorem 2.1]. Thus R^{i}f_{∗}O_{X} are reflexive sheaves by [11, Corollary 3.9]. Therefore it is enough to show that there exists an open set U of S such that R^{i}f_{∗}O_{X}|_{U} ≅ Ω_{U}^{i} and codim(S \ U) ≥ 2. Let D be the discriminant locus of f and f^{∘} the smooth part of f. We choose an open set as U := {x ∈ S; f is smooth at x or D is smooth at x}. Then, by [11, Theorem 2.6], R^{i}f_{∗}O_{X}|_{U} ≅ ^{ℓ}Gr^{0}(H_{i}), where H_{i} := R^{i}f_{∗}^{∘}ℂ. From Liouville’s Theorem, a general fibre of f is an Abelian variety. Hence ∧^{i}H_{1} ≅ H_{i}. There exists a morphism

∧^{i}R^{1}f_{∗}O_{X}|_{U} →^{α_{i}} R^{i}f_{∗}O_{X}|_{U}.

Then α_{i} is injective because R^{1}f_{∗}O_{X}|_{U} is torsion free. Considering exponents of the lower extension of H_{i}, α_{n} is not surjective if some α_{i} (0 < i < n) is not surjective. By [10, Corollary 7.6], R^{n}f_{∗}O_{X} ≅ ω_{S}. Combining R^{1}f_{∗}O_{X} ≅ Ω_{S}^{1}, α_{n} is isomorphism. Hence α_{i} (1 ≤ i ≤ n) are isomorphisms. □

Remark 4. The reflexiveness of R^{i}f_{∗}O_{X} is one of crucial point where we need projectivity. This comes from the following decomposition theorem.

Theorem 3.2 ([11, Theorem 3.1]). Let f : X → S be a surjective morphism between projective variety. If X is smooth, then

Rf_{∗}ω_{X} ∼_{q.i.s.} ∑ R^{i}f_{∗}ω_{X}[−i].

If the above theorem is valid for a projective morphism, we replace projectivity of X by projectivity of f in the assumptions of Theorem 1.2.

3.2. Theorem 1.2 is deduced from the following proposition.

Proposition 3.3. Let f : X → S be a projective Lagrangian fibration and ω a symplectic form such that a general fibre is a Lagrangian subvariety with respect to ω. Then there exists an open set U of S such that

R^{1}f_{∗}O_{X}|_{U} ≅ Ω_{U}^{1}

and codim(S \ U) ≥ 2.

Proof. We prove Proposition 3.3 in four steps. Step 1. First we consider the smooth part of f. Let U_{0} := S \ D, where D is the discriminant locus of f. We consider the following diagram:

T_{f^{−1}(U_{0})} → f^{∗}T_{U_{0}}  → 0
↓ ω
0 → f^{∗}Ω_{U_{0}}^{1} → Ω_{f^{−1}(U_{0})}^{1} → Ω_{f^{−1}(U_{0})/U_{0}}^{1} → 0.

From the above diagram, ω defines an isomorphism

f^{∗}T_{U_{0}} → Ω_{f^{−1}(U_{0})/U_{0}}^{1}

because every fibre of f is a Lagrangian subvariety by [14, Theorem 1]. Therefore we obtain isomorphisms

T_{U_{0}} ≅ f_{∗}Ω_{f^{−1}(U_{0})/U_{0}}^{1} ≅ F^{1}(H).

The dual of above isomorphism gives

(1)  R^{1}f_{∗}O_{X}|_{U_{0}} ≅ Ω_{U_{0}}^{1}.

Step 2. In order to prove Proposition 3.3, we extend the isomorphism (1) over codimension one point of S. Let s be a point of D such that S and D are smooth around s. In Step 2, 3 and 4, we will show that the isomorphism (1) is extended on a neighbourhood of s. First we treat the case that there exists a polydisk s ∈ Δ^{n} of S and the restriction morphism f : X_{Δ^{n}} := X ×_{S} Δ^{n} → Δ^{n} is a toroidal model of type I or II (See Definition 2.5). If f is a toroidal model of type I, then f is smooth and we obtain R^{1}f_{∗}O_{X_{Δ^{n}}} ≅ Ω_{Δ^{n}}^{1} around s. Thus we may assume that f is a toroidal model of type II. Let

X^{∘} := {x ∈ X_{Δ^{n}}|f is smooth at x}.

We consider the following diagram:

T_{X^{∘}} → f^{∗}T_{Δ^{n}}|_{X^{∘}} → 0
↓ ω
0 → f^{∗}Ω_{Δ^{n}}^{1}|_{X^{∘}} → Ω_{X^{∘}}^{1} → Ω_{X^{∘}/Δ^{n}}^{1} → 0.

HIGHER DIRECT IMAGES 9

From the above diagram, ω defines an isomorphism f^{∗}T_{Δ^{n}}|_{X^{∘}} → Ω_{X^{∘}/Δ^{n}}^{1}, since every fibre of f is a Lagrangian subvariety by [14, Theorem 1]. Let E := f^{∗}D. We investigate the relation between Ω_{X^{∘}/Δ^{n}}^{1} and Ω_{X_{Δ^{n}}/Δ^{n}}^{1}(log E)|_{X^{∘}}. Let us consider the following diagram:

0 →  f^{∗}Ω_{Δ^{n}}^{1}|_{X^{∘}}  →  Ω_{X^{∘}}^{1}  →  Ω_{X^{∘}/Δ^{n}}^{1}  → 0
↓ α  ↓ β  ↓ γ
0 → f^{∗}Ω_{Δ^{n}}^{1}(log D)|_{X^{∘}} → Ω_{X_{Δ^{n}}}^{1}(log E)|_{X^{∘}} → Ω_{X_{Δ^{n}}/Δ^{n}}^{1}(log E)|_{X^{∘}} → 0.

By Snake Lemma, we obtain Ω_{X^{∘}/S}^{1} ≅ Ω_{X_{Δ^{n}}/Δ^{n}}^{1}(log E)|_{X^{∘}} if Cokerα ≅ Cokerβ. Let x be a point of f^{−1}(D). From Definition 2.5, if we choose a suitable local parameter z_{i} at x and t_{i} at f(x), f is written as

z_{0}z_{1} = t_{1}, z_{i} = t_{i} (2 ≤ i),

on a suitable neighbourhood of x. The generator of Cokerα is f^{∗}dt_{1}/t_{1} and the generators of Cokerβ are dz_{0}/z_{0} and dz_{1}/z_{1}. The morphsim Cokerα → Cokerβ sends

f^{∗}dt_{1}/t_{1} ↦ dz_{0}/z_{0} + dz_{1}/z_{1}.

