Division Theorems for Exact Sequences

Division Theorems for Exact Sequences

Qingchun Ji Fudan University

The 10th Pacific Rim Geometry Conference December 4, 2011, Osaka

Skoda’s Division Theorem

Skoda’s division theorem is an analogue of Hilbert’s Nullstellensatz, but the remarkable feature of effectiveness makes it very powerful.

This theorem has many important applications in complex

differential geometry and algebraic geometry, including deformation invariance of plurigenera and effective versions of the

Nullstellensatz.

The statement of Skoda’s theorem is the following:

LetΩbe a pseudoconvex domain in Cⁿ, ψ∈PSH(Ω) , g₁,· · ·, g_r ∈ O(Ω), then for everyf ∈ O(Ω)with

Z

Ω

|f|²|g|^{−2(q+qε+1)}e^−ψdV <+∞,

there exist holomorphic functionsh1,· · ·, hr∈ O(Ω)such that f =X

gihi on Ω and

Z

Ω

|h|²|g|^−2q(1+ε)e^−ψdV ≤ 1 +ε ε

Z

Ω

|f|²|g|^{−2(q+qε+1)}e^−ψdV where|g|²=P

i|g_i|²,|h|² =P

i|h_i|², q= min{n, r−1} and ε >0 is a constant.

This theorem was generalized by Skoda and Demailly to (generic) surjective homomorphisms between holomorphic vector bundles by solving∂-equations.

We will talk about how to establish division theorem for general holomorphic homomorphisms.

We establish division theorems for the homomorphisms in an exact sequence of holomorphic vector bundles (among which the last one is surjective).

We consider a complex of holomorphic vector bundles overM, E→^Φ E⁰ →^Ψ E⁰⁰ (∗)

i.e. Φ∈Γ(M,Hom(E, E⁰)),Ψ∈Γ(M,Hom(E⁰, E⁰⁰))such that Ψ◦Φ = 0. E, E⁰, E⁰⁰ are assumed to be endowed with Hermitian structures.

We define for anyx∈M

E(x) = min{((Ψ^∗Ψ + ΦΦ^∗)ξ, ξ)|ξ∈E_x⁰,|ξ|= 1}

whereΦ^∗,Ψ^∗ are the adjoint ofΦ andΨrespectively w.r.t. the given Hermitian structures.

It is easy to see that the above complex is exact atx∈M if and only ifE(x)>0.

When the complex (*) is exact,Φ^∗(Ψ^∗Ψ + ΦΦ^∗)⁻¹|_KerΨ is a smooth lifting ofΦ,So it is possible to establish division theorems by solving a coupled system consisting of

∂g =∂[Φ^∗(Ψ^∗Ψ + ΦΦ^∗)⁻¹f]andΦg= 0 wheref ∈Γ(E⁰) satisfyingΨf = 0.

Ifgis a solution of this system, then

h^def= Φ^∗(Ψ^∗Ψ + ΦΦ^∗)⁻¹f−g∈Γ(E)and Φh=f.

In the special case whereΦ is surjective andE⁰ is equipped with the quotient Hermitian structure thenΨ = 0,ΦΦ^∗=Id_E⁰,and the above system reduces to

∂g=∂(Φ^∗f) on the subbundleKerΦ.

The difficulty of this method for our case is thatKerΦis no longer a subbundle ofE,so it amounts to solving∂-equations for

solutions valued in a subsheaf, it seems that it is not easy to give sufficient conditions for the solvability of this system.

Main Results

Theorem 1.Let (M, ω)be a K¨ahler manifold and letE, E⁰, E⁰⁰ be Hermitian holomorphic vector bundles overM,La Hermitian line bundle overM. All the Hermitian structures may have singularities in a subvarietyZ $M andΦ⁻¹(0)⊆Z. Suppose that (*) is generically exact overM, M\Z is weakly

pseudoconvex and that the following conditions hold onM\Z:

1. E ≥_m0, m≥min{n−k+ 1, r},1≤k≤n;

2. the curvature ofHom(E, E⁰) satisfies (F^Hom(E,E

0)

XX Φ,Φ)≤0 for everyX∈T^1,0M;

3. the curvature ofL satisfies

√−1(ςc(L)−∂∂ς−τ⁻¹∂ς∧∂ς)≥√

−1q(ς+δ)∂∂ϕ.

