Ms.u-tokyo.ac.jp

ON IMAGES OF WEAK FANO MANIFOLDS II
Abstract. We consider a smooth projective surjective morphismbetween smooth complex projective varieties. We give a Hodge the-oretic proof of the following well-known fact: If the anti-canonicaldivisor of the source space is nef, then so is the anti-canonicaldivisor of the target space. We do not use mod p reduction ar-guments. In addition, we make some supplementary comments onour paper: On images of weak Fano manifolds.
We will work over C, the complex number field. The following the- orem is the main result of this paper. It is a generalization of [D,Corollary 3.15 (a)].
Theorem 1.1 (Main theorem). Let f : X → Y be a smooth projec-
tive surjective morphism between smooth projective varieties. Let D be
an effective Q-divisor on X such that (X, D) is log canonical, SuppD
is a simple normal crossing divisor, and SuppD is relatively normal
crossing over Y . Let ∆ be a (not necessarily effective) Q-divisor on Y .
Assume that −(KX + D) − f ∗∆ is nef. Then −KY − ∆ is nef.
By putting D = 0 and ∆ = 0 in Theorem 1.1, we obtain the following Date: 2012/1/5, version 1.20.
2010 Mathematics Subject Classification. Primary 14J45; Secondary 14N30, Key words and phrases. anti-canonical divisors, weak positivity.
Corollary 1.2. Let f : X → Y be a smooth projective surjective mor-
phism between smooth projective varieties. Assume that −KX is nef.
Then −KY is nef.
By putting D = 0 and assuming that ∆ is a small ample Q-divisor, we can recover [KMM, Corollary 2.9] by Theorem 1.1. Note that The-orem 1.1 is also a generalization of [FG, Theorem 4.8].
Corollary 1.3 (cf. [KMM, Corollary 2.9]). Let f : X → Y be a smooth
projective surjective morphism between smooth projective varieties. As-
sume that −KX is ample. Then −KY is ample.
Note that Conjecture 1.3 in [FG] is still open. The reader can find some affirmative results on Conjecture 1.4 in [FG, Section 4].
Conjecture 1.4 (Semi-ampleness conjecture). Let f : X → Y be a
smooth projective surjective morphism between smooth projective vari-
eties. Assume that −KX is semi-ample. Then −KY is semi-ample.
In this paper, we give a proof of Theorem 1.1 without mod p reduc- tion arguments. Our proof is Hodge theoretic. We use a generalizationof Viehweg’s weak positivity theorem following [CZ]. In our previouspaper [FG], we just use Kawamata’s positivity theorem. We note thatTheorem 1.1 is better than [FG, Theorem 4.1] (see Theorem 2.3 below).
We also note that Kawamata’s positivity theorem (cf. [FG, Theorem2.2]) and Viehweg’s weak positivity theorem (and its generalization in[C, Theorem 4.13]) are obtained by Fujita–Kawamata’s semi-positivitytheorem, which follows from the theory of the variation of (mixed)Hodge structure. We recommend the readers to compare the proof ofTheorem 1.1 with the arguments in [FG, Section 4]. By the Lefschetzprinciple, all the results in this paper hold over any algebraically closedfield k of characteristic zero. We do not discuss the case when the char-acteristic of the base field is positive.
Acknowledgments. The first author was partially supported by the
Grant-in-Aid for Young Scientists (A) ♯20684001 from JSPS. The sec-
ond author was partially supported by the Research Fellowships of the
Japan Society for the Promotion of Science for Young Scientists. The
authors would like to thank Professor Sebastien Boucksom for inform-
ing them of Berndtsson’s results [B]. They also would like to thank the
Erwin Schr¨
odinger International Institute for Mathematical Physics in In this section, we prove Theorem 1.1. We closely follow the argu- Lemma 2.1. Let f : Z → C be a projective surjective morphism from
a (d + 1)-dimensional smooth projective variety Z to a smooth pro-
jective curve C. Let B be an ample Cartier divisor on Z such that
Rif∗OZ(kB) = 0 for every i > 0 and k ≥ 1. Let H be a very
ample Cartier divisor on C such that Bd+1 < f ∗(H − KC) · Bd and
Bd+1 ≤ f ∗H · Bd. Then
(f∗OZ(kB))∗ ⊗ OC(lH) is generated by global sections for l > k ≥ 1. Proof. By the Grothendieck duality RHom(Rf∗OZ(kB), ω• ) ≃ Rf ∗RHom(OZ (kB), ω•Z (f∗OZ(kB))∗ ≃ Rdf∗OZ(KZ/C − kB) for k ≥ 1 and i ̸= d. We note that f∗OZ(kB) is locally free and(f∗OZ(kB))∗ is its dual locally free sheaf. Therefore, we have H1(C, (f∗OZ(kB))∗ ⊗ OC((l − 1)H)) ≃ H1(C, Rdf∗OZ(KZ/C − kB) ⊗ OC((l − 1)H))≃ Hd+1(Z, OZ(KZ − f∗KC − kB + f∗(l − 1)H)) for k ≥ 1. By the Serre duality, Hd+1(Z, OZ(KZ − f ∗KC − kB + f ∗(l − 1)H)) H0(Z, OZ(kB + f ∗KC − f ∗(l − 1)H)). (kB + f ∗KC − f ∗(l − 1)H) · Bd < 0 if l − 1 ≥ k. Thus, we obtain H0(Z, OZ(kB + f ∗KC − f ∗(l − 1)H)) = 0 H1(C, (f∗OZ(kB))∗ ⊗ OC((l − 1)H)) = 0 for k ≥ 1 and l > k. Therefore, (f∗OZ(kB))∗ ⊗ OC(lH) is generatedby global sections for k ≥ 1 and l > k.
