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Pathologies of the volume function
John Lesieutre(with Yuan Fu, Heming Liu, Junsheng Shi)
July 8, 2020
Iitaka dimension
I X smooth projective variety over k = k , D a divisor.
I Theorem: There are constants Ci such that for largedivisible m, C1m
κ < h0(X ,mD) < C2mκ.
I This κ is the Iitaka dimension of D.
I Key case is Kodaira dimension D = KX .
Numerical invariance
I This is not a numerical invariant: it can happen thatD ≡ D ′ but different Iitaka dimension.
I Basic example: torsion and non-torsion classes on anelliptic curve.
I On threefold: two numerically equivalent divisors, onerigid and one which moves in a pencil.
I We want a numerically invariant version ν
Numerical dimension: sections
I Fix sufficiently ample A.
I Look at growth of h0(bmDc+ A), extract a number:
I
κ+σ (D) = min
{k : lim sup
m→∞
h0(bmDc+ A)
mk<∞
}κσ(D) = max
{k : lim sup
m→∞
h0(bmDc+ A)
mk> 0
}κ−σ (D) = max
{k : lim inf
m→∞
h0(bmDc+ A)
mk> 0
}I These are numerical invariants.
I Bonus: makes sense for R-divisors. Roundup vs downdoesn’t matter.
Numerical dimension: sections
I Fix sufficiently ample A.
I Look at growth of h0(bmDc+ A), extract a number:I
κ+σ (D) = min
{k : lim sup
m→∞
h0(bmDc+ A)
mk<∞
}κσ(D) = max
{k : lim sup
m→∞
h0(bmDc+ A)
mk> 0
}κ−σ (D) = max
{k : lim inf
m→∞
h0(bmDc+ A)
mk> 0
}I These are numerical invariants.
I Bonus: makes sense for R-divisors. Roundup vs downdoesn’t matter.
Volume
I Volume of D is limm→∞
h0(mD)
md/d !.
I Asymptotic growth rate of number of sections.
I Extends to a continuous function on Eff(X )
An example
I X the blow-up of P2 at 9 very general points.
I Then h0(−mKX ) = 1, but h0(−mKX + A) ∼ Cm.
I So κ(−KX ) = 0 but κσ(−KX ) = 1.
I Abundance: κ(KX ) = κσ(KX ) for any klt X .
Numerical dimension in the wild
Backstory
Conjecture (Nakayama, 2002)
Theorem (2012-2016, 2016-2019)Suppose that D is a pseudoeffective divisor and that A isample. Then there exist constants C1,C2 and a positiveinteger ν(D) so that:
C1mν(D) ≤ h0(bmDc+ A) ≤ C2m
ν(D)
I We are going to see that this is false!
I As far as I know, all uses in literature have been correctedby Fujino ’20.
Some definitions
I N1(X ) = N1(X )⊗ R is divisors on X modulo numericalequivalence.
I Eff(X ) is the cone in N1(X ) spanned by classes ofeffective divisors.
I Eff(X ) is the closure of this cone.
General remarks
I Suppose X is a variety and φ : X 99K X is apseudoautomorphism (i.e. isomorphism in codimension 1).
I φ∗ preserves h0(L), hence volume, as well as variouspositive cones.
I Look at φ∗ : N1(X )→ N1(X ), assume λ1 > 1.
I Perron–Frobenius: the eigenvector D+ lies on theboundary of Eff(X ) (and Mov(X )).
First example
I Let X be (1, 1), (1, 1), (2, 2) complete intersection inP3 × P3.
I This is a smooth CY3, Picard rank 2.
I Studied by Oguiso in connection withKawamata–Morrison conj.
The example
I It has some birational automorphisms coming fromcovering involutions.
I Action on N1(X ) given by
τ ∗1 =
(1 60 −1
), τ ∗2 =
(−1 06 1
), φ∗ =
(35 6−6 −1
)I Composition has infinite order: λ = 17 + 12
√2.
I Nef cone bounded by H1, H2.
I Psef cone bounded by (1±√
2)H1 + (1∓√
2)H2.
I Let D1 = c1H1 + c2H2 be divisor in this class.
Cones
The trick
I Here’s the good news. For any line bundle whatsoever onX , you can compute h0(D).
