THE CANONICAL VOLUME OF 3-FOLDS OF GENERAL TYPE WITH χ ≤ 0
JUNGKAI A. CHEN AND MENG CHEN
Abstract. We prove that the canonical volume K3 ≥ 1 30 for all
3-folds of general type with χ(O) ≤ 0. This bound is sharp.
1. Introduction
Let V be a nonsingular projective 3-fold of general type. According to Mori’s Minimal Model Program (see for instance [15, 16, 19]), V has at least one minimal model X which is normal projective with at worst Q-factorial terminal singularities. Denote by K3 := K3
X. Since it
is uniquely determined by the birational equivalence class of V , K3 is
usually referred to as the canonical volume of V , also written as Vol(V ). In the study of 3-folds of general type, a major difficulty arises when K3
is only a small rational number, rather than an integer. For example, among known ones by Fletcher-Reid (cf. [10], p151), Vol(V ) could be as small as 1
420. It is a fact that the birational invariant Vol(V ) strongly
affects the geometry of V . So a natural and interesting question is to find the sharp lower bound v3of K3among all those nonsingular 3-folds
V of general type.
There have been some relevant known results already:
• There exists a constant v3 > 0 such that Vol(V ) ≥ v3 for
all threefolds of general type. This is proved by Hacon and Mckernan [12], Takayama [21] and Tsuji [22];
• It is proved by the second author [5] that Vol(V ) ≥ 1
3 for all
3-folds of general type with pg(V ) := dim H3(V, OV) ≥ 2 and
the bound “1
3” is sharp.
In this paper we would like to prove the following:
Theorem 1.1. Let V be a nonsingular projective 3-fold of general type with χ(OV) ≤ 0. Then
(i) Vol(V ) ≥ 1 30.
(ii) When Vol(V ) = 1
30, V has the invariants: pg(V ) = 1, q(V ) = 0,
χ(OV) = 0, P2(V ) = 1, P3(V ) = 2, P4(V ) = 3 and P5(V ) = The first author was partially supported by National Science Council of Taiwan. The second author was supported by both the Program for New Century Excel-lent TaExcel-lents in University (#NCET-05-0358) and the National Outstanding Young Scientist Foundation (#10625103).
4. Furthermore any minimal model of V has exactly 3 virtue baskets of singularities (in the sense of Reid): 1 × 1
2(1, −1, 1),
1 × 1
3(1, −1, 1), 1 × 15(1, −1, 1).
The next example shows that the lower bound of Vol(V ) in Theorem 1.1(i) is optimal.
Example 1.2. (cf. [10], p151) The canonical hypersurface X28 ⊂
P(1, 3, 4, 5, 14) has the canonical volume K3 = 1
30, pg = 1, q = 0,
χ(OX28) = 1. X28 has 3 terminal singularities: 1 ×
1 2(1, −1, 1), 1 × 1 3(1, −1, 1), 1 × 1 5(1, −1, 1).
This note also contains some effective results. For example we will prove the following:
Corollary 1.3. Let V be a nonsingular projective 3-fold of general type with q := h1(O
V) > 0. Then Vol(V ) ≥ 221 .
The paper is organized as the following. In section 2, we study the pluricanonical maps. We obtained, in Theorem 2.5, a lower bound > 1
30 when the plurigenera are large. In section 3, we consider irregular
threefolds. Combining results obtained in these two sections, the only unknown case has the information: χ(OX) = 0, q(X) = 0, pg(X) = 1
and P5(X) < 5. Thus in sections 4 we classify all possible types of
singularities and hence are able to complete the proof of the main theorem.
Throughout the paper ∼ means linear equivalence while ≡ denotes the numerical one.
2. Bounding K3 via ϕ m
In order to get an effective lower bound of K3 we need to study the
m-canonical map ϕm. Let X be a minimal projective 3-fold of general
type (admitting at worst Q-factorial terminal singularities) with Pm0 = Pm0(X) := dimCH
0(X, O
X(m0KX)) ≥ 2
for some integer m0 > 0, where KX is a canonical divisor of X.
2.1. Set up for ϕm0. We study the m0-canonical map ϕm0 : X 99K
PPm0−1 which is only a rational map. First of all we fix an effective
Weil divisor Km0 ∼ m0KX. By Hironaka’s big theorem, we can take
successive blow-ups π : X0 → X such that:
(i) X0 is smooth;
(ii) the movable part of |m0KX0| is base point free;
(iii) the support of π∗(K
m0) is of simple normal crossings.
Set gm0 := ϕm0 ◦ π. Then gm0 is a morphism by assumption. Let
X0 −→ Bf −→ Ws 0 be the Stein factorization of g
m0 with W0 the image
of X0 through g
m0. In summary, we have the following commutative
X X0 W0 B -? ? @ @ @ @ @ R --- -f s π ϕm0 gm0
We recall the definition of π∗(K
X) and denote by r(X) the Cartier
index of X. Then r(X)KX0 = π∗(r(X)KX) + Eπ where Eπ is a sum of
exceptional divisors. One defines π∗(K
X) := KX0− 1
r(X)Eπ. So,
when-ever we take the round up of mπ∗(K
X), we always have dmπ∗(KX)e ≤
mKX0 for any integer m > 0. We may write m0KX0 =Q π∗(m0KX) +
Em0 = Mm0+Zm0, where Mm0 is the movable part of |m0KX0|, Zm0 the
fixed part and Em0 an effective Q-divisor which is a Q-sum of distinct
exceptional divisors. We may also write m0π∗(KX) =Q Mm0 + Em0 0,
where E0
m0 = Zm0 − Em0 is an effective Q-divisor.
