Algebraic Geometry3 (2) (2016) 261–263 doi:10.14231/AG-2016-012
Corrigendum
Direct images of relative pluricanonical bundles
(Algebraic Geometry 3, no. 1 (2016), 50-62) Osamu Fujino
Abstract
We replace Step 3 of the proof of the local freeness of f∗0ωX⊗m0/Y0 in Theorem 1.6 in [Direct images of relative pluricanonical bundles, Algebr. Geom.3(2016), no. 1, 50–62]
because we misused Abramovich–Karu’s lemma on the base change of weakly semistable morphisms.
In this short note, we will freely use the notation in the proof of the local freeness off∗0ωX⊗m0/Y0
in Theorem 1.6 in [Fuj16].
Observation. We put Σ = Y0\UY0. Let P be a closed point of Y0. Let H be a general very ample Cartier divisor onY0 passing throughP. We assume dimY =n>3 and that the number of irreducible components of Σ passing throughP is equal to n. Then Supp Σ|H is not a simple normal crossing divisor on H at P. Therefore, we can not apply [AbK00, Lemma 6.2] to the base change ofX†→ Y0 by H ,→Y0. If the number of the irreducible components of Σ passing through P is less than n, then Σ|H is a simple normal crossing divisor on H sinceH is general.
In this case, by [AbK00, Lemma 6.2], the base change XH† =X†×Y0 H → H of X† → Y0 by H ,→Y0 is weakly semistable.
By the above observation, we see that the argument in Step 3 of the proof of the local freeness of f∗0ωX⊗m0/Y0 in Theorem 1.6 in [Fuj16] is not true as stated (see also the remark below). All we have to do is to replace Step 3 of that proof with the following new argument.
Step 3: Local freeness via the flat base change theorem. We take an arbitrary closed point P ∈Y0. We take general very ample Cartier divisors H1, H2, . . . , Hn−1, where n= dimY, such thatC =H1∩H2∩ · · · ∩Hn−1 is a smooth projective curve passing throughP. We put
Si =H1∩ · · · ∩Hi
for every 16i6n−1. Of course,S1 =H1 and Sn−1 =C. Thus we have a sequence of Cartier divisors
Y0 =:S0 ⊃S1⊃ · · · ⊃Sn−1 =C .
Received 12 February 2016, accepted in final form 22 February 2016.
2010 Mathematics Subject Classification14D06 (primary), 14E30 (secondary).
Keywords:local freeness, weakly semistable morphisms.
This journal is cFoundation Compositio Mathematica2016. This article is distributed with Open Access under the terms of theCreative Commons Attribution Non-Commercial License, which permits non-commercial reuse, distribution, and reproduction in any medium, provided that the original work is properly cited. For commercial re-use, please contact theFoundation Compositio Mathematica.
The author was partially supported by Grant-in-Aid for Young Scientists (A) 24684002 from JSPS.
O. Fujino We put XS†
i =X†×Y0 Si for every i. Then, by adjunction, XS†
i is Gorenstein for every i. Note that X† is Gorenstein and that XS†
i+1 is a Cartier divisor on XS†
i for every i. By [AbK00, Lemma 6.2], the morphism XS†
i → Si is weakly semistable outside P for every 1 6i 6 n−1.
In particular, XS†
i is normal outside (f†)−1(P) for every 16i6n−1. Since XS†
i is Gorenstein and codimX†
Si(f†)−1(P) > 2 for every 1 6 i 6 n−2, the scheme XS†
i is normal for every 1 6 i 6 n−2. Note that f†:X† → Y0 is equidimensional. By [Kar00, Lemma 2.12(2)], the morphism XS†n−1 → Sn−1 is weakly semistable because Supp Σ|Sn−1 = P. Therefore, XS†n−1 is normal. Thus, XS†
i is normal for every i. We note that XS†
i has only rational Gorenstein singularities outside (f†)−1(P) for every i because XS†
i → Si is weakly semistable outside P for every i. We consider the pair (XS†n−2, XS†n−1). Note thatXS†n−1 has only rational Gorenstein singularities because XS†n−1 → Sn−1 is weakly semistable (see [AbK00, Lemma 6.1]). We also note that a normal variety has only rational Gorenstein singularities if and only if it has only canonical Gorenstein singularities. By the inversion of adjunction (see [KM98, Theorem 5.50]), the pair (XS†
n−2, XS†
n−1) is plt in a neighborhood of XS†
n−1. Therefore, XS†
n−2 has only canonical Gorenstein singularities. Apply the same argument to the pair (XS†
i−1, XS†
i) for n−2 > i > 2 inductively. Then XS†
i has only canonical Gorenstein singularities for every i. We put XeSi = Xe ×Y0 Si for every i. Since (X†, XS†
1) is plt by the inversion of adjunction as above, (X,e XeS1) is also plt by the negativity lemma (see, for example, [KM98, Proposition 3.51]). Thus, XeS1 is normal (see [KM98, Proposition 5.51]). By the negativity lemma and adjunction again, XeS1 has only canonical singularities. By repeating this process n−1 times, we obtain that XeC = Xe×Y0Chas only canonical singularities. SinceXeC →Cis a dominant morphism from a normal irreducible varietyXeC onto a smooth curveC, it is flat. In particular,XeC →Cis equidimensional.
Therefore, fe: Xe → Y0 is equidimensional by the choice of C. Since Xe is Cohen–Macaulay and Y0 is smooth, feis flat (see [Har77, Chapter III, Exercise 10.9] and [AlK70, Chapter V, Proposition 3.5]). Moreover, O
Xe(mK
Xe) is flat over Y0 for every integer m since O
Xe(mK
Xe) is Cohen–Macaulay by [Fuj16, Lemma 2.4] and feis equidimensional (see [AlK70, Chapter V, Proposition 3.5]). By applying [Fuj16, Lemma 4.1] and the base change theorem (see [Har77, Chapter III, Theorem 12.11]) toXeC →C, we obtain that
dimH0 Xey,O
Xe mKX/Ye 0
|
Xey
is independent of y∈Y0 for every positive integer m. By the base change theorem (see [Har77, Chapter III, Corollary 12.9]), we obtain thatf∗0ω⊗mX0/Y0 'fe∗O
Xe(mK
X/Ye 0) is locally free for every positive integer m.
Remark. Let k be the number of the irreducible components of Σ passing through P. Then XS†
i →Si is not weakly semistable for n+ 1−k6i6n−2 because Supp Σ|Si is not a simple normal crossing divisor on Si for n+ 1−k 6 i 6 n−2. On the other hand, we always have Supp Σ|Sn−1 =P. Therefore, XS†n−1 →Sn−1 is weakly semistable by [Kar00, Lemma 2.12(2)].
Acknowledgements
The author would like to thank Professor Kenji Matsuki, whose questions and comments made him find a mistake in [Fuj16].
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Corrigendum: Direct images of relative pluricanonical bundles References
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Osamu Fujino [email protected]
Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
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