If x ∈ X^{∘}, z_{0} ≠ 0 or z_{1} ≠ 0. Hence Cokerα → Cokerβ is an isomorphsim and we obtain Ω_{X^{∘}/Δ^{n}}^{1} ≅ Ω_{X_{Δ^{n}}/Δ^{n}}^{1}(log E)|_{X^{∘}}. We conclude that

f^{∗}T_{Δ^{n}} ≅ Ω_{X_{Δ^{n}}/Δ^{n}}^{1}(log E),

because codim(X \ X^{∘}) ≥ 2. Since f_{∗}Ω_{X_{Δ^{n}}/Δ^{n}}^{1}(log E) ≅ ^{u}F^{1}(H), we obtain T_{Δ^{n}} ≅ uF(H). The dual of this isomorphism gives

R^{1}f_{∗}O_{X_{Δ^{n}}} ≅ Ω_{Δ^{n}}^{1}.

Step 3. In this step, we treat that f^{−1}(s) satisfies Theorem 4.1 (1). By Theorem 4.1 (1), there exists a following diagram:

X_{Δ^{n}} →^{η} X̃/G ←^{ν} X̃
↓ ↓ ↓
Δ^{n} = Δ̃^{n}/G ← Δ̃^{n},

where f̃: X̃ → Δ̃^{n} is a toroidal model of type I or II. Let U_{1} be the smooth locus of X̃/G. Since η is isomorphism on U_{1} and codim((X̃/G) \ U_{1}) ≥ 2, ω|_{U_{1}} is extended on X̃/G as holomorphic form. We denote by η_{∗}ω by this form. Since ν is étale in codimension one, ν^{∗}(η_{∗}ω) is a symplectic form on X̃ and a general fibre of f̃ is a Lagrangian subvariety with respect to ν^{∗}(η_{∗}ω). From Step 2, ν^{∗}(η_{∗}ω) defines a G-equivariant isomorphsim

R^{1}f̃_{∗}OX̃ ≅ Ω1 ; Δ̃^{n}.

The G-invariant part of the above morphism gives

R^{1}f_{∗}O_{X_{Δ^{n}}} ≅ Ω_{Δ^{n}}^{1}.

10 DAISUKE MATSUSHITA

Step 4. In this step, we treat that f^{−1}(s) satisfies Theorem 4.1 (2). From Theorem 4.1 (2), there exists a following diagram:

X_{Δ^{n}} = X̃/G ←^{τ} X̃
↓ ↓ ↓
Δ^{n} = Δ̃^{n}/G ← Δ̃^{n},

where τ is an étale morphism. Since τ is étale, ω̃ := τ^{∗}ω is a symplectic form and a general fibre of f̃ is a Lagrangian subvariety with respect to ω̃. If ω̃ defines an isomorphism R^{1}f̃_{∗}OX̃ → Ω1 ; Δ̃^{n}, we obtain a R^{1}f_{∗}O_{X_{Δ^{n}}} ≅ Ω_{Δ^{n}}^{1} by taking a Ginvariant part. Thus we consider the Abelian fibration f̃: X̃ → Δ̃^{n}. By Theorem 4.1 (2), there is a birational morphism ν : W → X̃ such that f̃ ∘ ν : W → Δ̃^{n} satisfies the assumptions (1) and (2) of Lemma 2.6. Let E := ((f̃ ∘ ν)^{∗}D̃)_{red} and (f̃∘ ν)^{∗}D̃ = ∑ a_{i}E_{i}. First we show that ν^{∗}ω̃ defines the morphism

(2)  (f̃∘ ν)^{∗}TΔ̃^{n} → Ω_{W/Δ̃^{n}}^{1}(log E) ⊗ ∑(a_{i} − 1)E_{i}.

Let W^{∘} := W \ Sing(f̃∘ ν)^{−1}(D̃). We consider the following diagram:

0 → T_{W^{∘}/Δ̃^{n}} → T_{W^{∘}} → (f̃∘ ν)^{∗}TΔ̃^{n}|_{W^{∘}}
↓
(f̃∘ ν)^{∗}Ω1 ; Δ̃^{n}|_{W^{∘}}  →  Ω_{W^{∘}}^{1}  →  Ω_{W^{∘}/Δ̃^{n}}^{1}  → 0
↓  ↓  ↓ j
0 → (f̃∘ ν)^{∗}Ω1 ; Δ̃^{n}(log D̃)|_{W^{∘}} → Ω_{W}^{1} (log E)|_{W^{∘}} → Ω_{W/Δ̃^{n}}^{1}(log E)|_{W^{∘}} → 0.

Since codim(W \ W^{∘}) = 2 and both sheaves are locally free, existence of the morphism (2) is followed by the following Claim.

Claim 3.4. (1) The image of T_{W^{∘}} → (f̃∘ ν)^{∗}TΔ̃^{n}|_{W^{∘}} contains ((f̃∘ ν)^{∗}TΔ̃^{n} ⊗

∑(1 − a_{i})E_{i})|_{W^{∘}}.

(2) For every element of α ∈ T_{W^{∘}/Δ̃^{n}}, ν^{∗}ω̃(α, ∗) = 0 in Ω_{W^{∘}/Δ̃^{n}}^{1}/Kerj.

Proof. (1) Let w be a point of (f̃∘ ν)^{−1}(D̃). On a suitable neighbourhood of w, f̃∘ ν is written as

z_{0}^{a_{0}} ⋯ z_{k}^{a_{k}} = t_{1}, z_{k+i} = t_{i} (i ≥ 2),

where z_{i} are local parameter at w and t_{i} are local parameter at f̃ ∘ ν(w). The morphism T_{W} → (f̃∘ ν)^{∗}TΔ̃^{n} is written as

\frac{∂}{∂z_{j}} ↦ a_{j}z_{0}^{a_{0}} ⋯ z_{j}^{a_{j}−1} ⋯ z_{k}^{a_{k}} \frac{∂}{∂t_{1}} .

If w ∈ W^{∘}, at most only one of z_{i} (0 ≤ i ≤ k) is zero and we are done.

(2) Let w a point of W^{∘} and F_{w} the irreducible component of the fibre of (f̃∘ν)(w) which contains w. By direct calculation,

(Ω_{W^{∘}/Δ̃^{n}}^{1}/Kerj) ⊗ k(w) ≅ Ω_{F_{w},w}^{1}.

T_{W^{∘}/Δ̃^{n}} is isomorphic to the dual of Ω_{W^{∘}/Δ̃^{n}}^{1}/Kerj. Thus

T_{W^{∘}/Δ̃^{n}} ⊗ k(w) ≅ T_{F_{w},w}.

By [14, Theorem 1], the restriction of ν^{∗}ω̃ to T_{F_{w},w} is zero. Hence ν^{∗}ω̃(α, ∗) = 0 in the double dual of Ω_{W/Δ̃^{n}}^{1}/Kerj. Since Ω_{W/Δ̃^{n}}^{1}/Kerj is torsion free, we have ν^{∗}ω̃(α, ∗) = 0 in Ω_{W/Δ̃^{n}}^{1}/Kerj. □ Next we consider the push forward of the morphism (2) by ν. Let X̃^{∘} := {x̃ ∈ X̃; f̃ is smooth at x̃} and V the open set of X̃ such that ν^{−1}(V) → V is an isomorphism. Note that a_{i} = 1 if E_{i} ∩ ν^{−1}(X^{∘}) ≠ ∅. By similar argument in Step 2, we obtain

Ω_{W/Δ̃^{n}}^{1}(log E) ⊗ (∑(a_{i} − 1)E_{i})|_{ν^{−1}(X̃^{∘})} ≅ Ω_{ν^{−1}(X̃^{∘})/Δ̃^{n}}^{1}.