If dim(B) ≥ 2, a general member S of |Mm0| is a nonsingular
pro-jective surface of general type by Bertini’s theorem and by the easy addition formula for Kodaira dimension.
If dim(B) = 1, a general fiber S of f is an irreducible smooth pro-jective surface of general type, still by the easy addition formula for Kodaira dimension. We may write
Mm0 =
am0
X
i=1
Si ≡ am0S
where the Si is a smooth fiber of f for all i and am0 ≥ Pm0(X) − 1.
In both cases we call S a generic irreducible element of |Mm0|.
De-note by σ : S −→ S0 the blow-down onto the smooth minimal model
S0.
2.2. Assumptions. We need some assumptions to estimate K3.
(1) Keep the same notations as above, we define p =
(
1 if dim(B) ≥ 2 am0 if dim(B) = 1.
(2) Take a generic irreducible element S of |Mm0|. Assume that |G|
is a movable complete linear system on S. Also assume that a generic irreducible element C of |G| is smooth.
(3) Assume there is a rational number β > 0 such that π∗(K X)|S−
βC is numerically equivalent to an effective Q-divisor on S. Set α = (m − 1 − m0
p − β1)ξ and α0 := dαe.
Under Assumptions 2.2, one has K3 ≥ p m0 π∗(K X)2· S ≥ pβ m0 (π∗(K X) · C). (2.1)
So it suffices to estimate the rational number ξ := (π∗(K
X) · C)X0.
We need the following theorem to study the lower bound of ξ: Theorem 2.3. Let m > 0 be an integer. Under Assumptions 2.2, the inequality
ξ ≥ deg(KC) + α0 m
holds if one of the following conditions is satisfied: (i) α0 ≥ 2;
(ii) α > 0 and C is an even divisor on S. Proof. We consider the sub-linear system
|KX0 + d(m − 1)π∗(KX) −1
pE
0
m0e| ⊂ |mKX0|.
Take a generic irreducible element S of |Mm0|. Noting that (m −
1)π∗(K
X) − 1pEm0 0 − S ≡ (m − 1 −
m0
p )π∗(KX) is nef and big
when-ever α > 0, the Kawamata-Viehweg vanishing theorem [23, 14] gives the surjective map
H0(X0, K X0 + d(m − 1)π∗(KX) −1 pE 0 m0e) −→ H0(S, K S+ d(m − 1)π∗(KX) − S − 1 pE 0 m0e|S). (2.2)
Now consider a generic irreducible element C ∈ |G|. By assumption there is an effective Q-divisor H on S such that
1 βπ
∗(K
X)|S ≡ C + H.
By the vanishing theorem again, whenever m − 1 − m0
p − 1 β > 0 which yields that ((m − 1)π∗(KX) − S − 1 pE 0 m0)|S− C − H ≡ (m − 1 − m0 p − 1 β)π ∗(K X)|S
is nef and big, we have the surjective map H0(S, K S+ d((m − 1)π∗(KX) − S − 1 pE 0 m0)|S− He) −→ H0(C, K C + D) (2.3) where D := d((m − 1)π∗(K X) − S −1pEm0 0)|S− C − He|C is a divisor
on C. Noting that C is nef on S, we have deg(D) ≥ α and thus deg(D) ≥ α0.
Whenever either deg(D) ≥ 2 or C is an even divisor and m − 1 −
m0
p − β1 > 0 (deg(D) ≥ 2 automatically follows), |KC + D| is base
|KS+d((m − 1)π∗(KX) − S −1pEm0 0)|S− He|. Applying Lemma 2.7 of
[7] to surjective maps (2.2) and (2.3), one has
mπ∗(KX)|S ≥ Nm and (Nm· C)S ≥ 2g(C) − 2 + deg(D).
So mξ ≥ deg(KC) + α0. We are done. ¤
Remark 2.4. A technical problem in utilizing Theorem 2.3 is to ver-ify Assumptions 2.2. To avoid unnecessary redundancy, we only copy several technical results here without proof. Note that the most com-plicated situation is the one with dim(B) = 1, in which case we set b := g(B), the geometric genus.
(1) Usually we will take G ≤ σ∗(K
S0) whenever pg(S) ≥ 2 or G =
2σ∗(K
S0), 4σ∗(KS0) otherwise;
(2) When g(B) > 0, it is proved in Lemma 3.4 of [8] that π∗(K
X)|S ∼ σ∗(KS0).
So one may take β = 1 or 1 2 or 12.
(3) When g(B) = 0, it is proved in Lemma 3.3 of [8] that π∗(K
X)|S− eβnσ∗(KS0)
is numerically equivalent to an effective Q-divisor for a sequence of positive rational numbers { eβn} with eβn 7→ m0p+p. So one may
take β = eβn or 12βen or 14βen accordingly.
