Effective version of Matsusaka’s big theorem

An important problem of algebraic geometry is to find effective bounds m0 such that multiplesmLof an ample line bundle become very ample form>m₀. From a theoretical point of view, this problem has been solved by Matsusaka [Mat72] and Koll´ar-Matsusaka [KoM83]. Their result is that there is a bound m₀ = m₀(n, Lⁿ, Lⁿ⁻¹·K_X) depending only on the dimension and on the first two coefficients Lⁿ and Lⁿ⁻¹·K_X in the Hilbert polynomial of L. Unfortunately, the original proof does not tell much on the actual dependence of m₀ in terms of these coefficients. The goal of this section is to find effective bounds for such an integer m₀, along the lines of [Siu93]. However, one of the technical lemmas used in [Siu93] to deal with dualizing sheaves can be sharpened. Using this sharpening of the lemma, Siu’s bound will be here substantially improved. We first start with the simpler problem of obtaining merely a nontrivial section inmL. The idea, more generally, is to obtain a criterion for the ampleness of mL−B when B is nef. In this way, one is able to subtract from mL any multiple of K_X which happens to get added by the application of Nadel’s vanishing theorem (for this, replace B by B plus a multiple of K_X+ (n+ 1)L).

(9.1) Proposition. Let L be an ample line bundle over a projective n-fold X and let B be a nef line bundle over X. Then K_X+mL−B has a nonzero section for some integer m such that

m6nLⁿ⁻¹·B

Lⁿ +n+ 1.

Proof. Letm₀ be the smallest integer > n^Ln−1_Ln^·B. Thenm₀L−B can be equipped with a singular hermitian metric of positive definite curvature. Let ϕ be the weight of this metric. By Nadel’s vanishing theorem, we have

H^q(X,^O(K_X +mL−B)⊗^I(ϕ)) = 0 for q>1,

thus P(m) = h⁰(X,^O(K_X +mL −B) ⊗^I(ϕ)) is a polynomial for m > m₀. Since P is a polynomial of degree n and is not identically zero, there must be an integer m∈[m₀, m₀+n] which is not a root. Hence there is a nontrivial section in

H⁰(X,^O(K_X+mL−B))⊃H⁰(X,^O(K_X+mL−B)⊗^I(ϕ))

for some m∈[m₀, m₀+n], as desired.

(9.2) Corollary. If L is ample and B is nef, then mL−B has a nonzero section for some integer

m6nLⁿ⁻¹·B+Lⁿ⁻¹·K_X

Lⁿ +n+ 1 .

Proof. By Fujita’s result 9.3 a), K_X + (n+ 1)L is nef. We can thus replace B by B+K_X+ (n+ 1)L in the result of Proposition 9.1. Corollary 9.2 follows.

(9.3) Remark. We do not know if the above Corollary is sharp, but it is certainly not far from being so. Indeed, for B = 0, the initial constant n cannot be replaced by anything smaller than n/2 : take X to be a product of curves C_j of large genus g_j and B = 0; our bound for L = ^O(a₁[p₁])⊗. . .⊗^O(a_n[p_n]) to have |mL| 6= ∅ becomes m 6 P

(2g_j −2)/a_j +n(n+ 1), which fails to be sharp only by a factor 2 when a₁ = . . .=a_n = 1 and g₁ ≫g₂ ≫. . .≫g_n →+∞. On the other hand, the additive constant n+ 1 is already best possible when B= 0 and X =Pⁿ. So far, the method is not really sensitive to singularities (the Morse inequalities are indeed still true in the singular case as is easily seen by using desingularizations of the ambient variety). The same is true with Nadel’s vanishing theorem, provided that KX is replaced by the L² dualizing sheaf ω_X (according to the notation introduced in Remark 6.22, ωX = KX(0) is the sheaf of holomorphic n-forms u on Xreg such that iⁿ²u∧u is integrable in a neighborhood of the singular set). Then Proposition 9.1 can be generalized as

(9.4) Proposition. Let L be an ample line bundle over a projective n-fold X and let B be a nef line bundle over X. For every p-dimensional (reduced) algebraic subvariety Y of X, there is an integer

m6pL^p−1·B·Y

L^p·Y +p+ 1

such that the sheaf ω_Y ⊗^O_Y(mL−B) has a nonzero section.

To proceed further, we need the following useful “upper estimate” aboutL²dualizing sheaves (this is one of the crucial steps in Siu’s approach; unfortunately, it has the effect of producing rather large final bounds when the dimension increases).

(9.5) Proposition. Let H be a very ample line bundle on a projective algebraic mani-fold X, and let Y ⊂X be ap-dimensional irreducible algebraic subvariety. If δ =H^p·Y is the degree of Y with respect to H, the sheaf

Hom ω_Y,^O_Y((δ−p−2)H) has a nontrivial section.