Since ν^{∗}ω̃ is nondegenerate on ν^{−1}(V), the morphism (2) is an isomorphism on ν^{−1}(X̃^{∘} ∩ V). Thus we obtain an isomorphism

(3)  f̃^{∗}TΔ̃^{n}|_{X̃^{∘}∩V} → ν_{∗}(Ω_{W/Δ̃^{n}}^{1}(log E) ⊗ ∑(a_{i} − 1)E_{i})|_{X̃^{∘}∩V} .

By Theorem 4.1 (2), every fibre of f̃ is reduced. Hence X^{∘} = X \Sing(f̃^{−1}(D̃)) and codim(X̃ \X̃^{∘}) ≥ 2. Combining with codim(X̃ \V) ≥ 2, codim(X̃ \(X̃^{∘} ∩V)) ≥ 2. Since f̃^{∗}TΔ̃^{n} is locally free, we obtain

f̃^{∗}TΔ̃^{n} ⊕ F ≅ ν_{∗}(Ω_{W/Δ̃^{n}}^{1}(log E) ⊗ ∑(a_{i} − 1)E_{i}),

where F is a torsion sheaf. Hence

TΔ̃^{n} ⊕ f̃_{∗}F ≅ (f̃∘ ν)_{∗}(Ω_{W/Δ̃^{n}}^{1}(log E) ⊗ ∑(a_{i} − 1)E_{i}).

By Lemma 2.6, (f̃ ∘ ν)_{∗}(Ω_{W/Δ̃^{n}}^{1}(log E) ⊗ ∑(a_{i} − 1)E_{i}) ≅ ^{u}F^{1}(H). Since ^{u}F(H) is locally free and rank(^{u}F(H)) = rank(TΔ̃^{n}), we conclude that TΔ̃^{n} ≅ ^{u}F^{1}(H). The dual of this isomorphism gives R^{1}f̃_{∗}OX̃ ≅ Ω1 ; Δ̃^{n}. We complete the proof of Proposition 3.3. □

3.3. Proof of Theorem 1.2. Under the assumption of Theorem 1.2, R^{1}f_{∗}O_{X} is a reflexive sheaf by [11, Corollary 3.9]. Hence we obtain R^{1}f_{∗}O_{X} ≅ Ω_{S}^{1} by Proposition 3.3. Then by Lemma 3.1, we obtain the assertion of Theorem 1.2. □

Proof of Corollary 1.3. By [12, Theorem 2] and [13, Theorem 1], f : X → S is a Lagrangian fibration and S is a Fano manifold. From [11, Corollary 3.2],

h^{k}(X, ω_{X}) = ∑_{p+q=k} h^{q}(S, R^{p}f_{∗}ω_{X}).

Since ω_{X} ≅ O_{X}, the right hand side of the above equation is the sum of Hodge numbers of S by Theorem 1.2. By Hodge conjugate h^{k}(X, O_{X}) = h^{0}(X, Ω_{X}^{k}).

Moreover

h^{0}(X, Ω_{X}^{k}) = \{ \begin{matrix}0 & k ≡ 1 (mod 2) \\ 1 & k ≡ 0 (mod 2) \end{matrix}

since X is an irreducible symplectic manifold. From the assumption that S is projective, h^{p,p}(S) ≥ 1. Combining the above results, we obtain the assertion of Corollary 1.3. □

Proof of Corollary 1.5. We classify S by using the classification of algebraic surfaces. First we prove that S does not contain curves whose selfintersection number are negative. Assume the contrary. Then there is a birational morphism π : S → S̄ which contracts curves to points. The composition morphism π ∘ f is a Lagrangian fibration and not equidimensional. This contradicts to [14, Theorem 1]. Next we investigate the irregularity and Kodaira dimension of S. From [8, Theorem 1.1], κ(S) ≤ 0. By similar argument in the proof of Corollary 1.3, 2q(S) = q(X). Since κ(X) = 0, q(X) ≤ 4 and q(S) ≤ 2. If q(S) = 2, X is an Abelian variety. From the following Claim, S is an Abelian surface.

Claim 3.5. Let X be an Abelian variety and f : X → S be a fibre space. Then S is an Abelian variety.

Proof. Let F be a general fibre of f. Since K_{F} is trivial, F is an Abelian variety by [7, Theorem 12]. We consider the following exact sequence:

0 → F → X → X/F → 0

Then the quotient morphism is nothing but X → S. Thus X/F ≅ S and S is an Abelian variety. □ We go back to the proof of Corollary 1.5. Since S does not contain curves whose self-intersection number are negative and κ(S) ≤ 0, S is a ℙ^{1}-bundle over an elliptic curve or a hyperelliptic surface if q(S) = 1. If q(S) = 0,, S is ℙ^{2} or ℙ^{1} × ℙ^{1}. From these classification, we obtain assertions of Corollary 1.5. □

4. Classification of singular fibres over codimension one points

4.1. Statements and Set up. We classify singular fibres over codimension one points of a projective Lagrangian fibration.

Theorem 4.1. Let f : X → S be a projective Lagrangian fibration and D the discriminant locus of f. Assume that dim X = 2n, S and D is smooth at s. Then there exists a polydisk s ∈ Δ^{n} of S and the restriction morphism f_{Δ^{n}} : X_{Δ^{n}} := X ×_{S} Δ^{n} → Δ^{n} has one of the following two properties:

(1) There is a toroidal model f̃: X̃ → Δ̃^{n} of type I or II (See Definition 2.5) and an action of a cyclic group G on X̃ which satisfy the following diagram.

X_{Δ^{n}} →^{η} X̃/G ←^{ν_{1}}  X̃
↓ ↓ ↓
Δ^{n} = Δ̃^{n}/G ←^{ν_{2}} Δ̃^{n}

where η is a resolution of of X̄/G, ν_{1} is étale in codimension one and the branch locus of ν_{2} coincides D.

(2) There is an Abelian fibration f̃: X̃ → Δ̃^{n} and an action of a cyclic group G on X̃ which satisfy the following properties:

(a) X and X̃ satisfy the following diagram:

X_{Δ^{n}} = X̃/G ←^{τ} X̃
↓ ↓ ↓
Δ^{n} = Δ̃^{n}/G ←^{ν} Δ̃^{n}

where τ is an étale morphism and the branch locus of ν coincides D.

(b) For the discriminant locus D̃ of f̃, f̃^{∗}D̃ is a reduced divisor.