Let us start the proof of Theorem 1.1.
Proof of Theorem 1.1. We prove the following claim.
Claim. Let π : C → Y be a projective morphism from a smooth pro-
jective curve C and let L be an ample Cartier divisor on C. Then
(−π∗KY − π∗∆ + 2εL) · C ≥ 0 for any positive rational number ε.
Let us start the proof of Claim. We fix an arbitrary positive rational number ε. We may assume that π(C) is a curve, that is, π is finite.
We consider the following base change diagram where Z = X ×Y C. Then g : Z → C is smooth, Z is smooth,Supp(p∗D) is relatively normal crossing over C, and Supp(p∗D) is asimple normal crossing divisor on Z. Let A be a very ample Cartierdivisor on X and let δ be a small positive rational number such that0 < δ ≪ ε. Since −(KX +D)−f ∗∆+δA is ample, we can take a generaleffective Q-divisor F on X such that −(KX + D) − f ∗∆ + δA ∼Q F .
Then we have KX/Y + D + F ∼Q δA − f∗KY − f∗∆. KZ/C + p∗D + p∗F ∼Q δp∗A − g∗π∗KY − g∗π∗∆. Without loss of generality, we may assume that Supp(p∗D + p∗F ) isa simple normal crossing divisor, p∗D and p∗F have no common irre-ducible components, and (Z, p∗D + p∗F ) is log canonical. Let m be asufficiently divisible positive integer such that mδ and mε are integers,mp∗D, mp∗F , and m∆ are Cartier divisors, and m(KZ/C + p∗D + p∗F ) ∼ m(δp∗A − g∗π∗KY − g∗π∗∆). Note that g : Z → C is smooth, every irreducible component ofp∗D + p∗F is dominant onto C, and the coefficient of any irreduciblecomponent of m(p∗D + p∗F ) is a positive integer with ≤ m. Therefore,we can apply the weak positivity theorem (cf. [C, Theorem 4.13]) andobtain that g∗OZ(m(KZ/C + p∗D + p∗F )) ≃ g∗OZ(m(δp∗A − g∗π∗KY − g∗π∗∆)) is weakly positive over some non-empty Zariski open set U of C. Forthe basic properties of weakly positive sheaves, see, for example, [V,Section 2.3]. Therefore, E1 := Sn(g∗OZ(m(δp∗A − g∗π∗KY − g∗π∗∆))) ⊗ OC(nmεL) ≃ Sn(g∗OZ(mδp∗A)) ⊗ OC(−nmπ∗KY − nmπ∗∆ + nmεL) is generated by global sections over U for every n ≫ 0. On the otherhand, by Lemma 2.1, if mδ ≫ 0, then we have that E2 := OC(nmεL) ⊗ Sn((g∗OZ(mδp∗A))∗) is generated by global sections because 0 < δ ≪ ε and p∗A is ample onZ. We note that E2 ≃ Sn(OC(mεL) ⊗ (g∗OZ(mδp∗A))∗). OC → E := E1 ⊗ E2 which is surjective over U . By using the non-trivial trace map Sn(g∗OZ(mδp∗A)) ⊗ Sn((g∗OZ(mδp∗A))∗) → OC, −→ OC(−nmπ∗KY − nmπ∗∆ + 2nmεL), where β is induced by the above trace map. We note that g∗OZ(mδp∗A)is locally free and Sn((g∗OZ(mδp∗A))∗) ≃ (Sn(g∗OZ(mδp∗A)))∗. (−nmπ∗KY −nmπ∗∆+2nmεL)·C = nm(−π∗KY −π∗∆+2εL)·C ≥ 0. Since ε is an arbitrary small positive rational number, we obtain π∗(−KY − ∆) · C ≥ 0. This means that −KY − ∆ is nef on Y .