Observe that if Y is a smooth hypersurface of degree δ in (X, H) = (P^p+1,^O(1)), then ω_Y = ^O_Y(δ − p− 2) and the estimate is optimal. On the other hand, if Y is a smooth complete intersection of multidegree (δ1, . . . , δr) in P^p+r, then δ = δ1. . . δr

whilst ω_Y =^O_Y(δ₁+. . .+δ_r−p−r−1) ; in this case, Proposition (9.5) is thus very far from being sharp.

Proof. Let X ⊂ P^N be the embedding given by H, so that H = ^OX(1). There is a linear projection P^{n ≻} P^p+1 whose restriction π : Y → P^p+1 to Y is a finite and regular birational map of Y onto an algebraic hypersurface Y^′ of degree δ in P^p+1. Let s ∈ H⁰(P^p+1,^O(δ)) be the polynomial of degree δ defining Y^′. We claim that for any small Stein open set W ⊂P^p+1 and anyL² holomorphic p-form u onY^′∩W, there is a L² holomorphic (p+ 1)-form eu on W with values in^O(δ) such that eu_↾Y^′_∩W =u∧ds. In fact, this is precisely the conclusion of the Ohsawa-Takegoshi extension theorem [OT87], [Ohs88] (see also [Man93] for a more general version); one can also invoke more standard local algebra arguments (see Hartshorne [Har77], Th. III-7.11). As KP^p+1 = ^O(−p−2), the form ue can be seen as a section of ^O(δ −p− 2) on W, thus the sheaf morphism u7→u∧dsextends into a global section of ^Hom ω_Y^′,^O_Y^′(δ−p−2)

. The pull-back by π^⋆ yields a section of ^Hom π^⋆ω_Y^′,^O_Y((δ−p−2)H)

. Since π is finite and generically 1 : 1, it is easy to see that π^⋆ω_Y^′ =ω_Y. The Proposition follows.

By an appropriate induction process based on the above results, we can now improve Siu’s effective version of the Big Matsusaka Theorem [Siu93]. Our version depends on a constant λ_n such that m(K_X + (n+ 2)L) +G is very ample for m > λ_n and every nef line bundle G. Corollary (8.5) shows thatλ_n 6 ³ⁿ⁺¹_n

−2n, and a similar argument involving the recent results of Angehrn-Siu [AS95] impliesλ_n 6n³−n²−n−1 forn>2.

Of course, it is expected that λ_n = 1 in view of the Fujita conjecture.

(9.6) Effective version of the Big Matsusaka Theorem. Let L and B be nef line bundles on a projective n-fold X. Assume that L is ample and let H be the very ample line bundle H =λ_n(K_X+ (n+ 2)L). Then mL−B is very ample for

m>(2n)^{(3n−1−1)/2}(Lⁿ⁻¹·(B+H))^(3n−1+1)/2(Lⁿ⁻¹·H)³ⁿ⁻²(n/2−3/4)−1/4

(Lⁿ)³ⁿ⁻²(n/2−1/4)+1/4 . In particular mL is very ample for

m>C_n (Lⁿ)³ⁿ⁻²

n+ 2 + Lⁿ⁻¹·K_X Lⁿ

3ⁿ⁻²(n/2+3/4)+1/4

with Cn = (2n)^{(3n−1−1)/2}(λn)³ⁿ⁻²(n/2+3/4)+1/4.

Proof. We use Proposition (9.4) and Proposition (9.5) to construct inductively a sequence of (non necessarily irreducible) algebraic subvarieties X = Y_n ⊃ Y_n−1 ⊃ . . . ⊃ Y₂ ⊃ Y₁ such thatYp =S

jYp,j isp-dimensional, andY_p−1 is obtained for eachp>2 as the union of zero sets of sections

σp,j ∈H⁰(Yp,j,^OYp,j(mp,jL−B))

with suitable integers mp,j > 1. We proceed by induction on decreasing values of the dimension p, and find inductively upper bounds m_p for the integers m_p,j.

By Corollary (9.2), an integer mn for mnL−B to have a section σn can be found with

mn 6nLⁿ⁻¹·(B+K_X + (n+ 1)L)

Lⁿ 6nLⁿ⁻¹·(B+H)

Lⁿ .

Now suppose that the sections σ_n, . . ., σ_p+1,j have been constructed. Then we get inductively a p-cycle Ye_p = P

µ_p,jY_p,j defined by Ye_p = sum of zero divisors of sections σ_p+1,j in Ye_p+1,j, where the mutiplicity µ_p,j on Y_p,j ⊂ Y_p+1,k is obtained by multiplying the corresponding multiplicity µp+1,k with the vanishing order of σp+1,k along Yp,j. As cohomology classes, we find

Ye_p ≡X

(m_p+1,kL−B)·(µ_p+1,kY_p+1,k)6m_p+1L·Ye_p+1. Inductively, we thus have the numerical inequality

Ye_p 6m_p+1. . . m_nL^n−p.