(c) There is a birational morphism ν : W → X̃ such that f̃∘ ν : W → Δ̃^{n} satisfies the assumptions (1) and (2) of Lemma 2.6.

4.2. Theorem 4.1 follows from the following two Propositions.

Proposition 4.2. Let f : X → S be a projective Lagrangian fibration. Assume that the discriminant locus D of f and S are smooth at s. Then there is a polydisk s ∈ Δ^{n} of S and the restriction f_{Δ_{n}} : X_{Δ^{n}} := X ×_{S} Δ^{n} → Δ^{n} has the following properties:

(1) There is a toroidal model f^{+} : X^{+} → Δ̃^{n} of type I or II and the action of a cyclic group G on X^{+} and Δ̃^{n} which commutes with f^{+}. They satisfy the following diagram:

X_{Δ^{n}} −−→^{α} X^{+}/G
↓  ↓
Δ^{n} = Δ̃^{n}/G,

where α is a bimeromorphic map and the branch locus of Δ̃^{n} → Δ^{n} coincides D.

(2) Let D̃ be the discriminant locus of f^{+}. For every point s̃ of D̃, the fibre X+ ; s̃ of f^{+} over a point s̃ is G-stable and

dim F^{1}Gr_{1}^{W} H^{1}(X+ ; s̃, ℂ)^{G} ≥ n − 1.

Proposition 4.3. Under the situation of Proposition 4.2, the restriction f_{Δ^{n}} : X_{Δ^{n}} → Δ^{n} satisfies one of the following:

14 DAISUKE MATSUSHITA

(1) If f^{+} is a toroidal model of type I, then f_{Δ^{n}} satisfies Theorem 4.1 (1) or (2).

(2) If f^{+} is a toroidal model of type II, then f_{Δ^{n}} is satisfies Theorem 4.1 (1).

4.3. Proof of Proposition 4.2 (1). Let ν : Y → X be a birational morphism such that E = (f ∘ ν)^{∗}D is a simple normal crossing divisor. Let U be an open set of D such that f ∘ ν|_{(f∘ν)^{−1}(U)} is a topologically locally trivial deformation and every fibre of f ∘ ν|_{(f∘ν)^{−1}(U)} is a simple normal crossing variety. We choose a neighborhood Δ^{n}(t_{1}, ⋯, t_{n}) of s such that D ∩ Δ^{n} are smooth and Δ^{n} ∩ U ≠ ∅. We may assume that D is defined by t_{1} = 0 on Δ^{n}. Let (f ∘ ν)^{∗}D = ∑ e_{i}E_{i} and e := L.C.M.{e_{i}; E_{i} ∩ (f ∘ ν)^{−1}(U) ≠ ∅}. We define the e-fold cyclic cover

d : Δ̃^{n} ∋ (u, t_{2}, ⋯, t_{n}) ↦ (t_{1} = u^{e}, t_{2}, ⋯, t_{n}) ∈ Δ^{n}.

We denote by Ŷ the normalization Y ×_{Δ^{n}} Δ̃^{n}. Then Ŷ is projective and the fibre g^{−1}(u) of the Abelian fibration g : Ŷ → Δ̃^{n} at a general point u of d^{−1}(D) has a reduced component. Hence g satisfies assumptions of Proposition 2.1 except the monodromy condition.

4.4. Let T̃ be the monodromy matrix of R^{1}g_{∗}^{∘}ℂ, where g^{∘} is smooth part of g. We prove that T̃ is unipotent and the rank of T̃ − I is at most one. Let s′ be a point of U. We choose an unit disk Δ^{1} in Δ^{n} such that Δ^{1} ∩ D = {s′} and put Δ̃^{1} = d^{−1}(Δ^{1}). We concentrate the restriction morphism

gΔ̃^{1} : ŶΔ̃^{1} := Ŷ ×Δ̃^{n} Δ̃^{1} → Δ̃^{1},

because the monodromy matrix of R^{1}(gΔ̃^{1})_{∗}^{∘}ℂ is T̃, where g∘ ; Δ̃^{1} is smooth part of gΔ̃^{1}. By [9, Theorem 11^{∗}], if we take a suitable resolution ỸΔ̃^{1} of ŶΔ̃^{1}, every fibre of g̃Δ̃^{1} : ỸΔ̃^{1} → Δ̃^{1} is reduced and simple normal crossing. We need the following Lemmas.

Lemma 4.4. Every point s′ of U,

dim F^{1}Gr_{1}^{W} H^{1}(Y_{s′}, ℂ) ≥ n − 1,

where Y_{s′} is the fibre of f ∘ ν at s′.

Lemma 4.5. Let ZΔ̃^{1} be a resolution of ŶΔ̃^{1}. We denote by Y_{s′} the fibre of f ∘ ν at s′ and by Z_{Δ̃^{1},s̃′} the fibre of ZΔ̃^{1} → Δ̃^{1} at s̃′ := d^{−1}(s′). Then the induced morphism H^{1}(Y_{s′}, ℂ) → H^{1}(Z_{Δ̃^{1},s̃′}, ℂ) is injection.

We prove Lemma 4.4 and Lemma 4.5 later. From the above Lemmas,

dim F^{1}Gr_{1}^{W} H^{1}(Ỹs̃′, ℂ) ≥ n − 1.

Since g̃ is a semistable degeneration, T̃ is unipotent. We consider the Clemens-Schmid exact sequence.

0 → Gr_{1}^{W} (H^{1}(Ỹ_{Δ̃^{1},s̃′}, ℂ)) → Gr_{1}^{W} (Ỹ_{Δ̃^{1},ũ}, ℂ)) →^{T̃−I} Gr_{1}^{W} (Ỹ_{Δ̃^{1},ũ}, ℂ)),

where Ỹ_{Δ̃^{1},ũ} is a general fibre of g̃Δ̃^{1}. From the above exact sequence,

dim Gr_{1}^{W} H^{1}(Ỹ_{Δ̃^{1},ũ}, ℂ) ≥ 2n − 2,

Moreover, H^{1}(Ỹ_{Δ̃^{1},ũ}, ℂ) carries the limit Hodge structure:

0 ⊂ W_{0} = Im(T̃ − I) ⊂ W_{1} = Ker(T̃ − I) ⊂ W_{2} = H^{1}(Ỹ_{Δ_{1},ũ}, ℂ).

Since dim W_{0} = dim W_{2}/W_{1}, the rank of T̃ − I is at most one.

4.5. From the above argument, Ŷ satisfies the assumptions of Proposition 2.1 and there exists a toroidal model f^{+} : X^{+} → Δ̃^{n} of type I or II such that X^{+} is bimeromorphic to Ŷ. The Galois group G of the covering d : Δ̃^{n} → Δ^{n} acts on Ŷ and therefore act on X̃ bimeromorphically. This action commutes with f^{+}. By Lemma 2.4, the action of G is biholomorphic and X^{+}/G is bimeromorphic to X_{Δ^{n}} over Δ^{n}.