(‡) Note that a special situation (with m0 = 1) of this theory has
already appeared in [5], as 2.8 and Lemma 3.4. Now we are ready to estimate ξ and K3.
Theorem 2.5. Let V be a nonsingular projective 3-fold of general type. Then (i) Vol(V ) ≥ 1 22 if P4(V ) ≥ 5; (ii) Vol(V ) ≥ 1 25 if P5(V ) ≥ 5, pg(V ) > 0 and dim(B) ≥ 2. (iii) Vol(V ) ≥ 8 45 if P5(V ) ≥ 5, pg(V ) > 0 and dim(B) = 1.
Proof. Take a minimal model X of V . We study |mK| on X. Keep the same set up as in 2.1.
Part (i). We can take m0 = 4. We will study according to the value
of dim(B). Take a generic irreducible element S of |M4|.
If dim(B) = 3, we know that p = 1 by definition. In this case we know S ∼ M5 and that |S| gives a generically finite morphism. Set
G := S|S. Then |G| is base point free and ϕ|G| gives a generically
finite map. So a generic irreducible element C of |G| is a smooth curve of genus ≥ 2. If ϕ|G| gives a birational map, then dim ϕ|G|(C) = 1
for a general member C. The Riemann-Roch and Clifford’s theorem on C says C2 = G · C ≥ 2. If ϕ
|G| gives a generically finite map
[6] gives C2 ≥ 2h0(S, G) − 4 ≥ 4. Anyway we have C2 ≥ 2. So
deg(KC) = (KS + C)C > 2C2 ≥ 4. We see deg(KC) ≥ 6 because it is
even. One may take β = 1
4 since 4π∗(KX)|S ≥ C. Now if we take a
very big m such that α > 1 then Theorem 2.3 gives: mξ ≥ deg(KC) + (m − 1 − m0− 1
β)ξ. This gives ξ ≥ 2
3. If we take m = 11. Then α = 2ξ > 1. Theorem 2.3
says ξ ≥ 8
11. So inequality (2.1) gives K3 ≥ 221.
If dim(B) = 2, we know that |G| := |S|S| is composed with a pencil
of curves. A generic irreducible element C of |G| is a smooth curve of genus ≥ 2, so deg(KC) ≥ 2. Furthermore we have h0(S, G) ≥
h0(X0, S) − 1 ≥ 4. So G ≡ eaC for ea ≥ h0(S, G) − 1 ≥ 3. This means
4π∗(K
X)|S ≥ S|S ≥numerically 3C. So we may take β = 34. Now take a
very big m. Theorem 2.3 gives ξ ≥ 6
19. Take m = 10. Then α ≥ 2219 > 1. We get ξ ≥ 2 5. So inequality (2.1) gives K3 ≥ 3 40 > 1 22.
If dim(B) = 1 and b = g(B) > 0, there is an induced fibration f : X0 −→ B. Recall that S is a general fiber of f and S can be a
nonsingular surface of general type of any numerical type. One has p = a4 ≥ P4 ≥ 5 by the Riemann-Roch and Clifford’s theorem. Set
G := 4σ∗(K
S0). Because |4KS0| is base point free by Bombieri [2], |G| =
|4σ∗(K
S0)| is also base point free. Denote by C a generic irreducible
element of |G|. Then C is smooth and deg(KC) = (KS + C)C ≥
(π∗(K X)|S+ C)C = (σ∗(KS0) + C)C = 20σ ∗(K S0) 2 ≥ 20. By Remark 2.4(2), we can take β = 1 4 since π∗(KX)|S ∼ 14(4σ∗(KS0)). Now if we
take a very big m, Theorem 2.3 gives ξ ≥ 100
29. Inequality (2.1) gives
K3 ≥ 125 116 >
1 22.
If dim(B) = 1 and b = g(B) = 0, we have p ≥ P5 − 1 ≥ 4. Take
m0 = 4. We study two cases separately: (a) pg(S) > 0; (b) pg(S) = 0.
First we consider case (a). We still set G := 2σ∗(K
S0). Take a
generic irreducible element C of |G|. By an established theorem (see Bombieri [2], Reider [20], Catanese-Ciliberto [3], and P. Francia [11] or directly refer to Theorem 3.1 in the survey article by Ciliberto [9]), |2KS0| is always base point free. We get deg(KC) = (KS + C)C >
C2 ≥ 4. So actually deg(K
C) ≥ 6 due to its evenness. We only
have to find a suitable β. Remark 2.4(3) says that one can find a sequence of positive rational numbers { eβn} with eβn 7→ p+4p ≥ 12 such
that π∗(K
X)|S − eβnσ∗(KS0) is numerically equivalent to an effective
Q-divisor. Take βn := 12βen. Then π∗(KX)|S − βnC is numerically
equivalent to an effective Q-divisor. We know βn 7→ 14 whenever p = 4.
When m is very big, Theorem 2.3 gives ξ ≥ 1. So inequality (2.1) gives K3 ≥ 1
4 > 221 .