I Pull it back some number of times, it’s ample, and thencompute h0 for ample using Riemann-Roch+Kodairavanishing!
I HRR on CY3:
χ(D) =D3
6+
D · c2(X )
12.
Let’s do it!
I Suppose our ample is A = M1D1 + M2D2.
I We need to compute h0(bmDc+ A).
I How many times to pull back? Looks like a mess, butthere’s an invariant quadratic form: the product of thecoefficients when you work in the eigenbasis.
φ∗ =
(λ 00 λ−1
)
Let’s do it! cont
I So mD1 + A is (m + M1,M2).
I The pullback that’s ample has the two coefficientsroughly equal:(√
(m + M1)M2,√
(m + M1)M2
)I Then h0(bmD1c+ A) ≈ Cm3/2.
Wehler
I Let X be a hypersurface in (P1)4 of bidegree (2, 2, 2, 2).This is a Calabi–Yau threefold, satisfying theKawamata–Morrison conjecture [Cantat–Oguiso].
I Bir(X ) ∼= (Z/2Z)∗4
I What do the positive cones of divisors look like?
Two kinds of divisors on the psef boundary
I There are some distinguished semiample classes:π∗i (OP1(1)) + π∗j (OP1(1)).
I Similarly, orbit of these classes under Bir(X ). We’ll callthese “semiample type”. (NB: they aren’t semiample.)
I All these have vol(D + tA) ∼ Ct.
I There are also many eigenvector classes:
I All these have vol(D + tA) ∼ Ct3/2.
First picture: the eigenvectors
Plotting those divisors
Blue are eigenvectors, red are semiample type.
Plotting those divisors
Blue are eigenvectors, red are semiample type.
Volume near the boundary
I There are some “circles” on the boundary of Eff(X ) onwhich both eigenvectors and semiample type are dense.
I The former have vol(D + tA) ∼ Ct3/2, the latter havevol(D + tA) ∼ Ct.
I How is this possible?
I Because the volume function is so easy to computenumerically, we can plot it!
Volume near the boundary
Cross-section: ε = 0.002
I y -axis is log h0(vol(Dt+εA))log ε
, the estimated exponent of the
volume (not all the way down to 1 because of constants)
Cross-section: ε = 0.0005
Cross-section: ε = 5 · 10−5
Cross-section: ε = 5 · 10−6
Strange boundary classes
Theorem (probably)There exist classes D on Eff(X ) for which κ−σ (D) = 3/2 andκ+σ (D) = 2.
I This means roughly that h0(bmDc+ A) oscillatesbetween C1m
3/2 and C2m2 as m increases.
Strategy
I If D is “close” to one of our semiample type classes, thenvol(D + εA) will be larger than expected as long as ε isnot too small.
I But as ε shrinks, D isn’t close enough any more.
I Goal: find divisors which are unusually close to semiampletype at many different scales
I Analogous to Liouville type numbers∑∞
k=11
10k!which are
close to rationals at many scales.
I Behavior of h0(bmDc+ A) depends in a subtle way onDiophantine properties of D.
Location of the “hills”
(This is roughly a plot of all classes whose estimated exponentis < 1.4)
A divisor with some oscillation
To get these divisors. . .
I Choose a pseudoautomorphism φ : X 99K X with a λ = 1Jordan block whose leading eigenvector is a semiamplewith exponent 1.
I Choose a pseudoautomorphism ψ : X 99K X with λ > 1,so leading eigenvector D1 has 3/2 exponent.
I Choose a rapidly increasing sequence ni = n! (or maybeeven more).
I Fix an ample A and consider the sequence of divisors:
Sequence of divisors
I Consider the following sequence of divisors, normalized tohave length 1:I (φ∗)n1A
I (φ∗)n1(ψ∗)n2A
I (φ∗)n1(ψ∗)n2(φ∗)n3A
I · · ·
I After rescaling, these converge to a limit D+.
Why it works
I (φ∗)n3A will be within ε of a semiample type if n3 � 0.
I (φ∗)n1(ψ∗)n2(φ∗)n3A is still within ε of a semiample ifn3 � n2.
I That class is also very close to the limit D+.
I So D+ is close enough to a divisor of semiample type toget an increase in the volume at a suitable ε scale.
I Similarly, D+ is close enough to eigenvectors at suitablescales to get exponent near 3/2.