4.6. Proof of Lemma 4.4. We use same notations in Subsection 4.3. First we prove that there is a morphism

(4)  (f ∘ ν)^{∗}T_{U} → Ω_{E_{i}/U}^{1}

for the component E_{i} ∩ (f ∘ ν)^{−1}(U) ≠ ∅. Let ω be a symplectic form on X such that a general fibre of f is a Lagrangian submanifold with respect to ω. There is a following diagram

0 → T_{E_{i}} → T_{Y} → N_{E_{i}/Y} → 0
↓ ν^{∗}ω
0 → I_{E_{i}}/I_{E_{i}}^{2} → Ω_{Y}^{1} → Ω_{E_{i}}^{1} → 0.

We define T_{E_{i}} → Ω_{E_{i}}^{1} by ν^{∗}ω. By the definition of U, E_{i} → U is a smooth morphism. There is an exact sequence:

0 → T_{E_{i}/U} → T_{E_{i}} → (f ∘ ν)^{∗}T_{U} → 0.

By [14, Theorem 1], the restriction ν^{∗}ω to each component of a fibre of f ∘ ν is identically zero. Hence ν^{∗}ω(α, ∗) = 0 in Ω_{E_{i}/U}^{1} for every element α ∈ T_{E_{i}/U} and there exists the morphism (4).

16 DAISUKE MATSUSHITA

4.7. We denote by t_{j}, (2 ≤ j ≤ n) local parameters of U at s′. Let E_{i,s′} be the fibre of E_{i} → U over s′. From the morphism (4), α_{ij} := ν^{∗}ω(∂/∂t_{j}, ∗)|_{E_{i,s′}} defines an element of H^{0}(E_{i,s′}, Ω_{E_{i,s′}}^{1}). We prove that α_{ij}, (2 ≤ j ≤ n) are linearly independent in H^{0}(E_{i,s′}, Ω_{E_{i,s′}}^{1}) if E_{i} is non ν-exceptional. Assume the contrary. There exists a linear combination γ := ∑ λ_{j}∂/∂t_{j} such that γ ≠ 0 and ν^{∗}ω(γ, ∗)|_{E_{i,s′}} = 0 in H^{0}(E_{i,s′}, Ω_{E_{i,s′}}^{1}). From the definition of U, E_{i,s′} is not contain ν-exceptional locus if E_{i} is not ν-exceptional. Thus there is an open set V of E_{i,s} such that ν^{∗}ω is nondegenerate over V. For every point y ∈ V,

T_{E_{i,s′},y} = {α ∈ T_{Y,y}; ν^{∗}ω(α, β) = 0, ∀β ∈ T_{E_{i,s′},y}},

because ν(E_{i,s′}) is a Lagrangian subvariety. However, γ ∉ T_{E_{i,s′},y} and ν^{∗}ω(γ, ∗)|_{E_{i,s′}} = 0 in H^{0}(E_{i,s′}, Ω_{E_{i,s′}}^{1}). That derives a contradiction.

4.8. Let Y_{s′} be the fibre of f ∘ν at s′. From Lemma 2.7, there is an exact sequence

0 → F^{1}Gr_{1}^{W} H^{1}(Y_{s′}, ℂ) → ⊕_{i}H^{0}(E_{i,s′}, Ω^{1}) →^{β} ⊕_{i<j}H^{0}((E_{i,s′} ∩ E_{j,s′})^{∼}, Ω^{1})

From the definition, α_{i,j} = α_{i′,j} on E_{i,s′} ∩ E_{i′,s′}. Hence each element ⊕_{i}{α_{i,j}}, (2 ≤ j ≤ n) defines an element of F^{1}Gr_{1}^{W} H^{1}(Y_{s′}, ℂ) by the definition of β. Since ⊕_{i}{α_{i,j}}, (2 ≤ j ≤ n) are linearly independent, we obtain the assertion of Lemma 4.4. □

4.9. Proof of Lemma 4.5. We use same notations in Subsection 4.3. Let Y_{Δ^{1}} := Y ×_{Δ^{n}} Δ^{1}. Since Y_{Δ^{1}} and ZΔ̃^{1} are deformation retract to Y_{s′} and Z_{Δ̃^{1},s̃′} respectively, it is enough to prove that H^{1}(Y_{Δ^{1}}, ℂ) → H^{1}(ZΔ̃^{1}, ℂ) is injective. From the isomorphism ŶΔ̃^{1}/G ≅ Y_{Δ^{1}}, H^{1}(Y_{Δ^{1}}, ℂ) ≅ H^{1}(ŶΔ̃^{1}, ℂ)^{G} and H^{1}(Y_{Δ^{1}}, ℂ) → H^{1}(ŶΔ̃^{1}, ℂ) is injection. Moreover ŶΔ̃^{1} is a homology manifold because ŶΔ̃^{1} has only quotient singularities. ([18, Proposition 1.4]) Hence H^{1}(ŶΔ̃^{1}, ℂ) → H^{1}(ZΔ̃^{1}, ℂ) is injection by [3, Thèoréme 8.2.4]. □ We finish the proof of Proposition 4.2 (1). □

4.10. Proof of Proposition 4.2 (2). Let T be the monodromy matrix of R^{1}(f ∘ ν)_{∗}^{∘}ℂ around D and D̃ := d^{−1}(D). For the fibre of X+ ; s̃ of f^{+} over s̃ ∈ D̃, the action G on H^{1}(X+ ; s̃, ℂ) is determined by the monodromy matrix T. Thus the function

μ(s̃) : s̃ ↦ dim F^{1}Gr_{1}^{W} H^{1}(X+ ; s̃, ℂ)^{G}

is constant on D̃. Hence it is enough to prove μ(s̃) ≥ n − 1 for some point s̃ of D̃. We use same notations in Subsection 4.4. The restriction morphism X+ Δ̃^{1} := X^{+} ×Δ̃^{n} Δ̃^{1} → Δ̃^{1} is G-equivariantly bimeromorphic to gΔ̃^{1} : ŶΔ̃^{1} → Δ̃^{1}. We take a G-equivariant resolution of indeterminacy ZΔ̃^{1} of ŶΔ̃^{1} −−→ X+ Δ̃^{1}. Let s̃′ := d^{−1}(s′) and X+ ; s̃′ the fibre of f^{+} at s̃′. By Lemma 4.5, H^{1}(Y_{Δ^{1},s′}, ℂ) → H^{1}(Z_{Δ̃^{1},s̃′}, ℂ) is injection. Thus each element of H^{1}(Y_{Δ^{1},s′}, ℂ) defines a G-invariant element of