Finally we consider the case (b). We set G := 4σ∗(K
S0). The surface
element C of |G| is a smooth curve. Because
deg(KC) = (KS+ C)C ≥ (π∗(KX)|S+ C)C > C2 ≥ 16,
again we see deg(KC) ≥ 18. Similar to the case (a), we know that
π∗(K
X)|S − eβnσ∗(KS0) is numerically equivalent to an effective
Q-divisor for a rational number sequence { eβn} with eβn7→ p+4p ≥ 12. Take
βn := 14βen. Then π∗(KX)|S − βnC is numerically equivalent to an
ef-fective Q-divisor. We know βn 7→ 18 whenever p = 4. When m is very
big, Theorem 2.3 gives ξ ≥ 9
5. Take m = 11. Then α ≥ ξ > 1. One
gets ξ ≥ 20
11 > 95. So inequality (2.1) gives K3 ≥ 225 > 221.
Comparing what we have proved, we see K3 ≥ 1 22.
Part (ii). Take m0 = 5. We have p = 1 by definition. A general
member S ∈ |M5| is a nonsingular projective surface of general type.
Set G := S|S.
If ϕ5 is generically finite, then ϕ|G| is either birational or generically
finite of degree ≥ 2. We have the following argument:
(]) If ϕ|G|gives a birational map, then clearly h0(S, G) ≥
4 because S is of general type. Since pg(V ) > 0, we know
G ≤ (KX0+ S)|S = KS. And because G is nef, Lemma
2.1 of [6] says C2 ≥ 3h0(S, G) − 7 ≥ 5. When |S| gives
a generically finite map of degree ≥ 2, then Lemma 2.2 of [6] gives C2 ≥ 2h0(S, G) − 4.
Because h0(S, G) ≥ h0(X0, S)− 1 ≥ 4, the argument (]) says C2 ≥ 4.
We get deg(KC) = (KS + C)C > 2C2 ≥ 8 noting that KX0|S · C ≥
π∗(K
X)|S · C > 0 by the Hodge Index Theorem. Actually we have
deg(KC) ≥ 10 since it is even. On the other hand, we can take β = 15
since 5π∗(K
X)|S ≥ C. Now take a very big m. Theorem 2.3 gives
ξ ≥ 10
11. Take m = 13. Then α ≥ 2011 > 1. We get ξ ≥ 1213. So inequality
(2.1) gives ξ ≥ 12
25·13. In fact a similar calculation says ξ ≥ l+1l for all
l ≥ 12. Thus ξ ≥ 1 and (2.1) gives K3 ≥ 1 25.
If dim(B) = 2, we know that |G| := |S|S| is composed with a pencil
of curves. A generic irreducible element C of |G| is a smooth curve of genus ≥ 2, so deg(KC) ≥ 2. Furthermore we have h0(S, G) ≥
h0(X0, S) − 1 ≥ 4. So G ≡ eaC for ea ≥ h0(S, G) − 1 ≥ 3. This means
5π∗(K
X)|S ≥ S|S ≥numerically 3C. So we may take β = 35. Now take a
very big m. Theorem 2.3 gives ξ ≥ 6
23. Take m = 12. Then α ≥ 2623 > 1. We get ξ ≥ 1 3. Take m = 11. Then α ≥ 10 9 > 1. We get ξ ≥ 4 11. So inequality (2.1) gives K3 ≥ 1 25· 1211 > 251 .
Part (iii). Take m0 = 5. Parallel to the last parts in the proof of (i),
we can discuss according to the value of b = g(B). So we have more or less a redundant calculation as follows.
If b = g(B) > 0, there is an induced fibration f : X0 −→ B. Because
Riemann-Roch and Clifford’s theorem. Set G := 2σ∗(K
S0). Because
|2KS0| is base point free, |G| = |2σ∗(KS0)| is also base point free.
Denote by C a generic irreducible element of |G|. Then C is smooth and deg(KC) = (KS+C)C > 4σ∗(KS0)2 ≥ 4. So actually deg(KC) ≥ 6.
By Remark 2.4(2), we can take β = 1
2 since π∗(KX)|S ∼ 1
2(2σ∗(KS0)).
Now if we take a very big m, Theorem 2.3 gives ξ ≥ 3
2. Inequality (2.1)
gives K3 ≥ 3 4.
If b = g(B) = 0, we have p ≥ P5 − 1 ≥ 4. Take m0 = 5. Because
pg(V ) > 0, one sees pg(S) > 0. We still set G := 2σ∗(KS0). We have
deg(KC) ≥ 6. Similarly we only have to find a suitable β. Remark
2.4(3) says that one can find a sequence of positive rational numbers { eβn} with eβn7→ p+5p ≥ 49 such that π∗(KX)|S− eβnσ∗(KS0) is numerically
equivalent to an effective Q-divisor. Set βn := 12βen. Then π∗(KX)|S −
βnC is numerically equivalent to an effective Q-divisor. We know βn 7→ 2
9 whenever p = 4. When m is very big, Theorem 2.3 gives ξ ≥ 8
9. Take
m = 8. Then α ≥ 10
9 > 1. We get ξ ≥ 1. So inequality (2.1) gives
K3 ≥ 8
45. This completes the proof. ¤
Remark 2.6. One may remove extra condition: pg(V ) > 0 in Theorem
2.5 (ii) and (iii) to obtain parallel, but weaker results. We omit the details simply because it is not used in the proof of the main theorem.
