A NULLSTELLENSATZ FOR AMOEBAS

534 BOUCKSOM, FAVRE, and JONSSON
is represented by the effective zero divisor of the Jacobian determinant of the map
X π ′ → X π induced by F , assuming that this is holomorphic. Now, for each blowup
X π ,letC π be the set of nef classes α ∈ H 1,1
R
(X π) such that (α · K X ) ≤ 0.ThenC π is
a closed convex cone with compact basis and is not reduced to zero since K X is not
psef. It is, furthermore, invariant by S π . Indeed, if α ∈ H 1,1
R
(X π) is a nef class, we
have
(S π α · K X ) = (F ∗ α · K Xπ ) ≤ (F ∗ α · K X ) = (α · F ∗ K X ) ≤ (α · K X ).
We can thus assume that the nonzero eigenclasses ϑ n in the proof of Theorem 3.2
belong to C n , and we get (θ ∗ · K X ) ≤ 0.
The same argument does not work for θ ∗ since F ∗ K X ≤ K X does not hold in
general.
3.2. Spectral properties
Theorem 3.2 asserts the existence of eigenclasses for F ∗ and F ∗ with eigenvalue λ 1 .
We now further analyze the spectral properties under the assumption that λ 2 1 >λ 2.
THEOREM 3.5
Assume that λ 2 1 >λ 2. Then the nonzero nef Weil classes θ ∗ ,θ ∗ ∈ L 2 (X) such that
F ∗ θ ∗ = λ 1 θ ∗ and F ∗ θ ∗ = λ 1 θ ∗ are unique up to scaling. We have (θ ∗ · θ ∗ ) > 0 and
(θ ∗2 ) = 0. We rescale them so that (θ ∗ · θ ∗ ) = 1. LetH ⊂ L 2 (X) be the orthogonal
complement of θ ∗ and θ ∗ , so that we have the decomposition L 2 (X) = Rθ ∗ ⊕Rθ ∗ ⊕H.
The intersection form is negative definite on H, and ‖α‖ 2 :=−(α 2 ) defines a Hilbert
norm on H. The actions of F ∗ and F ∗ with respect to this decomposition are as
follows.
(i) The subspace H is F ∗ -invariant, and
⎧
F n∗ θ ∗ = λ n 1 θ ∗ ,
( ⎪⎨ λ2
) (
nθ∗
F n∗ θ ∗ = + (θ 2 ∗
λ ) λn 1
1 −
1
with h ⎪⎩
n ∈ H, ‖h n ‖=O(λ n/2
2 ),
‖F n∗ h‖=λ n/2
2 ‖h‖ for all h ∈ H.
( λ2
λ 2 1
) n
)
θ ∗ + h n
(ii) The subspace H is not F ∗ -invariant in general, but
⎧
F∗ ⎪⎨
nθ ∗ = λ n 1 θ ∗,
(
F∗ nθ λ2
) nθ ∗ =
∗ ,
λ 1
⎪⎩ ‖F∗ nh‖≤Cλn/2
2 ‖h‖ for some C>0 and all h ∈ H.