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H^{1}(Z_{Δ̃^{1},s̃′}, ℂ). Hence, by Lemma 4.4, dim F^{1}Gr_{1}^{W} H^{1}(Z_{Δ̃^{1},s̃′}, ℂ)^{G} ≥ n − 1. Because ZΔ̃^{1} → X+ Δ̃^{1} is G-equivariant bimeromorphic morphism, there is a G-equivariant isomorphism H^{1}(ZΔ̃^{1}, ℂ) ≅ H^{1}(X+ Δ̃^{1}, ℂ). Since ZΔ̃^{1} and X+ Δ̃^{1} are deformation retract to Z_{Δ̃^{1},s̃′} and X+ ; s̃′ respectively,

F^{1}Gr_{1}^{W} H^{1}(Z_{Δ̃^{1},s̃′}, ℂ)^{G} ≅ F^{1}Gr_{1}^{W} H^{1}(X+ ; s̃′, ℂ)^{G}

and we obtain μ(s̃′) ≥ n − 1. □

4.11. Proof of Proposition 4.3 (1). After investigation of singularities of X^{+}/G, we will construct a suitable resolution of X^{+}/G and a relative minimal model over Δ^{n}. Let D̃ be the discriminant locus of f^{+}, s̃ a point of D̃ and X+ ; s̃ the fibre of f^{+} at s̃. First we treat the case that the representation ρ : G → H^{0}(X+ ; s̃, ℂ) is trivial. Since every fibre of f^{+} is an Abelian variety, the action of G is fixed point free. Hence X^{+}/G is smooth. Moreover X^{+}/G is the unique relative minimal model over Δ^{n} since it has no rational curves. On the contrary, X_{Δ^{n}} is a relative minimal model over Δ^{n}, Thus X_{Δ^{n}} ≅ X^{+}/G and X satisfies Theorem 4.1 (1) if we put X̃ = X^{+}.

4.12. In the following, we may assume that the representation ρ is not trivial. For investigation of singularities of X^{+}/G, we need the following Lemma.

Lemma 4.6. Let D̃ be the discriminant locus of f^{+}, D the discriminant locus of f and X+ ; D̃ := (f^{+})^{−1}(D̃). Then X+ ; D̃ → D̃ decompose a G-equivariant smooth elliptic fibration X+ ; D̃ → A and a G-equivariant smooth Abelian fibration π : A → D̃. They satisfy the following diagram:

X+ ; D̃ → A →^{π} D̃
↓ ↓ ↓
X+ ; D̃/G → A/G → D,

where A/G → D is a smooth n − 1 dimensional Abelian fibration.

4.13. We prove Lemma 4.6 later. Since A/G → D is a smooth Abelian fibration, the action of G on each fibre of π is a translation. Let g be a generator of G and m the smallest integer such that the action of g^{m} on each fibre of π is trivial. We denote by H the subgroup generated by g^{m}. If H = {1}, the action of G on X^{+} is fix point free. It is proved that X_{Δ^{n}} satisfies the properties of Theorem 4.1 (1) by similar argument in 4.11.

4.14. We consider the case that H ≠ {1}. Since the action of H on each fibre of π is trivial and X+ ; D̃ → A is a smooth elliptic fibration, X+ ; D̃/H → A is a ℙ^{1}-bundle and every singular locus of X^{+}/H forms a multisection of this bundle. For every singular point p of X^{+}/H, (p, X^{+}/H) ≅ (surface cyclic quotient singularity) ×

18 DAISUKE MATSUSHITA

(ℂ^{2n−2}, 0). The list of surface cyclic quotient singularities which occur above is found in [1, Table 5 (pp 157)]. By definition, the action of G/H on each fibre of π is non trivial translation. Thus the action of G/H on X^{+}/H is fix point free and the quotient morphism X^{+}/H → X^{+}/G is étale. Hence X+ ; D̃/G → A/G is a ℙ^{1}-bundle and every singular locus also forms a multisection of this bundle. For every singular point p of X^{+}/G, (p, X^{+}/G) ≅ (surface cyclic quotient singularity) × (ℂ^{2n−2}, 0). We will construct a suitable resolution X^{+}/G and a relative minimal model over Δ^{n} according to the following two cases:

(1) Every singular point p of X^{+}/G, (p, X^{+}/G) ≅ (Du Val singularity) × (ℂ^{2n−2}, 0).

(2) Some singular point p of X^{+}/G, (p, X^{+}/G) ≅ (Du Val singularity)×(ℂ^{2n−2}, 0).

4.15. Case (1). Let η : X′ → X^{+}/G be the minimal resolution which is obtained from the minimal resolution of Du Val singularities. If we prove that X′ ≅ X_{Δ^{n}}, then X satisfies Theorem 4.1 (1) by setting X̃ = X^{+}. Since η is a crepant birational morphism, X′ is a relative minimal model over Δ^{n}. By construction, the morphism η^{∗}(X+ ; D̃/G) → A/G is a locally trivial deformation whose fibre is a Kodaira singular fibre of type I_{0}^{∗}, II^{∗}, III^{∗} or IV^{∗}. Since A/G → D is a smooth Abelian fibration, every irreducible component of η^{∗}(X+ ; D̃/G) is a ℙ^{1}-bundle over a smooth Abelian fibration. By Lemma 2.4, X′ is the unique minimal model over Δ^{n}. Hence X_{Δ^{n}} ≅ X′.

4.16. Case (2). Let w : W → X^{+}/H be the unique minimal resolution which is obtained from the minimal resolution of surface quotient singularities. We prove that there exists the unique relative minimal model X̃ of W over Δ̃^{n}/H. We denote by D_{H} the discriminant locus of f_{H}^{+} : X^{+}/H → Δ̃^{n}/H. Since the action of H on A is trivial, the morphism (f_{H}^{+} ∘ w)^{∗}D_{H} → D_{H} factors (f_{H}^{+} ∘ w)^{∗}D_{H} → A and A → D_{H}. From the construction of W, (f_{H}^{+} ∘ w)^{∗}D_{H} → A is a topologically locally trivial deformation and whose fibre is a tree of ℙ^{1}. (cf. [1, pp158]) We obtain a minimal model f̃ : X̃ → Δ̃^{n}/H of W over Δ̃^{n}/H by contracting some irreducible components of (f_{H}^{+} ∘ w)^{∗}D_{H} along its ruling (cf. [1, pp 158]). The morphism f̃^{∗}D_{H} → D_{H} factors f̃^{∗}D_{H} → A and A → D_{H}. Moreover f̃^{∗}D_{H} → A is a topologically locally trivial deformation whose fibre is a Kodaira singular fibre of type II, III or IV. Let ℓ be a rational curve on X̃. Then ℓ must be an irreducible component of a fibre of f̃^{∗}D_{H} → A. Hence ℓ is a fibre of the ruling of an irreducible component of f̃^{∗}D_{H}. Thus X̃ has no small contraction and X̃ is the unique relative minimal model.