3. On irregular 3-folds of general type
In this section, we study the canonical volume of irregular threefolds of general type. Let X be a nonsingular projective threefold of general type and a : X → alb(X) the Albanese map. By running the minimal model program, one easily see that the Albanese map factors through its minimal model. So we may and do assume that X is a minimal (KX
nef) threefold of general type with Q-factorial terminal singularities. In the study of pluricanonical systems on irregular threefolds, the most unpleasant case is when the Albanese map a : X → Alb(X) is surjective onto an elliptic curve E with general fiber F of type (K2
F, pg(F )) = (1, 2).
Theorem 3.1. Let X be a minimal 3-fold of general type with q(X) = 1 and the general fiber of a : X → Alb(X) is of (1, 2) type. Then the canonical volume Vol(X) = K3
X ≥ 19.
Before proving the main result, we would like to recall some notion and results in [4].
Definition 3.2. For any vector bundle E on an elliptic curve, we write E = ⊕Ei for its decomposition into indecomposable vector bundles.
We define ν(E) := min µ(Ei), where µ(Ei) = deg(Erk(Eii)).
Lemma 3.3. ([4], Lemma 4.8) Let E1, E2 be indecomposable vector
In particular ν(E2) ≥ ν(E1) if E1 → E2 is a surjective map of vector
bundles.
Definition 3.4. A coherent sheaf F on an abelian variety A is said to be IT0 if Hi(A, F ⊗ P ) = 0 for all i > 0 and all P ∈ Pic0(A).
Lemma 3.5. ([4], Lemma 4.10) Let E be an IT0 vector bundle on an
elliptic curve which admits a short exact sequence 0 → F → E → Q → 0
of coherent sheaves such that Q has generic rank ≤ 1. Then ν(E) ≥ min{1, ν(F )}.
3.6. Multiplication maps ϕm,n and ψm,n. Let Rm := H0(F, ωFm)
and Em := a∗ωXm. By Lemma 4.1 of [4], Em is an IT0 vector bundle of
rank Pm(F ) for all m ≥ 2. We also remark that ν(E1) ≥ 0 by the
semi-positivity theorem (see Viehweg [24]) and Atiyah’s description of vector bundles over elliptic curves (cf. [1]). We consider the multiplication map of pluricanonical systems on fibers
ϕm,n : Rm⊗Rn → Rm+n.
This induces a map
ψm,n : Em⊗En → Em+n.
Clearly if cokernel of ϕm,n has dimension ≤ r, then cokernel of ψm,n
has rank ≤ r.
3.7. Surfaces of (1,2) type. Let F be a nonsingular minimal projec-tive surface of general type with (K2
F, pg(F )) = (1, 2). It’s well-known
that |KF| has only one base point z and |2KF| is base point free (cf.
[2]).
We recall the following result in §5 of [4].
Lemma 3.8. ([4], p353, line -6) Assume that a general fiber F of the fibration a : X → Alb(X) is a surface of (1,2) type. Then ϕ1,m−1 :
R1⊗Rm−1 7→ R1Rm−1 ⊂ Rm has codimension ≤ 1 for all m ≥ 1 and
ϕ1,2 is surjective.
(†) Clearly R1R2 = R3 implies that R2R2, which contains R1R1R2 =
R1R3, has codimension ≤ 1 in R4.
Moreover, we have the following:
Lemma 3.9. Assume that a general fiber F of the fibration a : X → Alb(X) is a surface of (1,2) type. Then the multiplication map ϕ2,m−2 :
R2⊗Rm−2 7→ Rm is surjective whenever m ≥ 8 and has codimension
≤ 1 if m = 7.
Fix two sections s, s1 ∈ R2 with smooth curves C := div(s), C1 :=
div(s1) and assume that Z := div(s1) ∩ C consists of 4 distinct points,
we have the exact sequences:
0 → Rm−2 → Rs m → HrC 0(C, mKF|C) → 0, (3.1) 0 → Rm−2 → Rs1 m rC1 → H0(C 1, mKF|C1) → 0, (3.2) 0 → Rm−4 → Rs m−2 rC → H0(C, (m − 2)KF|C) → 0, (3.3)
thanks to the vanishing of H1(F, tK
F) for all t ≥ 0. We also have the
following exact sequence:
0 → H0(C, (m − 2)KF|C)→ Hes1 0(C, mKF|C)→ HrZ 0(Z, OZ). (3.4)
Among the four exact sequences, one can find the commutative relation: e s1◦ rC = rC◦ s1. Rm−2 −−−→s1 Rm rC y yrC H0(C, (m − 2)K F|C) −−−→ e s1 H0(C, mK F|C)
One knows es1◦ rC(Rm−2) = es1H0(C, (m − 2)KF|C) has codimension
≤ 4 in the space H0(C, mK
F|C) since dim H0(Z, OZ) = 4. So rC ◦
s1(Rm−2) has codimension d0 ≤ 4 in H0(C, mKF|C). Since
d0 = h0(C, mK
F|C) − dim rC(s1Rm−2)
≥ dim Rm− dim sRm−2− dim s1Rm−2.
It follows that sRm−2+ s1Rm−2 ⊂ Rm has codimension ≤ 4.
Moreover, we consider
0 → H0(C, (m − 4)K
F|C)→ Hes1 0(C, (m − 2)KF|C) rZ
→ H0(Z, OZ) → H1(C, (m − 4)KF|C).
