CATALOGUE 034 · PAPER 08 / 14 Overview
Schnell fiber spaces and good canonical models
Good minimal models, mixed intersections, and Schnell’s reduction
The results in manuscript order, followed by proof diagrams with cited inputs and an explanation of each argument.
Read the manuscript on GitHub1. Main results
The statements follow the manuscript’s numbering, order, hypotheses, and conclusions.
Theorem 1.1（Schnell’s zero-Kodaira fiber-space conclusion）
Suppose $X$ and $Y$ are smooth connected projective complex varieties and $f\colon X\to Y$ is surjective with connected fibers. Assume its smooth geometric generic fiber $F$ has $\kappa(F)=0$. If $H$ is an ample Cartier divisor on $Y$ and $m_0$ a positive integer such that $m_0K_X-f^*H$ is pseudo-effective, then
Corollary 1.2（Campana–Peternell）
For a smooth connected projective complex variety $X$, an effective Cartier divisor $D$ on $X$, and a positive integer $m_0$, pseudo-effectivity of $m_0K_X-D$ implies
Corollary 1.3（Schnell’s general fiber-space conclusion）
Suppose $f\colon X\to Y$ is surjective with connected fibers, with $X$ and $Y$ smooth connected projective complex varieties. Let $F$ be a very general smooth fiber. For an ample Cartier divisor $H$ on $Y$ and a positive integer $m_0$, assume $m_0K_X-f^*H$ is pseudo-effective. Then
There also exist positive integers $r,\ell_0$ such that, for every integer $\ell\ge\ell_0$,
The account below traces the arguments for these claims. No uniform values of $r,\ell_0$ are asserted in Corollary 1.3.
2. Proof of Theorem 1.1
If $\dim Y=0$, then $F=X$ and the conclusion is the hypothesis. Write $X\xleftarrow{p}W\xrightarrow{q}V$ for the common resolution, and put $d=\dim X$, $y=\dim Y>0$, and $h=fp$. In the diagram, $A$ is a very ample Cartier divisor on $V$.
1. Two outputs of model existence
Ampleness makes $f^*H$ pseudo-effective, so the numerical hypothesis implies pseudo-effectivity of $K_X$. Theorem 3.1 gives a $K_X$-negative birational contraction $X\dashrightarrow V$, with $V$ normal, projective, $\mathbb Q$-factorial and klt, and $K_V$ semiample. It also supplies the comparison below on a common resolution. This theorem is [LA, Corollary 11.2]; constructing the model is an external input.
Theorem 3.1 / §3 · pp. 7–8 ↗·[LA] Corollary 11.2 · pp. 73–74 ↗
2. Effectivity transfers the sections
Choose $D=rK_V$ Cartier and globally generated, and write $g\colon V\to Z$ for its morphism and $k=\dim Z$. Since $rE$ is effective Cartier, multiplication by its canonical section gives $H^0(V,D)\hookrightarrow H^0(X,rK_X)$. The section ratios are unchanged, hence $k\le\kappa(X)$. Independently, Lemma 2.2 uses $\kappa(F)=0$ to give $\kappa(X)\le y$. It remains to prove $k\ge y$.
3. Incompatible pairings with the same curve class
Assume $k<y$ and use the curve class $C=(q^*D)^k(q^*A)^{d-k-1}$ of Lemma 2.3. The pullback $B=m_0p^*K_X-h^*H$ is pseudo-effective, so part (1) gives $B\cdot C\ge0$. Part (2) uses $q_*E=0$ to give $E\cdot C=0$, while part (3) gives $q^*D\cdot C=0$ and $h^*H\cdot C>0$. The canonical comparison therefore yields
For strict positivity, $y>k$ supplies a cotangent direction from $h$ outside those from $gq$; directions from $A$ complete a transverse intersection. Pulling all factors of $C$ back from $V$ kills the exceptional term. Thus $k\ge y$ and $y\le k\le\kappa(X)\le y$. This is the good-model argument of Theorem 2.1; Theorem 3.1 supplies its model, completing Theorem 1.1.
Lemma 2.3(1)–(3) · pp. 5–6 ↗·Theorem 2.1, §2.3 / (2.7) · pp. 6–7 ↗
3. Proof of Corollary 1.2
Theorem 3.1 and section transfer also give $K_T\in\overline{\operatorname{Eff}}(T)\Rightarrow\kappa(T)\ge0$ for smooth projective complex $T$ (§4.1). This nonvanishing settles $\kappa(D)=0$. Figure 2 treats $\kappa(D)>0$.
1. Realize the Iitaka dimension as a base dimension
Resolve $|nD|$, take Stein factorization, and resolve the base as in [Sch22, §§4–6]. This gives $f_0\colon X_0\to Y_0$ with $\dim Y_0=\kappa(D)$ and pseudo-effective $a_0K_{X_0}-f_0^*H_0$ for an ample Cartier divisor $H_0$. On resolving the base, a sufficiently divisible big and nef pullback minus an effective divisor supplies the required ample divisor. Restriction to very general fibers, followed by nonvanishing, gives $\kappa(F_i)\ge0$ at every stage.
