Enoki injectivity theorem

In this section, we discuss Enoki’s injectivity theorem (see [Eno, Theorem 0.2]), which contains Koll´ar’s original injectivity theorem: The- orem3.6.2. We recommend the reader to compare the proof of Theorem 3.7.1 with the arguments in [Ko2, Section 2] and [Ko6, Chapter 9].

Theorem 3.7.1 (Enoki’s injectivity theorem). Let X be a compact K¨ahler manifold and let L be a semi-positive line bundle on X. Then, for any non-zero holomorphic section s of L^⊗k with some positive in- teger k, the multiplication homomorphism

×s :H^q(X, ω_X ⊗L^⊗l)−→H^q(X, ω_X ⊗L^⊗(l+k)), which is induced by ⊗s, is injective for every q≥0 and l > 0.

Let us recall the basic notion of the complex geometry. For details, see, for example, [Dem].

Definition 3.7.2 (Chern connection and its curvature form). Let X be a complex manifold and let (E, h) be a holomorphic hermitian vector bundle on X. Then there exists the Chern connection D = D_(E,h), which can be split in a unique way as a sum of a (1,0) and of a

3.7. ENOKI INJECTIVITY THEOREM 65

(0,1)-connection, D=D⁰_(E,h)+D⁰⁰_(E,h). By the definition of the Chern connection, D⁰⁰=D⁰⁰_(E,h) = ¯∂. We obtain the curvature form

Θ_h(E) := D_(E,h)² .

The subscripts might be suppressed if there is no risk of confusion.

LetLbe a holomorphic line bundle onX. We say thatLispositive (reps.semi-positive) if there exists a smooth hermitian metrich_L onL such that√

−1Θ_hL(L) is a positive (resp. semi-positive) (1,1)-form on X.

Definition 3.7.3 (Inner product). Let X be an n-dimensional complex manifold with the hermitian metric g. We denote by ω the fundamental form of g. Let (E, h) be a holomorphic hermitian vec- tor bundle on X, and u, v are E-valued (p, q)-forms with measurable coefficients, we set

kuk² =

∫

X

|u|²dV_ω, hhu, vii=

∫

X

hu, vidV_ω,

where |u| (resp. hu, vi) is the pointwise norm (resp. inner product) induced by g and h on Λ^p,qT_X^∗ ⊗E, and dV_ω = _n!¹ωⁿ.

Let us prove Theorem3.7.1.

Proof of Theorem 3.7.1. Throughout this proof, we fix a K¨ahler metricg onX. Lethbe a smooth hermitian metric onLsuch that the curvature √

−1Θh(L) = √

−1 ¯∂∂logh is a smooth semi-positive (1,1)- form on X. We putn = dimX. We introduce the space ofL^⊗l-valued harmonic (n, q)-forms as follows,

H^n,q(X, L^⊗l) :={u∈C^n,q(X, L^⊗l)|∆⁰⁰u= 0} for every q ≥0, where

∆⁰⁰ := ∆⁰⁰_(L⊗l,h^l) :=D_(L^00∗_⊗l,h^l)∂¯+ ¯∂D_(L^00∗_⊗l,h^l)

and C^n,q(X, L^⊗l) is the space of L^⊗l-valued smooth (n, q)-forms on X.

We note that D_(L⁰⁰ _⊗l,h^l) = ¯∂ and that D^00∗_(L⊗l,h^l) is the formal adjoint of D⁰⁰_(L⊗l,h^l). It is easy to see that ∆⁰⁰u= 0 if and only if

D_(L^00∗_⊗l,h^l)u= ¯∂u = 0

for u∈C^n,q(X, L^⊗l) since X is compact. It is well known that C^n,q(X, L^⊗l) = Im ¯∂⊕ H^n,q(X, L^⊗l)⊕ImD_(L^00∗⊗l,h^l)

and

Ker ¯∂ = Im ¯∂⊕ H^n,q(X, L^⊗l).

66 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

Therefore, we have the following isomorphisms, H^q(X, ω_X ⊗L^⊗l)'H^n,q(X, L^⊗l) = Ker ¯∂

Im ¯∂ ' H^n,q(X, L^⊗l).

We obtain H^q(X, ωX ⊗L^⊗(l+k))' H^n,q(X, L^⊗(l+k)) similarly.

Claim. The multiplication map

×s:H^n,q(X, L^⊗l)−→ H^n,q(X, L^⊗(l+k)) is well-defined.

If the claim is true, then the theorem is obvious. This is because su = 0 in H^n,q(X, L^⊗(l+k)) implies u = 0 for u ∈ H^n,q(X, L^⊗l). This implies the desired injectivity. Thus, it is sufficient to prove the above claim.