Since 3KF |C = KC and deg(KF|C) = 2, one sees that H1(C, (m −
4)KF|C) = 0 if m ≥ 8. When m = 7, H1(C, 3KF|C) = H1(C, KC) is
one dimensional. We can take a section s2 ∈ R2 such that s2 never
vanishing on Z. Set J = div(s2) ∩ C which can be a union of 4
dis-tinct points. As we have seen the map rJ : rC(s2Rm−2) → H0(J, OJ)
is either surjective when m ≥ 8 or having codimension ≤ 1 when m = 7. Together with surjectivity of rC, we see that rC(s2Rm−2) =
e
s2(H0(C, (m − 2)KF|C)) where
e
s2 : H0(C, (m − 2)KF|C) 7→ H0(C, mKF|C)
is defined by the multiplication of s2. This already means that
sRm−2+ s1Rm−2+ s2Rm−2 ⊂ Rm
has codimension 0 or ≤ 1 if m ≥ 8 or = 7 respectively. We are done. ¤ Now we prove Theorem 3.1.
Proof of Theorem 3.1. First of all, E2 is an IT0 vector bundle of
rank 4. So one has ν(E2) ≥ 14.
Consider the induced multiplication map ψ2,2 : E2⊗E2 → E4. Since
ϕ2,2 has image of codimension ≤ 1 by (†), it follows that ψ2,2 has
cokernel of rank ≤ 1. We consider the exact sequence 0 → Im(ψ2,2) → E4 → Coker(ψ2,2) → 0.
By Lemma 3.5 and Lemma 3.3, one has
µ(E4) ≥ min{ν(Im(ψ2,2)), 1} ≥ min{ν(E2⊗E2), 1}
= min{2ν(E2), 1} ≥
1 2.
Next, similarly, we consider ψ4,1, then we see that
ν(E5) ≥ min{ν(E4) + ν(E1), 1} ≥
1 2.
By considering ψ5,2, we see that ν(E7) ≥ min{ν(E5) + ν(E2), 1} ≥ 34.
Finally, we consider ψ7,2, then we have ν(E9) ≥ 1.
Now ν(E9) ≥ 1 implies that there is a line bundle L of degree 1 with
an injection L → E9. In particular, H0(X, 9KX⊗f∗L∨) has a section.
Thus 9KX ≥ F . Hence
9K3
X ≥ KX2 · F = KF2 = 1.
This completes the proof. ¤
Next we consider the case with (K2
F, pg(F )) = (2, 3). In this case,
ϕ2,m−2 : R2⊗Rm−2 → Rm is surjective for m ≥ 6 by the same
argu-ment( cf. remark at p190, [2]). Also one can check that ϕ2,2 : R2⊗R2 →
R4 has cokernel of dimension ≤ 1. Thus we are able to show the
fol-lowing:
Proposition 3.10. Let X be a minimal 3-fold of general type with q(X) = 1 and the general fiber of a : X → Alb(X) is of (2, 3) type. Then the canonical volume Vol(X) ≥ 1
6.
Proof. We have ν(E2) ≥ 16. By considering ψ2,2, we have ν(E4) ≥
min{2ν(E2), 1} ≥ 13. Then consider ψ2,4, ..., ψ2,10 inductively, we get
ν(E12) ≥ 1. Hence 12KX > F . So we have
12K3
X ≥ KX2 · F = KF2 = 2.
¤ Combining Theorem 3.1, results in [4], and Theorem 2.5, we are able to get a lower bound of the canonical volume for all those irregular threefolds.
Corollary 3.11. Let V be a nonsingular projective irregular 3-fold of general type. Then Vol(V ) ≥ 1
Proof. We consider 3-folds of general type with q(V ) > 0. Then there is a non-trivial Albanese map a : V → Alb(V ). If the general fiber has dimension ≤ 1, then by Proposition 2.9 of [4], |4KV + P | is
bira-tional for general P ∈ Pic0(V ). In particular, h0(V, O
V(4KV)⊗P ) ≥ 4.
However, it’s in fact ≥ 5 because otherwise it gives a birational map onto P3, which is not of general type. By the upper semicontinuity of
cohomology, we have P4(V ) = h0(V, O(4KV)) ≥ 5. Now by Theorem
2.5 (i), we get Vol(V ) ≥ 1 22.
We now assume that the Albanese map has 1-dimensional image. Let f : V → H be an induced fibration from the Stein factorization of a. We now consider the case that g(H) ≥ 2. We can take the relative minimal model of f , say h : X → H. So X is birational to V . By Theorem 1.4 of [17], KX/H := KX − h∗(KH) is nef. In particular KX
is nef and X is minimal. Because g(H) ≥ 2, we see that KX − 2F is
nef where F is a general fiber of h. So Vol(V ) = K3
X ≥ 2KF2 ≥ 2 > 221 .
Finally, we consider the case that g(H) = 1. We remark that g(H) = 1 if and only if q(V ) = 1 because if q(V ) ≥ 2, then either its Albanese image has dimension ≥ 2 or is a curve of genus ≥ 2.