2. Absorb positive fiber Kodaira dimension into the base
While $\kappa(F_i)>0$, the construction in [Sch22, Lemma 7.1] gives effective $L_i\sim r_iK_{X_i}+f_i^*(b_iH_i)$ with $\kappa(L_i)=\kappa(F_i)+\dim Y_i$. The numerical hypothesis survives because
Apply the divisor-to-fiber-space reduction to $L_i$ again: $\dim Y_{i+1}=\dim Y_i+\kappa(F_i)$. These dimensions strictly increase and are bounded by $\dim X$, so some finite stage has $\kappa(F_j)=0$. Section 4.1 identifies the Kodaira dimensions of very general and geometric generic fibers, allowing Theorem 1.1 to apply. Birational invariance gives $\kappa(X)=\dim Y_j\ge\dim Y_0=\kappa(D)$.
4. Proof of Corollary 1.3
First use Corollary 1.2 to obtain equality of Kodaira dimensions. Feed that equality into the Fujita–Mori criterion, then construct a section after subtracting the original ample divisor $H$.
1. Apply Corollary 1.2 to the effective twist
After §4.1 gives $\kappa(F)\ge0$, use the construction in [Sch22, Lemma 7.1] to choose effective $L\sim aK_X+f^*(bH)$ with $\kappa(L)=\kappa(F)+\dim Y$. Since $(m_0b+a)K_X-L\sim b(m_0K_X-f^*H)$ is pseudo-effective, Corollary 1.2 gives $\kappa(X)\ge\kappa(L)$. Easy addition gives the reverse bound, hence $\kappa(X)=\kappa(F)+\dim Y$.
Corollary 1.2 / §4.3 · pp. 2, 10 ↗·[Sch22] Lemma 7.1 · p. 3 ↗
2. Convert equality into a section and adjust the twist
Apply the Fujita–Mori criterion in [PS14, Lemma 4.6] with $N=\omega_X$. It supplies a big Cartier divisor $B$ on $Y$ and $0\ne\sigma\in H^0(X,uK_X-f^*B)$. Choose $0\ne\tau\in H^0(Y,vB-H)$ using bigness; then $r=uv$ and $s=\sigma^vf^*\tau$ give a nonzero section of $rK_X-f^*H$. Ampleness of $H$ supplies $0\ne\tau_\ell\in H^0(Y,(\ell-1)H)$ for every sufficiently large $\ell$, and $s^\ell f^*\tau_\ell$ is the required section.
When $Y$ is a point, the equality is tautological and the section assertion follows from nonvanishing in §4.1 and powers of a nonzero section.
5. Which papers supply which steps
This article records the LA input to Schnell. The catalogue summary records the connections among all 14 papers. The external rows list the principal reduction tools; routine background references are omitted.
| Source result | Used in | Input and role | Editorial check |
|---|---|---|---|
| Log abundance in characteristic zeroCorollary 11.2 · pp. 73–74 | Schnell: Theorem 3.1 → Theorem 2.1; §4.1 · pp. 7–8 | For smooth projective complex varieties with pseudo-effective $K$, supplies a semiample good minimal model and the effective exceptional comparison. Used for both the main theorem and nonvanishing. | Statement, comparison, and application hypotheses matched; the full model-existence proof is not verified. |
| Source result | Used in | Input and role | Editorial check |
|---|---|---|---|
| Christian Schnell, Singular metrics and a conjecture by Campana and PeternellarXiv:2202.01295v1 · §§4–10, Lemma 7.1 · pp. 2–4 | Schnell: §4.2–4.3 · pp. 9–10 | With nonvanishing as input, passes from an effective divisor to fiber spaces and uses a positive base twist to reduce to $\kappa(F)=0$. | Reduction, construction in Lemma 7.1, and use matched; deeper cited proofs are not verified. |
| Mihnea Popa–Christian Schnell, On direct images of pluricanonical bundlesarXiv:1405.6125v2 · Lemma 4.6 · p. 14 | Schnell: §4.3 · p. 10 | The Fujita–Mori criterion turns equality of Kodaira dimensions into a pluricanonical section minus a big pullback. | Lemma statement and application with $N=\omega_X$ matched; the original Fujita–Mori proofs are not verified. |
6. Return to the sources
- Lemma 2.3 · pp. 5–6 — The construction and strict positivity of the curve class.
- Theorem 3.1 · pp. 7–8 — Where the external model-existence theorem enters.
- §4.2–4.3 · pp. 9–10 — Iteration to zero-Kodaira fibers and the final section construction.
Pinned manuscript snapshot: adc7f1241b42e322a6451854ab7e4b4c146bf78a
Page numbers refer to the linked PDFs; external arXiv versions are fixed in the dependency table.