Remark 2.2. In Theorem 1.1, if −(KX + D) is semi-ample, then we
can simply prove that −KY is nef as follows. First, by the Stein factor-
ization, we may assume that f has connected fibers (cf. [FG, Lemma
2.4]). Next, in the proof of Theorem 1.1, we can take δ = 0 and ∆ = 0
when −(KX + D) is semi-ample. Then
g∗OZ(m(KZ/C + p∗D + p∗F )) ≃ OC(−mπ∗KY ) is weakly positive over some non-empty Zariski open set U of C. Thismeans that −mπ∗KY is pseudo-effective. Since C is a smooth projec-tive curve, −π∗KY is nef. Therefore, −KY is nef. In this case, we donot need Lemma 2.1. The proof given here is simpler than the proofof [FG, Theorem 4.1].
We apologize the readers of [FG] for misleading them on [FG, The- orem 4.1]. A Hodge theoretic proof of [FG, Theorem 4.1] is implicitlycontained in Viehweg’s theory of weak positivity (see, for example,[V]). Here we give a proof of [FG, Theorem 4.1] following Viehweg’sarguments.
Theorem 2.3 (cf. [FG, Theorem 4.1]). Let f : X → Y be a smooth
projective surjective morphism between smooth projective varieties. If
−KX is semi-ample, then −KY is nef.
Proof. By the Stein factorization, we may assume that f has connectedfibers (cf. [FG, Lemma 2.4]). Note that a locally free sheaf E on Y isnef, equivalently, semi-positive in the sense of Fujita–Kawamata, if andonly if E is weakly positive over Y (see, for example, [V, Proposition2.9 e)]). Since f is smooth and −KX is semi-ample, f∗OX(KX/Y −KX)is locally free and weakly positive over Y (cf. [V, Proposition 2.43]).
Therefore, we obtain that OY (−KY ) ≃ f∗OX(KX/Y − KX) is nef.
Note that our Hodge theoretic proof of [FG, Theorem 4.1], which depends on Kawamata’s positivity theorem, is different from the proofgiven above and plays important roles in [FG, Remark 4.2], which isrelated to Conjecture 1.4.
2.4 (Analytic method). Sebastien Boucksom pointed out that the fol-
lowing theorem, which is a special case of [B, Theorem 1.2], implies
[FG, Theorem 4.1] and [KMM, Corollary 2.9].
Theorem 2.5 (cf. [B, Theorem 1.2]). Let f : X → Y be a proper
smooth morphism from a compact K¨
complex manifold Y . If −KX is semi-positive (resp. positive), then −KY is semi-positive (resp. positive). The proof of [B, Theorem 1.2] is analytic and does not use mod p reduction arguments. For the details, see [B].
2.6 (Varieties of Fano type). Let X be a normal projective variety. If
there is an effective Q-divisor on X such that (X, ∆) is klt and that
−(KX + ∆) is ample, then X is said to be of Fano type.
In [PS, Theorem 2.9] and [FG, Corollary 3.3], the following statement Let f : X → Y be a proper surjective morphism between normal projective varieties with connected fibers. If X is of Fano type, thenso is Y .
It is indispensable for the proof of the main theorem in [FG] (cf. [FG, Theorem 1.1]). The proofs in [PS] and [FG] need the theory of thevariation of Hodge structure. It is because we use Ambro’s canonicalbundle formula or Kawamata’s positivity theorem. In [GOST], Okawa,Sannai, Takagi, and the second author give a new proof of the aboveresult without using the theory of the variation of Hodge structure.
It deeply depends on the minimal model theory and the theory of F -singularities.
We close this paper with a remark on [D]. By modifying the proof of Theorem 1.1 suitably, we can generalize [D, Corollary 3.14] withoutany difficulties. We leave the details as an exercise for the readers.
Corollary 2.7 (cf. [D, Corollary 3.14]). Let f : X → Y be a projective
surjective morphism from a smooth projective variety X such that Y
is smooth in codimension one. Let D be an effective Q-divisor on X
such that SuppDhor, where Dhor is the horizontal part of D, is a simple
normal crossing divisor on X and that (X, D) is log canonical over
the generic point of Y . Let ∆ be a not necessarily effective Q-Cartier
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Department of Mathematics, Faculty of Science, Kyoto University, E-mail address: [email protected] Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan.
E-mail address: [email protected]

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