If F is not of the type (1, 2), then |4KF| is birational according to
Bombieri’s classification. By Theorem 2.8 of [4], |4KV +P | is birational
for general P ∈ Pic0(X). So we get Vol(V ) ≥ 1
22 as above.
It remains to consider the case that F is of type (1.2). By Theorem 3.1, we have Vol(V ) ≥ 1
9 > 221. ¤
4. The case P5 < 5
4.1. First let us recall Reid’s plurigenera formula (cf. [18], p413) for a minimal 3-fold X of general type (with Q-factorial terminal singulari-ties): Pm(X) = 1 12m(m − 1)(2m − 1)K 3 X − (2m − 1)χ(OX) + l(m) (4.1)
where m is an integer > 1. The correction term is l(m) :=X Q lQ(m) := X Q m−1X j=1 bj(r − bj) 2r ,
where the sum PQ runs through all baskets Q of singularities of type
1
r(a, −a, 1) with the integer a coprime to r, 0 < a < r, 0 < b < r, ab ≡ 1
(mod r), bj the smallest residue of bj mod r. One can see easily that (b, r) = 1. Note by definition that the singularity 1
r(a, −a, 1) is a
ter-minal quotient one obtained by a cyclic group action on (C3, (0, 0, 0)):
ε(x, y, z) = (εax, ε−ay, z)
where ε is a fixed r-th primitive root of 1. Reid’s Theorem 10.2 in [18] says that the above baskets {Q} of singularities are in fact virtual (!) and that one need not worry about the authentic type of all those
terminal singularities on X, though X may have non-quotient terminal singularities. Iano-Fletcher [13] has shown that the set of baskets {Q} in Reid’s formula is uniquely determined by X.
In the next context we will always study those 3-folds X with the following conditions:
(*) pg = 1, χ(O) = 0 and P5 ≤ 4.
4.2. Reid’s formula (4.1) tells:
P5 > P4 > P3 > P2 > 0
whenever χ(O) = 0. So one gets P2 = 1, P3 = 2, P4 = 3 and P5 = 4.
We shall classify those X satisfying the condition that χ(OX) =
0, P2(X) = 1, P3(X) = 2.
Recall the plurigenera formula that Pm(X) = 1 12m(m − 1)(2m − 1)K 3 X + X Q m−1X j=1 bQj(rQ− bQj) 2rQ . We introduce b0 Q := ½ bQ, if bQ≤ 12rQ; rQ− bQ, if bQ> 12rQ.
Then it’s easy to see that bQj(rQ− bQj) = b0Qj(rQ− b0Qj) for j = 1, 2.
For m = 2, 3, we have 1 = P2(X) = 1 2K 3 X + X Q b0 Q(rQ− b0Q) 2rQ = 1 2K 3 X + 1 2 X Q b0Q−1 2 X Q b02 Q rQ , 2 = P3(X) = 5 2K 3 X + 3 2 X Q b0 Q− 5 2 X Q b02 Q rQ . By solving these, we get
X Q b0 Q = 3, X Q b02 Q rQ = 1 + K3 X. (4.1)
Moreover, the inequality 1 = P2(X) ≥ 1 2K 3+X r − 1 2r > n 4
implies n < 4, where n denotes the number of baskets. Thus n = 3, 2, 1. 4.3. Three basket case. First we consider the case n = 3. Assume that the basket Qi is of the type r1i(ai, −ai, 1) with aibi ≡ 1 (mod ri)
and 0 < bi < ri for i = 1, 2, 3. Since K3 > 0, one has:
1 = P2(X) > 3 X i=1 bi(ri− bi) 2ri ≥ 3 X i=1 ri− 1 2ri
and so 1 r1 + 1 r2 + 1 r3 > 1. (4.2)
One may assume r1 ≤ r2 ≤ r3. Then clearly, the only possible solution
for (r1, r2, r3) are (2, 3, 3), (2, 3, 4), (2, 3, 5) and (2, 2, r3).
(4.3.1.) The case (r1, r2, r3) = (2, 2, r3).
By (4.1), we have b0
1 = b02 = b03 = 1 and KX3 = r13. Hence b1 = b2 = 1,
and b3 = 1 or r3− 1. Easy computation shows that P4(X) = 4 (resp.
= 5) if r3 ≥ 3 (resp. r3 = 2). And also P5(X) = 6 (resp. = 7, = 9) if
r3 ≥ 4 (resp. r3 = 3, = 2).
(4.3.2.) The case (r1, r2, r3) = (2, 3, 3).
Computation shows that K3
X = 16 and P4 = 3, P5 = 5.
(4.3.3.) The case (r1, r2, r3) = (2, 3, 4).
Since b0
1 = b02 = b03 = 1. Then one gets a possible case:
(C1). (r1, r2, r3) = (2, 3, 4), K3 = 121, P2 = 1, P3 = 2,
P4 = 3 and P5 = 4.
(4.3.4.) The case (r1, r2, r3) = (2, 3, 5).
Similarly, by b0
1 = b02 = b03 = 1. So we have found another possible case:
(C2). (r1, r2, r3) = (2, 3, 5), b3 = 1 or 4, K3 = 301,
P2 = 1, P3 = 2, P4 = 3 and P5 = 4.