4.17. We prove that W and X̃ satisfy Theorem 4.1 (2). From the construction of W and X̃, it is easy to see that these satisfy Theorem 4.1 (2) (b) and (c). Let ν :

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W → X̃. From bimeromorphic morphisms w and ν, the action of G/H on X^{+}/H induces bimeromorphic actions on W and X̃. Since W is the unique minimal resolution of X^{+}/H and X̃ is the unique relative minimal model, all bimeromorphic maps W −−→ W which commutes with w and all bimeromorphic maps X̃ −−→ X̃ which commutes with f̃ are isomorphisms. Hence w, ν and (f_{H}^{+} ∘ w)^{∗}D_{H} → A are G/H-equivariant morphisms. We derive a contradiction assuming that the action of G/H on X̃ is not fixed point free. Let p be a fixed point and g an element of G/H. The fibre ν^{−1}(p) is contained in a fibre F of (f_{H}^{+} ∘ w)^{∗}D_{H} → A. Since the action of G/H on A is non trivial translation, F ∩ g(F) = ∅. That is a contradiction. Therefore X̃ → X̃/(G/H) is étale. Since X̃ is the unique minimal model, X̃/(G/H) is so and X̃/(G/H) ≅ X_{Δ^{n}}. Hence X̃ and X satisfy Theorem 4.1 (2) (a).

4.18. Proof of Lemma 4.6. Let F be a fibre of X+ ; D̃ → D̃. First we show that F/G is smooth and h^{0}(F/G, Ω_{F/G}^{1}) = n − 1. Let T be the representation matrix of ρ : G → AutH^{0}(F, Ω_{F}^{1} ). Since G is a finite cyclic group and Proposition 4.2 (2), T = diag(ζ, 1, ⋯, 1) under suitable coordinate of H^{0}(F, Ω_{F}^{1} ). Note that ζ ≠ 1 because ρ is not trivial representation. If we choose a suitable global coordinate of F, the action of G on F is written

(z_{1}, ⋯, z_{n}) ↦ T (z_{1}, ⋯, z_{n}) + (a_{1}, ⋯, a_{n}),

where (a_{1}, ⋯, a_{n}) is a point of F. The branch locus of the quotient map F → F/G forms smooth divisors of F and F/G is smooth. Since dim H^{0}(F, Ω_{F}^{1} )^{G} = n − 1, h^{0}(F/G, Ω_{F/G}^{1}) = n − 1.

4.19. Next we consider relative Jacobian of X+ ; D̃/G → D. Since h^{0}(F/G, Ω_{F/G}^{1}) = n − 1, there is a n − 1-dimensional smooth Abelian fibration A′ → D which factor X+ ; D̃/G → D. If we prove that α : X+ ; D̃/G → A′ is surjective and every fibre is connected, then the stein factorisation of X+ ; D̃ → A′ gives the desired morphism X+ ; D̃ → A. Let us consider the following diagram:

X+ ; D̃ → B_{1}
↓  ↓
X+ ; D̃/G → B_{2} → A′ → D,

where X+ ; D̃/G → B_{2} is the stein factorisation of α and X+ ; D̃ → B_{1} is the stein factorization of X+ ; D̃ → B_{2}. Let p be a point of D and Ap^{′} (resp. B_{1,p}, B_{2,p}, X_{p}^{+}) the fibre of A′ → D (resp. B_{1} → D, B_{2} → D, X+ ; D̃ → D) at p. Note that X_{p}^{+} is an Abelian variety and X_{p}^{+} → B_{1,p} is surjective with connected fibres. By Claim 3.5, B_{1,p} is an Abelian variety. Since B_{1,p} → B_{2,p} is a finite morphism, κ(B_{2,p}) ≤ κ(B_{1,p}) = 0 by [7, Corollary 9]. From [7, Theorem 13], there is an étale morphism B̃_{2,p} → B_{2,p} and B̃_{2,p} is isomorphic to the product of an Abelian variety and a

20 DAISUKE MATSUSHITA

variety of general type. Thus 0 ≤ κ(B̃_{2,p}) = κ(B_{2,p}). Hence κ(B̃_{2,p}) = κ(B_{2,p}) = 0 and B̃_{2,p} is an Abelian variety. Therefore B_{2,p} is an étale quotient of Abelian variety and there is a finite morphism B_{2,p} → Ap^{′} over an Abelian variety Ap^{′}. We conclude that B_{2,p} is an Abelian variety. By the universal properties of relative Jacobian, B_{2} ≅ A′. □ We finish the proof of Proposition 4.3 (1). □

4.20. Proof of Proposition 4.3 (2). We need the following Lemma.

Lemma 4.7. Let f : X → Δ^{n} be a toroidal model of type II. Assume that a finite cyclic group G acts on X and Δ^{n} and this action satisfies the following properties:

(1) f : X → Δ^{n} is G-equivariant.

(2) The morphism Δ^{n} → Δ^{n}/G ramifies along the discriminat locus D of f.

(3) For every point s of D,

dim F^{1}Gr_{1}^{W} H^{1}(X_{s}, ℂ)^{G} ≥ n − 1,

where X_{s} is the fibre at s. We denote by E_{i} each irreducible component of f^{∗}D. Let H be the subgroup of G defined by

H := {g ∈ G; g(E_{i}) = E_{i} for all i. }

Then the quotient X/H satisfies one of the following:

(1) The morphism X → X/H is étale.

(2) There exists a toroidal model f′ : X′ → Δ^{n}/H of type II which is bimeromorhpic to X/H → Δ^{n}/H. Moreover, there exists an action of G/H on X′ and Δ^{n}/H which satisfies the assumptions (1), (2) and (3) of Lemma 4.7.

4.21. We prove Lemma 4.7 later. By Proposition 4.2 (2), f^{+} : X^{+} → Δ̃^{n} satisfies assumptions of Lemma 4.7. If X^{+} → X^{+}/H is not étale, X′ satisfies assumptions of Lemma 4.7. Thus we may assume that X^{+} → X^{+}/H is étale by induction of the order of G. The fibre space f_{H}^{+} : X^{+}/H → Δ̃^{n}/H satisfies the properties of toroidal model of type II except the properties (4) of Definition 2.5. The dual graph Γ of every singular fibre of X^{+}/H is the Dynkin diagram of type Ã_{ℓ}. Hence the action of G/H on Γ is rotation or reflection.

4.22. First we assume that the action of G/H is rotation. From the definition of H, the action of G/H on Γ is faithful. Thus the action of G/H on X^{+}/H is fixed points free and X^{+} → X^{+}/G is étale. The quotient X^{+}/G satisfies the properties of toroidal model of type II except the properties (4) of Definition 2.5. By Lemma 2.4, the quotient X^{+}/G is the unique minimal model. We conclude that X^{+}/G ≅ X_{Δ^{n}} and X satisfies Theorem 4.1 (1) if we put X̃ := X^{+}.

HIGHER DIRECT IMAGES 21

4.23. Next we assume that the action of G/H on Γ is reflection. If the action is fixed point free, we prove that X^{+}/G ≅ X and X satisfies Theorem 4.1 (1) by similar argument in the case that the action of G/H on Γ is rotation. Thus we assume that G/H has fixed points. We fix some notations. Let D_{H} be the discriminat locus of f_{H}^{+} : X^{+}/H → Δ̃^{n}/H and (f_{H}^{+})^{∗}(D_{H}) = ∑ e_{i}E_{i}. The following three cases occur:

(1) Two of {E_{i}}_{i} are G/H-stable.

