On the uniform birationality of the pluriadjoint line bundles Tomoki Arakawa (Sophia University) tomoki-a[at]sophia.ac.jp

1

Background and Problems

X: smooth projective variety defined over C L: ample line bundle over X (X, L): polarized manifold

2

Matsusaka’s big theorem

P(X,L)(m) := χ(X, mL) =

dim ∑X

(−1)ihi(X, mL) ∈ Q[m]

i=0 : Hilbert polynomial of (X, L)



Theorem (Matsusaka, 1972) Fix P ∈ Q[m]. Then ∃m(P ) ≥ 1 s.t. for any pol. mfd. (X, L) with P(X,L) = P , Φ|mL| : X −→ PN



(N := h0(mL) − 1)

is an embedding for ∀m ≥ m(P ).





−→ Hilbert scheme theory 3

Study of KX + mL / m(KX + L) (KX := ∧dim X T ∗ X: canonical bundle of X) 



Fact Suppose KX + L is nef.

(i.e., (KX + L) · C ≥ 0 for ∀C: irred. curve on X)

1. |m(KX + L)| is base-point-free for ∀m ≫ 0.

(By BPF)

2. |m(KX + L)| ̸= ∅ for ∀m ≫ 0. (By Shokurov’s nonvanishing th.)



4







Conjecture (Fujita, 1985) (X, L): polarized manifold w/ dim X = n



m ≥ n + 1 =⇒ |KX + mL| is free.



∼ (Pn, O(1)). Remark. KX + nL is nef unless (X, L) = −→ KX + (n + 1)L is nef.

Results. • (Reider, 1988) dim X = 2 =⇒ OK • (Kawamata, 1995) dim X = 3, 4 =⇒ OK • (Angehrn-Siu, 1995) n(n + 1) m≥ + 1 =⇒ |KX + mL| is free. 2 5









Conjecture (Ionescu, 1990) (X, L): polarized manifold KX + L is nef. =⇒ H 0(X, KX + L) ̸= 0.

Results. dim X ≤ 3 =⇒ OK. (by Fukuma, H¨ oring) 







Conjecture (Ambro, Kawamata) X: normal projective variety B: effective Q-divisor on X s.t. (X, B) is KLT D: nef Cartier divisor on X If D − (KX + B) is nef-big, then H 0(X, D) ̸= 0.

Results. • Kawamata (2000); dim X = 2 or X:minimal 3-fold • H¨ oring (2010); X: a normal proj. 3-fold with at most Q-factorial canonical singularities, B = 0, and D − KX is a nef-big Cartier div. 6





Problem (Fukuma, 2007) Mn := {r ∈ Z>0 | h0 (X, r(KX + L)) > 0 for every n-dim. pol. mfd. (X, L) w/ κ(KX + L) ≥ 0} { min Mn if Mn ̸= ∅ m(n) := ∞ if Mn = ∅

−→ m(n) =?





Remark. • κ(KX + L) := lim supm→∞ log h0 (X, m(KX + L))/ log m : Kodaira dimension of KX + L i.e. h0 (X, m(KX + L)) ∼ mκ(KX +L) • Recall : KX + L is nef =⇒ κ(KX + L) ≥ 0 (by Nonvanishing th.)

Results. (Fukuma) m(1) = m(2) = m(3) = 1, m(4) ≤ 6. 7

Main theorems 



Main Theorem 1 (A.) Suppose KX + L is nef.



n(n + 1) m≥ + 2 =⇒ H 0(X, OX (m(KX + L))) ̸= 0. 2







Main Theorem 2 (A.) m(n) < ∞ for ∀n. i.e., ∃m(n) ≥ 1 s.t. if (X, L): n-dim. pol. mfd. w/ κ(KX + L) ≥ 0, then m ≥ m(n) =⇒ H 0(X, OX (m(KX + L))) ̸= 0. More precisely, if m ≥ m(n), then the rational map Φ|m(KX +L)| is birationally

equivalent to the Iitaka fibration associated to KX + L.