Proof of Claim. By the Nakano identity (see, for example, [Dem, (4.6)]), we have

kD_(L^00∗_⊗l,h^l)uk² +kD⁰⁰uk² =kD^0∗uk²+hh√

−1Θ_h^l(L^⊗l)Λu, uii holds for L^⊗l-valued smooth (n, q)-form u, where Λ is the adjoint of ω∧ ·and ω is the fundamental form ofg. If u∈ H^n,q(X, L^⊗l), then the left hand side is zero by the definition ofH^n,q(X, L^⊗l). Thus we obtain kD^0∗uk² =hh√

−1Θ_hl(L^⊗l)Λu, uii= 0 since

√−1Θ_hl(L^⊗l) =√

−1lΘ_h(L)

is a smooth semi-positive (1,1)-form on X. Therefore, D^0∗u = 0 and h√

−1Θ_h^l(L^⊗l)Λu, uih^l = 0, whereh, ih^l is the pointwise inner product with respect to h^l and g. By Nakano’s identity again,

kD_(L^00∗_⊗(l+k),h^l+k)(su)k² +kD⁰⁰(su)k²

=kD^0∗(su)k²+hh√

−1Θ_h^l+k(L^⊗(l+k))Λsu, suii

Note that we assumed u ∈ H^n,q(X, L^⊗l). Since s is holomorphic, D⁰⁰(su) = ¯∂(su) = 0 by the Leibnitz rule. We know that

D^0∗(su) = − ∗∂¯∗(su) =sD^0∗u= 0

sincesis a holomorphicL^⊗k-valued (0,0)-form andD^0∗u= 0, where∗is the Hodge star operator with respect tog. Note thatD^0∗is independent of the fiber metrics. So, we have

kD_(L^00∗_⊗(l+k),h^l+k)(su)k² =hh√

−1Θ_h^l+k(L^⊗(l+k))Λsu, suii.

3.7. ENOKI INJECTIVITY THEOREM 67

We note that h√

−1Θ_h^l+k(L^⊗(l+k))Λsu, suih^l+k

= l+k

l |s|²_h^kh√

−1Θ_hl(L^⊗l)Λu, uih^l = 0

where h , ih^l+k (resp. |s|h^k) is the pointwise inner product (resp. the pointwise norm of s) with respect to h^l+k and g (resp. with respect to h^k). Thus, we obtain D_(L^00∗_⊗(l+k),h^l+k)(su) = 0. Therefore, we know that

∆⁰⁰_{(L⊗(l+k),hl+k)}(su) = 0, equivalently, su ∈ H^n,q(X, L^⊗(l+k)). We finish

the proof of the claim.

Thus we obtain the desired injectivity theorem.

The above proof of Theorem 3.7.1, which is due to Enoki, is ar- guably simpler than Koll´ar’s original proof of his injectivity theorem (see Theorem 3.6.2) in [Ko2].

We include Kodaira’s vanishing theorem for compact complex man- ifolds and its proof based on Bochner’s technique for the reader’s con- venience.

Theorem 3.7.4 (Kodaira vanishing theorem for complex mani- folds). Let X be a compact complex manifold and let L be a positive line bundle on X. Then H^q(X, ω_X ⊗L) = 0 for every q >0.

Proof. We take a smooth hermitian metric h on L such that

√−1Θ_h(L) = √

−1 ¯∂∂logh is a smooth positive (1,1)-form on X. We define a K¨ahler metric g on X associated to ω := √

−1Θ_h(L). As we saw in the proof of Theorem3.7.1, we have

H^q(X, ωX ⊗L)' H^n,q(X, L)

where n = dimX and H^n,q(X, L) is the space of L-valued harmonic (n, q)-forms on X. We take u∈ H^n,q(X, L). By Nakano’s identity, we have

0 = kD_(L,h)^00∗ uk²+kD⁰⁰uk²

=kD^0∗uk²+hh√

−1Θ_h(L)Λu, uii. On the other hand, we have

h√

−1Θ_h(L)Λu, uih =q|u|²h.

Therefore, we obtain 0 = kuk² when q ≥ 1. Thus, we have u = 0.

This means that H^n,q(X, L) = 0 for every q ≥ 1. Therefore, we have

H^q(X, ω_X ⊗L) = 0 for every q≥1.

68 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

It is a routine work to prove Theorem3.7.5by using Theorem3.7.1.

More precisely, Theorem 3.6.2 for compact K¨ahler manifolds, which is a special case of Theorem 3.7.1, induces Theorem 3.7.5 by the usual argument as in [EsVi3] and [Ko6].

Theorem 3.7.5 (Torsion-freeness and vanishing theorem). Let X be a compact K¨ahler manifold and let Y be a projective variety. Let π : X → Y be a surjective morphism. Then we obtain the following properties.

(i) Rⁱπ_∗ω_X is torsion-free for every i≥0.

(ii) If H is an ample line bundle on Y, then H^j(Y, H⊗Rⁱπ_∗ω_X) = 0 for every i≥0 and j >0.

For related topics, see [Take2], [Oh], [F30], and [F31]. See also [F37]. We close this section with a conjecture.

Conjecture 3.7.6. Let X be a compact K¨ahler manifold (or a smooth projective variety) and let D be a reduced simple normal cross- ing divisor on X. Let L be a semi-positive line bundle on X and let s be a non-zero holomorphic section of L^⊗k on X for some positive in- teger k. Assume that (s = 0) contains no strata of D, that is, (s= 0) contains no log canonical centers of (X, D). Then the multiplication homomorphism

×s:H^q(X, ω_X ⊗ OX(D)⊗L^⊗l)→H^q(X, ω_X ⊗ OX(D)⊗L^⊗(l+k)), which is induced by ⊗s, is injective for every q≥0 and l > 0.