Khan, Adeel A.

Algebraic K-theory of quasi-smooth blow-ups and cdh descent

Delarue, François

Lagoutière, Frédéric

Vauchelet, Nicolas

Convergence analysis of upwind type schemes for the aggregation equation with pointy potential

Jacoboni, Lison

Kropholler, Peter

Soluble groups with no $\protect \mathbb{Z}\wr \protect \mathbb{Z}$ sections

Chhaibi, Reda

Ekren, Ibrahim

A Hörmander condition for delayed stochastic differential equations

Huybrechts, Daniel

Annales Henri Lebesgue
3 (2020) 273-274
DANIEL HUYBRECHTS
CORRIGENDUM: FINITENESS OF
POLARIZED K3 SURFACES AND
HYPERKÄHLER MANIFOLDS
CORRIGENDUM : FINITUDE DES
SURFACES K3 ET DES VARIÉTÉS
HYPERKÄHLERIENNES POLARISÉES
In the proof of Proposition 2.8 in [Huy18] we consider isometric embeddings
ϕ : T (S0) 
 //T (S0)⊕ Z · e
with ϕ(σ) ∈ C · σ⊕C · σ̄⊕C · e. At this point Lemma 2.5 is evoked, which, however,
assumes the more restrictive and unrealistic condition (2.1) ϕC(σ) = λ · σ + µ · e, i.e.
ϕC(σ) ∈ C · σ ⊕ C · e.
This is fixed by instead using the following lemma which is proved with essentially
the same techniques as the original [Huy18, Lemma 2.5].
Lemma 0.1. — Assume that σ ∈ T ⊗ C defines a general Hodge structure of K3
type on T such that Kσ is a subfield of a CM field K. Then there exist at most
finitely many isometric embeddings ϕ : T   //T ⊕ Z · e satisfying
(0.1) ϕC(σ) = λ · σ + λ′ · σ̄ + ν · e.
Proof. — Consider an isometric embedding ϕ satisfying (0.1). We think of ϕ in
terms of the integral matrix (aij|bj), where ϕ(γi) =
∑
aij · γj + bi · e with γi a
basis of T (S) as above. Then writing σ = ∑µi · γi the image ϕC(σ) corresponds
DOI: https://doi.org/10.5802/ahl.32
274 D. HUYBRECHTS
to (aij|bj) · (µi) and (0.1) becomes the system of equations
∑
aijµj = λµi + λ′µ̄i,
i = 1, . . . , n, and ∑ bjµj = ν. This shows λ, λ′, ν ∈ K. Indeed, there is at most one
solution λ, λ′, which then is contained in K, unless det
(
µi µ̄i
µj µ̄j
)
= 0, i.e. µiµ̄j ∈ R,
for all i 6= j. But then 1 = σ1 =
∑(µiγi.γ1) implies µ̄j = ∑µiµ̄j(γi.γ1) ∈ R for all
j, which would yield the contradiction σ = σ̄. Eventually use that Kσ = Q(µi) and
that the CM field K is closed under complex conjugation.
Next observe that, as aij, bj ∈ Z, there exists an N ∈ Z independent of ϕ such that
Nλ,Nλ′, Nν ∈ OK . As ϕ is an isometry, one also has (σ.σ̄) = (ϕ(σ).ϕ(σ̄)), which
translates into
(0.2) |λ|2 + |λ′|2 + |ν|2(d/(σ.σ̄)) = 1.
As any embedding g : K   //C commutes with complex conjugation, g applied
to (0.2) also shows |g(λ)|2 + |g(λ′)|2 + |g(ν)|2(d/g(σ.σ̄)) = 1. Observe that g(σ.σ̄) > 0
and that, therefore, the last summand is non-negative. Indeed, choose z ∈ C such that
(zσ, z̄σ̄) = 1. Then |g(z)|2 g(σ.σ̄) = (g(zσ), g(z̄σ̄)) = 1, as g commutes with complex
conjugation. Hence, forNλ,Nλ′ ∈ OK one has |g(Nλ)| 6 N and |g(Nλ′)| 6 N for all
g : K   //C. By Minkowski theory there are only finitely many such Nλ,Nλ′ ∈ OK .
As there are only finitely many possibilities for λ and λ′, it suffices to show that they
determine ϕ essentially uniquely. Indeed, if ϕ and ϕ′ are both isometric embeddings
satisfying (0.1) with the same λ, λ′, then σ ∈ Ker(ψ − ψ′)⊗C, where ψ, ψ′ : T //T
are the compositions of ϕ, ϕ′ with the projection to T . Hence, using the assumption
that the Hodge structure is general, one finds ψ = ψ′. Using that ϕ and ϕ′ are both
isometric embeddings allows one to conclude. 
BIBLIOGRAPHY
[Huy18] Daniel Huybrechts, Finiteness of polarized K3 surfaces and hyperkähler manifolds, Ann.
Henri Lebesgue 1 (2018), 227–246. ↑273
Manuscript received on 28th August 2019,
accepted on 5th October 2019.
Recommended by Editor Chenyang Xu.
Published under license CC BY 4.0.
This journal is a member of Centre Mersenne.
Daniel HUYBRECHTS
Mathematisches Institut, Universität Bonn,
Endenicher Allee 60, 53115 Bonn, Germany
huybrech@math.uni-bonn.de
ANNALES HENRI LEBESGUE


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Algebraic K-theory of quasi-smooth blow-ups and cdh descent

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Soluble groups with no \protect \mathbb{Z}\wr \protect \mathbb{Z} sections

A Hörmander condition for delayed stochastic differential equations

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