Then for every∂-closed (n, k−1)-formf which is valued in L⊗E⁰ with Ψf = 0 andkfk ς+δ

(ς+δ)ςE−|Φ|2ς2 <+∞, there exists a

∂-closed(n, k−1)-form h valued inL⊗E such thatΦh=f and khk 1

ς+τ

≤ kfk ς+δ (ς+δ)ςE−|Φ|2ς2 .

In the above statement, q= max

M\Z rankB_Φ, ϕ= logkΦk,0< ς, τ ∈C^∞(M) andδ is a measurable function onM satisfying E(ς+δ)≥ ||Φ||²ς.

BΦ is the second fundamental form of the holomorphic line bundle Span_C{Φ} in Hom(E, E⁰).

A Hermitian holomorphic vector bundle(E, h) is said to be

m-tensor semi-positive(semi-negative) if the curvatureF (of Chern connection ) satisfies√

−1F(η, η)≥0(≤0)for every η=η_αi∂z^∂

α ⊗e_i∈T^1,0M⊗E with rank(η_αi)≤m where z₁,· · ·, z_n are holomorphic coordinates of M,{e₁,· · · , e_r}is a holomorphic frame ofE and mis a positive integer. In this case, we writeE ≥_m0(E≤_m0).

LetE be a holomorphic vector bundle over M, Z $M be a subvariety, andh be a Hermitian structure onE|_M\Z.If for each z∈Z, there exist a neighborhood U of z, a smooth frame

{e₁,· · ·, er}overU and some constant κ >0such that the matrix h

h_ij(w)−κδ_iji

is semi-positive for every w∈U\Z where h_ij :=h(e_i, e_j) andδ_ij is the Kronecker delta, then we callh a singular Hermitian structure onE which has singularities inZ.

The curvature of the Chern connection of a Hermitian holomorphic vector bundle is said to be semi-negative in the sense of

Griffiths(Nakano) if and only if it is1-tensor(min{n, r}-tensor) semi-negative.

Hence a sufficient condition for(F^Hom(E,E

0)

XX Φ,Φ)≤0 is given by(since we always assumeE ≥_m0 for some positive integerm):

E⁰ is semi-negative in the sense of Griffiths.

Theorem 1 applied to

ς = 1, τ = constant>0, and δ=|Φ|²E⁻¹, we obtain the following corollary

Corollary1.If the condition 3 in theorem 2 is replaced by

√−1c(L)≥√

−1q(|Φ|²E⁻¹+ 1)∂∂ϕ,

then for every∂-closed(n, k−1)-formf which is valued in L⊗E⁰ with

Ψf = 0 and kfkE+|Φ|2 E2

<+∞

there is a∂-closed(n, k−1)-form hvalued in L⊗E such that Φh=f and the following estimate holds

khk ≤ kfk_E+|Φ|2 E2

.

LetM be a complex manifold and E be a holomorphic vector bundle of rankr overM. The Koszul complex associated to a sections∈Γ(E^∗)is defined as follows

0→detE→ ∧^{dr r−1}E ^d→ · · ·^r−1 → O^d1 _M →0

where the boundary operators are given by the interior product d_p=sy,1≤p≤r.

It gives a complex since we havedp−1◦dp = 0 for1≤p≤r.

We will apply theorem 1 to

Φ =sy∈Γ(M,Hom(∧^pE,∧^p−1E).

We can show by direct computation that (F^{Hom(∧pE,∧p−1E)}

XX Φ,Φ) =

r p−1

(F_XX^E∗ s, s)

whereX∈Tx^1,0M, x∈M, which implies that the condition 2 in theorem 1 holds as soon asE is assumed to be semi-positive in the sense of Griffiths.