(2) One of {E_{i}}_{i} and one of {E_{i} ∩ E_{j}}_{i<j} are G/H-stable.

(3) Two of {E_{i} ∩ E_{j}}_{i<j} are G/H-stable

4.24. We investigate the fixed locus. We prove that each irreducible component of fixed locus of G/H on X^{+}/H forms a section of the ruling of E_{i}. First we treat the case (1). Let s be a point of D_{H}, (X^{+}/H)_{s} the fibre of f_{H}^{+} at s with reduced structure and E_{s} a stable component of (X^{+}/H)_{s}. It is enough to prove that each irreducibe component of fixed locus of G/H on E_{s} forms a sections of the ruling of E_{s}. Since f_{H} : X^{+}/H → Δ̃^{n}/H satisfies the properties of toroidal model of type II except (4), E_{s} is a ℙ^{1}-bundle over an n − 1-dimensional Abelian variety A. The action of G/H on E_{s} induces the action on A. By Lemma 2.7, the natural inclusion E_{s} → (X^{+}/H)_{s} induce an injection F^{1}Gr_{1}^{W} H^{1}((X^{+}/H)_{s}, ℂ) → H^{0}(E_{s}, Ω_{E_{s}}^{1}). Since dim F^{1}Gr_{1}^{W} H^{1}((X^{+}/H)_{s}, ℂ)^{G/H} ≥ n − 1, every element of H^{0}(E_{s}, Ω_{E_{s}}^{1}) is G/H-invariant. Hence the action of G/H on A is translation. Moreover the action on A is trivial, because we assume that the action G/H on X^{+}/H is not free. Hence the fixed locus forms a section of the ruling. The other cases are treated similarly.

4.25. By the definition of H, the action of G/H on Γ is faithful. Since the action G/H on Γ is reflection, the order of G/H is two. From Subsection 4.24, we see that every singular fibre of X^{+}/G → Δ^{n} is a chain of several copies of a ℙ^{1}-bundle over an Abelian variety and every singular point of X^{+}/G is locally isomorphic to the product of Du Val singularity of type A_{1} and ℂ^{2n−2}. The minimal resolution X′ of X^{+}/G is obtained by blowing up along singular locus. Every singular fibre of w : X′ → Δ^{n} is a tree of several copies of a ℙ^{1}-bundle over an Abelian variety and equisingular along the discriminat locus. By Lemma 2.4, X′ is the unique minimal model over Δ^{n}. Thus X_{Δ^{n}} ≅ X′ and X satisfies Theorem 4.1 (1).

4.26. Proof of Lemma 4.7. We begin with investigation of the action of H on each E_{i}. Let s be a point of D, E_{i,s} the fibre of E_{i} → D over s and X_{s} the fibre of f at s. From Definition 2.5, E_{i,s} is a ℙ^{1}-bundle over an Abelian variety A_{i}. The action of H on E_{i,s} induces the action of H on A_{i}. By Lemma 2.7, the natural inclusion E_{i,s} → X_{s} induces the injection F^{1}Gr_{1}^{W} H^{1}(X_{s}, ℂ) ≅ H^{0}(E_{i,s}, Ω_{E_{i,s}}^{1}).

22 DAISUKE MATSUSHITA

By assumption (3), dim F^{1}Gr_{1}^{W} H^{1}(X_{s}, ℂ)^{G} ≥ n − 1. Hence every element of H^{0}(E_{i,s}, Ω_{E_{i,s}}^{1}) is H-equivariant. Therefore the action of H on A_{i} is translation. If the action of H on A_{i} are not trivial for every i, the action of H on E_{i,s} is fixed point free and X → X/H is étale.

4.27. We consider the case that the action of H on A_{i} is trivial for some i. In this case, f has H-equivariant section and f_{H} : X/H → Δ^{n}/H has a section. If we prove that the monodromy matrix T of R^{1}(f_{H})_{∗}ℂ is unipotent and the rank of T − I is one, there exists a toroidal model X′ → Δ^{n}/H which is bimeromorphic to X/H by Proposition 2.1. We consider the following diagram:

0 → H^{1}(X_{s}, ℂ) → H^{1}(X_{t}, ℂ)
↓ h^{∗}  ↓ h^{∗}
0 → H^{1}(X_{s}, ℂ) → H^{1}(h(X_{t}), ℂ),

where X_{t} is a general fibre of f and h is a generator of H. Since the action of H on Gr_{0}^{W} H^{1}(X_{s}, ℂ) is trivial, the action of H on H^{1}(X_{s}, ℂ) is trivial. Moreover h^{∗} is represented by T on H^{1}(X_{t}, ℂ). Hence the rank of T − I is one. Since T is quasi-unipotent, T is unipotent

4.28. We prove that X′ satisfies Lemma 4.7 (2). The quotient group G/H acts on X′ bimeromorphically. By Lemma 2.4, this action is holomorphic. Let C be a smooth curve of Δ^{n}/H which is stable under the action of G/H and which intersects transversally with the discriminant locus of f′ at s. We consider the restrictions (X/H)_{C} := X/H ×_{Δ̃^{n}/H} C and XC^{′} := X′ ×_{Δ̃^{n}/H} C. Let Z be a G/H-equivariant indeterminancy of (X/H)_{C} −−→ XC^{′}. Because (X/H)_{C} has only quotient singularities, there is an injection H^{1}((X/H)_{C}, ℂ) → H^{1}(Z_{C}, ℂ). Since (X/H)_{C} and Z_{C} are deformation retract to (X^{+}/H)_{s} and Z_{s} respectively, H^{1}((X/H)_{s}, ℂ) → H^{1}(Z_{s}, ℂ) is an injection. Moreover

F^{1}Gr_{1}^{W} H^{1}((X/H)_{s}, ℂ)^{G/H} ≅ F^{1}Gr_{1}^{W} H^{1}(X_{s}, ℂ)^{G}.

Hence

dim F^{1}Gr_{1}^{W} H^{1}(Z_{s}, ℂ)^{G/H} ≥ n − 1,

from the assumption (3). Since Z_{C} → XC^{′} is a G/H-equivariant bimeromorphic morphism, there is a G/H-equivariant isomorphism H^{1}(XC^{′}, ℂ) → H^{1}(Z_{C}, ℂ). Hence F^{1}Gr_{1}^{W} H^{1}(Z_{s}, ℂ)^{G/H} ≅ F^{1}Gr_{1}^{W} H^{1}(Xs^{′}, ℂ)^{G/H} because XC^{′} is deformation retract to Xs^{′}. We obtain that X′ satisfies Lemma 4.7. □ We complete the proof of Theorem 4.1. □

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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