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.