In the case of Koszul complex, we have the following division theorem:

Theorem 2.Let (M, ω)be a K¨aler manifold and let E be a Hermitian holomorphic vector bundle overM,L a line bundle over M, s∈ Γ(E^∗). All the Hermitian structures may have singularities in a subvarietyZ $M .Assume thats⁻¹(0)⊆Z,and that M\Z is weakly pseudoconvex and that the following conditions hold on M\Z:

1. E ≥_m0, m≥min{n−k+ 1, r−p+ 1};

2. the curvature ofL satisfies

√−1(ςc(L)−∂∂ς−τ⁻¹∂ς∧∂ς)≥√

−1q(ς+δ)∂∂ϕ.

Then for any∂-closed(n, k−1)-form f which is valued in L⊗ ∧^p−1E,ifdp−1f = 0and kfk ς+δ

ςδ|s|2 <+∞ there is at least one

∂-closed(n, k−1)-form h valued inL⊗ ∧^pE such thatdph=f and the following estimate holds

khk 1 ≤ kfk ς+δ .

In the above statement,1≤p≤r, ϕ= log|s|,1≤k≤n,1≤p≤ n, q= min{n, r−1}, n= dim_CM, r = rank_CE,

0< ς, τ ∈C^∞(M) andδ ≥0 is a measurable function onM.

Similar to corollary 1, we have the following result

Corollary 2.Let (M, ω) be a K¨ahler manifold and letE be a Hermitian holomorphic vector bundle overM,L a line bundle over M, s∈ Γ(E^∗). All the Hermitian structures may have singularities in a subvarietyZ $M .Assume thats⁻¹(0)⊆Z,and that M\Z is weakly pseudoconvex and the following conditions hold on M\Z:

1. E ≥_m0, m≥min{n−k+ 1, r−p+ 1};

2. the curvature ofL satisfies√

−1c(L)≥√

−1q(1 +ε)∂∂ϕ.

Then for any∂-closed(n, k−1)-form f valued in L⊗ ∧^p−1E,if dp−1f = 0 andkfk_|s|−2 <+∞ there is at least one ∂-closed (n, k−1)-formh valued inL⊗ ∧^pE such that d_ph=f and the following estimate holds

khk²≤ 1 +ε

ε kfk²_|s|−2,

where1≤p≤r,1≤k≤n, ϕ= log|s|², q= min{n, r−1}, n=

Now we discuss the special case of Koszul complex over a domain Ω⊆Cⁿ.

Letg₁· · ·, g_r ∈ O(Ω), the Koszul complex associated to g= (g1· · ·, gr) is given by

0→ ∧^rO^{⊕r d}→ ∧^{r r−1}O^{⊕r d}→ · · ·^r−1 → ∧O^{d2 ⊕r d}→ O →¹ 0

where the boundary operators are defined byd_p =gy,1≤p≤r.

It is easy to see that for every

h= (hi1···ip)^r_i1···ip=1∈Γ(Ω,∧^pO^⊕r)(i.e. hi1···ip ∈ O(Ω)andhi1···ip

is skew symmetric ini₁,· · ·, i_p),we have d_ph= (f_i1···ip−1)^r_i

1···ip−1=1∈Γ(Ω,∧^p−1O^⊕r) with fi1···ip−1 = X

1≤ν≤r

gνhνi1···ip−1.

By introducing the singular Hermitian structure 1

(P

i|g_i|²)^q(1+ε)e^ψ

on the trivial line bundle, we get the following division theorem:

Corollary3.Let Ω⊆Cⁿ be a pseudoconvex

domain,g1· · ·, gr ∈ O(Ω), ψ∈PSH(Ω)andε >0a constant, then for every global section(f_i1···i`−1)^r_i

1···i<sub>`−1</sub>=1∈Γ(Ω,∧<sup>`−1</sup>O_Ω^⊕r) (1≤`≤r ) satisfying P

1≤ν≤r

gνfνi1···i_`−2 = 0and Z

Ω

|f|²|g|^{−2(q+qε+1)}e^−ψdV <+∞

there exists at least one(h_i1···i_`)^r_i

1···i<sub>`</sub>=1 ∈Γ(Ω,∧<sup>`</sup>O^⊕r_Ω ) such that f_i1···i`−1 = P

1≤ν≤r

g_νh_νi1···i`−1, and

R

Ω|h|²|g|^−2q(1+ε)e^−ψdV ≤ ^1+ε_ε R

Ω|f|²|g|^{−2(q+qε+1)}e^−ψdV, where|g|² =P

i|g_i|²,|h|²= P

i1<···<i_`

|h_i1···i_`|²,

|f|² = P

i1<···<i_`−1

f_i1···i_`−1

2, q = min{n, r−1}.