Now, for each component Yp,j, Proposition (9.4) shows that there exists a section of ω_Yp,j ⊗^O_Yp,j(m_p,jL−B) for some integer

mp,j 6pL^p−1·B·Yp,j

L^p·Y_p,j +p+ 16pmp+1. . . mnLⁿ⁻¹·B+p+ 1.

Here, we have used the obvious lower bound L^p−1·Y_p,j > 1 (this is of course a rather weak point in the argument). The degree of Yp,j with respect to H admits the upper bound

δp,j :=H^p ·Yp,j 6mp+1. . . mnH^p·L^n−p.

We use the Hovanski-Teissier concavity inequality ([Hov79], [Tei79, Tei82]) (L^n−p·H^p)^1p(Lⁿ)^1−1p ≤Lⁿ⁻¹·H

to express our bounds in terms of the intersection numbers Lⁿ and Lⁿ⁻¹ ·H only. We then get

δ_p,j 6m_p+1. . . m_n(Lⁿ⁻¹·H)^p (Lⁿ)^p−1 . By Proposition (9.5), there is a nontrivial section in

Hom ωYp,j,^OYp,j((δp,j−p−2)H) .

Combining this section with the section inω_Yp,j⊗^O_Yp,j(m_p,jL−B) already constructed, we get a section of ^OYp,j(mp,jL−B+ (δp,j−p−2)H) on Yp,j. Since we do not want H

to appear at this point, we replace B with B+ (δp,j −p−2)H and thus get a section σ_p,j of ^O_Yp,j(m_p,jL−B) with some integer m_p,j such that

mp,j 6pmp+1. . . mnLⁿ⁻¹·(B+ (δp,j −p−2)H) +p+ 1 6p mp+1. . . mnδp,jLⁿ⁻¹·(B+H)

6p(mp+1. . . mn)²(Lⁿ⁻¹·H)^p

(Lⁿ)^p−1 Lⁿ⁻¹·(B+H).

Therefore, by putting M =n Lⁿ⁻¹·(B+H), we get the recursion relation mp 6M (Lⁿ⁻¹·H)^p

(Lⁿ)^p−1 (mp+1. . . mn)² for 2 6p6n−1,

with initial value mn 6 M/Lⁿ. If we let (mp) be the sequence obtained by the same recursion formula with equalities instead of inequalities, we get m_p 6 m_p with m_n−1 = M³(Lⁿ⁻¹·H)ⁿ⁻¹/(Lⁿ)ⁿ and

m_p = Lⁿ

Lⁿ⁻¹·H m²_p+1m_p+1 for 26p6n−2. We then find inductively

m_p 6m_p =M^3n−p(Lⁿ⁻¹·H)^3n−p−1(n−3/2)+1/2

(Lⁿ)^3n−p−1(n−1/2)+1/2 . We next show thatm₀L−B is nef for

m₀ = max m₂, m₃, . . . , m_n, m₂. . . m_nLⁿ⁻¹·B .

In fact, let C ⊂ X be an arbitrary irreducible curve. Either C = Y_1,j for some j or there exists an integer p = 2, . . . , n such that C is contained in Y_p but not in Y_p−1. If C ⊂Y_p,jrY_p−1, then σ_p,j does not vanish identically on C. Hence (m_p,jL−B)_↾C has nonnegative degree and

(m₀L−B)·C >(m_p,jL−B)·C ≥0.

On the other hand, if C =Y1,j, then

(m₀L−B)·C >m₀−B·Ye₁ >m₀−m₂. . . m_nLⁿ⁻¹·B>0.

By the definition of λn (and by Corollary (8.5) showing that such a constant exists), H +G is very ample for every nef line bundle G, in particular H +m₀L−B is very ample. We thus replace again B with B+H. This has the effect of replacing M with M =n Lⁿ⁻¹ ·(B+ 2H)

and m₀ with

m0 = max mn, m_n−1, . . . , m2, m2. . . mnLⁿ⁻¹·(B+H) .

The last term is the largest one, and from the estimate on mp, we get

m0 6M^{(3n−1−1)/2}(Lⁿ⁻¹·H)⁽³ⁿ⁻²−1)(n−3/2)/2+(n−2)/2(Lⁿ⁻¹·(B+H)) (Lⁿ)⁽³ⁿ⁻²−1)(n−1/2)/2+(n−2)/2+1

6(2n)^{(3n−1−1)/2}(Lⁿ⁻¹·(B+H))^(3n−1+1)/2(Lⁿ⁻¹·H)³ⁿ⁻²(n/2−3/4)−1/4

(Lⁿ)³ⁿ⁻²(n/2−1/4)+1/4

(9.7) Remark. In the surface case n= 2, one can take λn = 1 and our bound yields mL very ample for

m>4(L·(K_X+ 4L))²

L² .

If one looks more carefully at the proof, the initial constant 4 can be replaced by 2. In fact, it has been shown recently by Fern´andez del Busto that mL is very ample for

m > 1 2

(L·(K_X + 4L) + 1)²

L² + 3

,

and an example of G. Xiao shows that this bound is essentially optimal (see [FdB96]).