8

Sketch of the proof of Main Theorem 1 • For simplicity, we assume that KX + L is ample. Fix x ∈ X. • µ0 = µ(X, KX + L) : the volume of X with respect to KX + L h0 (X, m(KX + L)) := n! · lim sup mn m→∞ • Since KX + L is nef-big, µ0 = (KX + L)n ≥ 1. • Fix 0 < ε ≪ 1. Taking m0 ≫ 0, ∃σ0 (̸= 0) ∈ H (X, (m0 (KX + L)) ⊗ 0

√ ⌈ n µ0 (1−ε)m0 ⌉ mx ).

• h0 :=

1 : singular hermitian metric on KX + L. |σ0 |2/m0

• α0 := sup {α ∈ Q>0 | I(hα0 )x = OX,x } . 9

Since (

∑n

i=1 |zi |

2 )−n

is not locally integrable at O ∈ Cn, n α0 ≤ √ . n µ0 (1 − ε) n ∴ α0 ≤ √ + λ n µ0

(0 < ∃λ ≪ 1/n).

Suppose Supp(OX /I(hα0 0 )) is isolated at x. If m ≥ n + 2, by Nadel’s vanishing theorem, H 1 (X, OX (m(KX + L)) ⊗ I(hα0 0 )) = 0. Hence H 0 (X, OX (m(KX + L))) −→ H 0 (X, OX (m(KX + L)) ⊗ OX /I(hα0 0 )) is surjective. ∴ ∃σ ∈ H 0 (X, OX (m(KX + L))) with σ(x) ̸= 0.

10

If Supp(OX /I(hα0 0 )) is not isolated at x, we construct a filtration: X ) X1 ) · · · ) Xr ) Xr+1 = {x}, α0 , α1 , . . . , αr > 0, with ni αi ≤ √ + λ ni µi

(ni := dim Xi ,

µi := µ(Xi , (KX + L)|Xi )).

Note that µi = µ(Xi , (KX + L)|Xi ) = (KX + L)ni · Xi ≥ 1.

(∵ KX + L: ample)

∴ αi ≤ ni + λ. Then m≥

r ∑

αi + 1 =⇒ ∃σ ∈ H 0 (X, OX (m(KX + L))) s.t. σ(x) ̸= 0.

i=0

Since αi ≤ ni + λ, ∴ H 0 (X, OX (m(KX + L))) ̸= 0

( ) n(n + 1) ∀m ≥ +2 . 2 11

Remark on the proof of Main Theorem 2 



Main Theorem 2 (A.) m(n) < ∞ for ∀n.

i.e., ∃m(n) ≥ 1 s.t. if (X, L): n-dim. pol. mfd. w/ κ(KX + L) ≥ 0, then m ≥ m(n) =⇒ H 0 (X, OX (m(KX + L))) ̸= 0. More precisely, if m ≥ m(n), then the rational map Φ|m(KX +L)| is birationally equivalent to the Iitaka fibration associated to KX + L.



We can reduce the problem to the case : KX + L is big (but not nef). −→ µi ∈ Q>0 By induction on n and Hilbert scheme argument, we have a constant C(n) > 0 s.t. if (X, L): n-dim. polarized mfd. w/ κ(KX + L) = n, µ(X, KX + L) ≥ C(n).

12



References • U. Angehrn and Y.-T. Siu: Effective freeness and point separation for adjoint bundles, Invent. Math. 122 (1995), 291–308. • T. Arakawa: Effective nonvanishing of pluri-adjoint line bundles, Tokyo Journal of Mathematics Vol. 38 (2015), no. 1. • T. Arakawa: On the uniform birationality of the pluriadjoint line bundles (preprint). • Y. Fukuma: On the dimension of global sections of adjoint bundles for polarized 3-folds and 4-folds, J. Pure Appl. Algebra 211 (2007), 609–621. • A. H¨ oring: On a conjecture of Beltrametti and Sommese, Journal of Algebraic Geometry 21 (2012), 721–751. • Y. Kawamata: On effective nonvanishing and base point freeness, Kodaira’s issue, Asian J. Math. 4, (2000), 173–181. • H. Tsuji: Pluricanonical systems of projective varieties of general type. II, Osaka J. Math. 44 (2007), no. 3, 723–764.

13