Particularly, if|g| 6= 0 holds on Ωthen the Koszul complex induces an exact sequence on global sections.

The special case ofp= 1 recovers Skoda’s division theorem.

LetΩbe a domain inCⁿ,andΦbe aq×p matrix of holomorphic functions onΩ, p≥q. We denote by δ_i1···iq theq×q minors ofΦ, i.e.

δi1···iq =det







Φ_1i1 · · · Φ_1iq ... . .. ... Φ_qi1 · · · Φ_qiq





,

where1≤i₁ < i₂ <· · ·< i_q ≤p.There are ^p_q

distinct minors of orderq.

In complex Euclidean spaces, we also have the following division theorem.

Corollary 4.Letψ∈PSH(Ω), f ∈ O^q(Ω),ifΩ⊆Cⁿ is pseudoconvex and there exists a constantα >1 such that

Z

Ω

|f|² ( P

i1<···<iq

|δ_i1···iq|²)^βe^−ψdV <+∞,

whereβ= min{n, ^p_q

−1} ·α+ 1.Then there is at least one h∈ O^p(Ω)which solves the equations Φh=f.

The Case ε = 0

The technique of Skoda triple which was introduced by Varolin.

Definition A Skoda triple (ϕ, F, q)consists of a positive integer q andC² functions ϕ: (1,∞)→R, F : (1,∞)→R such that

x+F(x)>0,[x+F(x)]ϕ⁰(x) +F⁰(x) + 1>0 and

[x+F(x)]ϕ⁰⁰(x) +F⁰⁰(x)<0 hold for everyx >1.

It is easy to see that(εlogx,0, q) is a Skoda triple where εis a positive constant andq is a positive integer.

The notion of Skoda triple is quite useful to produce examples of division theorems.

Theorem 3LetΩ⊆Cⁿ be a pseudoconvex domain, g_i ∈ O(Ω)(1≤i≤p),ψ∈PSH(Ω). We assume that

kgk<1 holds onΩ.

For everyf ∈V`−1O(Ω)^⊕p, if gyf = 0

and Z

Ω

kfk² b

a(b−1)kgk<sup>−2(q`+1)</sup>e<sup>ϕ◦ξ−ψ</sup>dV <∞, then there exists anu∈V`O(Ω)^⊕p such thatιgu=f and Z

Ω

kuk² 1

(a+λ)kgk^−2q`e^ϕ◦ξ−ψ ≤ Z

Ω

kfk² b

a(b−1)kgk^−2(q`+1)e^ϕ◦ξ−ψ.

In the above statement,

p∈N,1≤`≤p, ξ= 1−logkgk², a=ξ+F◦ξ,

b= aϕ⁰◦ξ+F⁰ ◦ξ+ 1

qa` + 1, λ= Λ◦ξ, Λ(x) = −(1 +F⁰(x))²

F⁰⁰(x) + (x+F(x))ϕ⁰⁰(x), (ϕ, F, q) is a Skoda triple and

q=

(min{p−1, n}, `= 1;

min{p−`+ 1, n}, `≥2.

For the Skoda triple(εlogx,0, q), we have

Corollary 5LetΩ⊆Cⁿ be a pseudoconvex domain, gi ∈ O(Ω)(1≤i≤p),ψ∈PSH(Ω). We assume that

kgk<1 holds onΩ.