4.4. Two basket case. Consider the case n = 2. We may assume r1 ≤ r2. Also recall that b01+ b02 = 3 by (4.1). We will distinquish the
following two cases. (4.4.1.) b0 1 = 1, b02 = 2. By (4.1), 1 + K3 X = r11 + 4 r2. Hence we have 5 r1 ≥ 1 r1 + 4 r2 > 1. It follows that r1 < 5.
If r1 = 2, then one gets r42 = K3 + 12 and hence r2 < 8.
Not-ing (b2, r2) = 1 and b02 ≤ 12r2, one sees that r2 = 5, 7. Whenever
(r1, r2) = (2, 5), then computation shows that KX3 = 103, P4 = 4, P5 = 7.
Whenever (r1, r2) = (2, 7), we have found the possible case:
(C3). (r1, r2) = (2, 7), b2 = 2 or 5, K3 = 141 , P2 = 1,
P3 = 2, P4 = 3 and P5 = 4.
If r1 = 3, then r42 = 23 + KX3 > 23. This gives r2 < 6. The only
possibility is (r1, r2) = (3, 5) since 2 = b02 ≤ 21r2 and (b02, r2) = 1.
Computation shows that K3
X = 152, P4 = 3 and P5 = 5.
If r1 = 4, then similarly we have r2 ≤ 5. The only possibility is
r2 = 5. So K3 = 201 and P3 = 2, P4 = 3, P5 = 4. We have found the
possible case:
(C4). (r1, r2) = (4, 5), b2 = 2 or 3, K3 = 201 , P2 = 1,
(4.4.2.) b0 1 = 2, b02 = 1. By (4.1), 1 + K3 X = r41 + 1 r2. Hence we have 5 r1 ≥ 1 r1 + 4 r2 > 1. It
follows that r1 < 5. However, 2 = b01 ≤ 12r1 and (2, r1) = 1 gives a
contradiction.
4.5. One basket case. By (4.1), one has b0 = 3 and 9
r = 1 + KX3 > 1.
Hence r < 9. Moreover b0 ≤ r
2 and (b0, r) = 1, so it follows that r = 7, 8.
If r = 7, one gets K3
X = 27, P4 = 4, P5 = 7.
If r = 8, one gets K3
X = 18, P4 = 3, P5 = 5.
We summarize all the possible cases with Pg = 1, P5 < 5:
Corollary 4.6. Let X be a minimal projective 3-fold of general type with χ(OX) = 0, Pg(X) = 1 and P5(X) = 4. Then X has at most 3
baskets of singularities of type 1
r(a, −a, 1) and one of the following 4
situations occurs:
(C1). (r1, r2, r3) = (2, 3, 4), K3 = 121 ;
(C2). (r1, r2, r3) = (2, 3, 5), b3 = 1 or 4, K3 = 301;
(C3). (r1, r2) = (2, 7), b2 = 2 or 5, K3 = 141 ;
(C4). (r1, r2) = (4, 5), b2 = 2 or 3, K3 = 201 .
Example 1.2 shows that the situation (C2) really do occur. We give another example to show the existence of (C3), (C4).
Example 4.7. (1) ([10], p153) The canonical hypersurface X12,15⊂ P(1, 3, 4, 5, 6, 7)
has two terminal singularities: 1 × 1
7(4, −4, 1), 1 × 12(1, −1, 1). The
canonical volume is 1
14. This example corresponds to (C3).
(2) ([10], p151) The canonical hypersurface X21 ⊂ P(1, 3, 4, 5, 7)
has two terminal singularities: 1 × 1
4(1, −1, 1), 1 × 15(3, −3, 1). The
canonical volume is 1
20. This example corresponds to (C4).
It is interesting to ask:
Question 4.8. Does (C1) really occur? 4.9. Proof of Theorem 1.1.
Proof. Let X be a minimal projective 3-fold of general type (admitting at worst Q-factorial terminal singularities) with χ(OX) ≤ 0. Recall
that one has
χ(OX) = 1 − q + h2(OX) − pg
where the irregularity q := h1(O
X) and the geometric genus pg :=
h3(O
X). Since Vol(X) ≥ 13 whenever pg ≥ 2 by [5], and Vol(X) ≥ 221
whenever q > 0 by Corollary 3.11, we may assume, from now on, that pg ≤ 1 and q = 0. Therefore the assumption χ(OX) ≤ 0 implies pg = 1,
h2(O
Whenever P5(X) ≥ 5, Theorem 2.5 (ii) and (iii) says Vol(X) > 251.
Whenever P5(X) ≤ 4, pg > 0 and χ(OX) = 0, Corollary 4.6 says that
Vol(X) ≥ 1
30. Furthermore Vol(X) = 301 implies that X corresponds
exactly to the situation (C2) in the list of Corollary 4.6. This completes
the proof. ¤
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1981. Adv. Studies in Math. 1, Kinokunya-North-Holland Publ. 1983, 329-353 Taida Institute for Mathematical Sciences, National Center for Theoretical Sci-ences, Taipei Office, and Department of Mathematics, National Taiwan University, Taipei, 106, Taiwan
E-mail address: [email protected]
Institute of Mathematics and Key Laboratory of Mathematics for Nonlinear Sciences (Ministry of Education), Fudan University, Shanghai, 200433, People’s Republic of China