For everyf ∈V`−1

O(Ω)^⊕p, if ιgf = 0 and Z

Ω

kfk²(1−logkgk²)^ε

kgk^2(q`+1) e^−ψdV <∞,

then there exists someu∈V`O(Ω)^⊕p such that ιgu=f and Z

Ω

kuk²(1−logkgk²)^ε−1

kgk<sup>2q`</sup> e<sup>−ψ</sup> ≤ q`+ε+ 1 ε

Z

Ω

kfk²(1−logkgk²)^ε kgk<sup>2(q`+1)</sup> e<sup>−ψ</sup> wherep∈N,1≤`≤p, ε > ois a constant andq is the constant

In the case`= 1,we see that under the assumption that kgk<1 onΩ, the integrability condition in corollary 5 is weaker than that in Skoda’s division theorem.

We know by definition that(0,−¹₂e^−ε(x−1), q) is another example of Skoda triples whereεis a positive constant and q is the constant as above. Our previous theorem applied to the Skoda triple(0,−¹₂e^ε(x−1), q) gives the following result.

Corollary 6LetΩ⊆Cⁿ be a pseudoconvex domain, g_i ∈ O(Ω)(1≤i≤p),ψ∈PSH(Ω). We assume that

kgk<1 holds onΩ.

For everyf ∈V`−1O(Ω)^⊕p, if gyf = 0 and Z

Ω

kfk²kgk^−2(q`+1)e^−ψdV <∞,

then there exists someu∈V`O(Ω)^⊕p such that ιgu=f and Z

Ω

kuk²kgk^2(−q`+ε)e^−ψ ≤Cε

Z

Ω

kfk²kgk<sup>−2(q`+1)</sup>e<sup>−ψ</sup> wherep∈N,1≤`≤p, εandCε are both positive constants(Cε is determined byε) andq is the constant as above.

Basic Estimates

The Basic Estimate 1Let(M, ω)be a K¨ahler manifold, and let E be a Hermitian holomorphic vector bundle over M,L a

Hermitian holomorphic line bundle overM. The Hermitian

structures of these bundles may have singularity alongΦ⁻¹(0) and ΩbM\Φ⁻¹(0)is a pseudoconvex domain with smooth boundary.

Assume that the following conditions hold onΩ : 1. E ≥_m0, m≥min{n−k+ 1, r},1≤k≤n;

2. the curvature ofHom(E, E⁰) satisfies (F^Hom(E,E

0)

XX Φ,Φ)≤0 for everyX∈T^1,0M;

3. the curvature ofL satisfies

√−1(ςc(L)−∂∂ς−τ⁻¹∂ς∧∂ς)≥√

−1q(ς+δ)∂∂ϕ.

Then the following estimate

|Φ|⁻²Φ^∗u+∂^∗v

2

Ω,ς+τ + ∂v

2

Ω,ς ≥ kuk²

Ω,^{ς(λδ+λς−ς)}

(ς+δ)|Φ|2

n,k−1

and everyv∈A^n,k(Ω, L⊗E)∩Dom(∂^∗),wherec(L) denotes the Chern form,q= max

Ω rankB_Φ, ϕ= log|Φ|²,0< ς ∈C^∞(Ω)and λ, δ, τ are measurable functions onΩsatisfying λ, τ >0, ς+δ ≥0.

The Basic Estimate 2LetΩbe a bounded pseudoconvex domain with smooth boundary andg_i ∈ O(Ω)∩C^∞( ¯Ω)(1≤i≤p) without common zeros onΩ.Let¯ ϕ1, ϕ2 ∈C²( ¯Ω),0< a∈C²( ¯Ω) and1< b,0< λbe measurable functions on Ω. Assume that

ϕ2 =ϕ1+logkgk²,

a∂α∂β¯ϕ1−∂α∂β¯a−λ⁻¹∂αa∂β¯a≥q`ab∂α∂β¯logkgk².

Then for anyh∈V`−1

O(Ω)^⊕p satisfying X

1≤ν≤r

gνhνi1···ip−1 = 0

and anyv∈Dom ¯∂^∗_ϕ1 ⊆V`

L²_0,1(Ω, ϕ1)^⊕p satisfying ∂v¯ = 0,we have

k√

a+λ ¯g

||g||² ∧h+√

a+λ∂¯_ϕ^∗₁vk²_ϕ1 ≥ Z

Ω

(b−1)a

b khk²e^−ϕ2dV.

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Thank You!