Dynamical Degree Comparison Conjecture

• The Dynamical Degree Comparison Conjecture is a framework that equates arithmetic degrees with first dynamical degrees for dominant rational self-maps on smooth projective varieties.
• It includes multiple formulations, such as the Kawaguchi–Silverman Conjecture, asserting that for points with Zariski-dense orbits, the arithmetic complexity matches the system’s maximal geometric growth.
• Key techniques involve canonical heights, nef-divisor methods, and spectral gap arguments, with proven cases in abelian varieties, surfaces, and affine dynamics.

The Dynamical Degree Comparison Conjecture most commonly denotes the Kawaguchi–Silverman prediction that, for a dominant rational self-map of a smooth projective variety over $\overline{\mathbf Q}$, the arithmetic degree of a point with Zariski-dense forward orbit equals the first dynamical degree of the map. In the literature, the same label is also used for a comparison between cohomological and numerical dynamical degrees, and, in the affine-space setting of Dang–Favre, for the assertion that the first dynamical degree is an algebraic integer of bounded degree. These formulations are distinct, but all compare asymptotic growth rates arising from iteration and all treat dynamical degree as the geometric benchmark against which other invariants are measured (Silverman, 2015, Truong, 2016, Shao et al., 18 Sep 2025).

1. Basic invariants and the arithmetic–geometric comparison

For a smooth projective variety $X$ of dimension $d$, a dominant rational map

$f\colon X\dashrightarrow X,$

and an ample divisor $H$, the first dynamical degree is defined by

$$\delta(f)=\lim_{n\to\infty}\bigl((f^n)^*H\cdot H^{d-1}\bigr)^{1/n}.$$

The limit exists, is independent of the choice of $H$, and is a birational invariant. It measures the exponential growth of pull-backs of ample classes and hence the geometric complexity of iteration (Silverman, 2015).

On the arithmetic side, if $h_H$ is a Weil height associated to $H$ and $P$ has well-defined forward orbit, one considers

$X$0

When these agree, the common value

$X$1

is the arithmetic degree. Early formulations on global fields emphasized the upper arithmetic degree and proved the fundamental inequality

$X$2

together with a uniform height-growth estimate

$X$3

where $X$4 depends on $X$5, the height, the map, and $X$6 (Kawaguchi et al., 2012).

The guiding principle is that $X$7 measures the maximal geometric rate available to the system, while $X$8 measures the rate actually realized along a single orbit. The conjectural equality for dense orbits asserts that arithmetic complexity reaches geometric complexity whenever the orbit is sufficiently generic.

2. The Kawaguchi–Silverman formulation

In its standard form, the Kawaguchi–Silverman Conjecture states that if $X$9 is a smooth projective variety, $d$0 is dominant, and $d$1 has Zariski-dense forward orbit, then

$d$2

Many formulations also include the existence of the limit $d$3, its algebraicity, and finiteness of the set of all arithmetic degrees arising from $d$4 (Matsuzawa et al., 2017, Kawaguchi et al., 2012).

An early projective-space version, formulated for dominant rational self-maps

$d$5

predicted that the set

$d$6

is a finite set of algebraic integers and that $d$7 whenever the orbit of $d$8 is Zariski dense. The same work introduced a canonical height

$d$9

with $f\colon X\dashrightarrow X,$0 (Silverman, 2011).

The conjecture has also been extended from points to higher-dimensional cycles. For a codimension-$f\colon X\dashrightarrow X,$1 subvariety $f\colon X\dashrightarrow X,$2, one defines a $f\colon X\dashrightarrow X,$3-th arithmetic degree $f\colon X\dashrightarrow X,$4 using arithmetic intersections on the graphs of iterates, and one compares it with the $f\colon X\dashrightarrow X,$5-th dynamical degree

$f\colon X\dashrightarrow X,$6

The higher Kawaguchi–Silverman Conjecture predicts that if the forward orbit of $f\colon X\dashrightarrow X,$7 is Zariski dense in $f\colon X\dashrightarrow X,$8, then

$f\colon X\dashrightarrow X,$9

while a related product-formula conjecture predicts

$H$0

In particular, the case $H$1 recovers the original pointwise conjecture (Dang et al., 2019).

3. Proven cases and principal techniques

Substantial classes of maps are now known to satisfy the arithmetic version of the comparison conjecture.

Setting	Statement	Source
Morphisms with $H$2; regular affine automorphisms; positive-entropy surface automorphisms; monomial maps	Full conjecture verified in the forms listed as Conjecture 1 (a–c–d–e)	(Kawaguchi et al., 2012)
Abelian varieties	If $H$3 and $H$4 has Zariski-dense orbit, then $H$5	(Silverman, 2015)
Surjective endomorphisms on smooth projective surfaces	For every $H$6, the limit $H$7 exists; for dense orbit, $H$8	(Matsuzawa et al., 2017)
Product varieties under splitting hypotheses	For dense-orbit points, $H$9; if one factor is of general type, no dense orbit exists	(Sano, 2016)

The techniques vary with the geometry of the underlying variety. Canonical heights attached to eigendivisors are central when $$\delta(f)=\lim_{n\to\infty}\bigl((f^n)^*H\cdot H^{d-1}\bigr)^{1/n}.$$0 scales a nef or ample class. A general nef-height construction applies to morphisms $$\delta(f)=\lim_{n\to\infty}\bigl((f^n)^*H\cdot H^{d-1}\bigr)^{1/n}.$$1 when

$$\delta(f)=\lim_{n\to\infty}\bigl((f^n)^*H\cdot H^{d-1}\bigr)^{1/n}.$$2

yielding a canonical height $$\delta(f)=\lim_{n\to\infty}\bigl((f^n)^*H\cdot H^{d-1}\bigr)^{1/n}.$$3 with $$\delta(f)=\lim_{n\to\infty}\bigl((f^n)^*H\cdot H^{d-1}\bigr)^{1/n}.$$4, and therefore $$\delta(f)=\lim_{n\to\infty}\bigl((f^n)^*H\cdot H^{d-1}\bigr)^{1/n}.$$5 when $$\delta(f)=\lim_{n\to\infty}\bigl((f^n)^*H\cdot H^{d-1}\bigr)^{1/n}.$$6 (Kawaguchi et al., 2012).

In affine and surface settings, the arguments also use weak lower canonical heights, $$\delta(f)=\lim_{n\to\infty}\bigl((f^n)^*H\cdot H^{d-1}\bigr)^{1/n}.$$7-adic neighborhoods of periodic points at infinity, and classification-theoretic input. For surfaces, a Perron–Frobenius argument furnishes a nef $$\delta(f)=\lim_{n\to\infty}\bigl((f^n)^*H\cdot H^{d-1}\bigr)^{1/n}.$$8-divisor $$\delta(f)=\lim_{n\to\infty}\bigl((f^n)^*H\cdot H^{d-1}\bigr)^{1/n}.$$9 satisfying

$H$0

and the proof is then organized by the Enriques–Kodaira classification, with separate analyses for Kodaira dimensions $H$1, $H$2, $H$3, and $H$4 (Matsuzawa et al., 2017).

4. The abelian-variety theorem as a model case

For abelian varieties, the comparison conjecture admits a particularly explicit structure theorem. If $H$5 is an abelian variety and

$H$6

is dominant, then $H$7 is in fact a morphism and decomposes uniquely as

$H$8

where $H$9 is an isogeny and $h_H$0 is translation by a point $h_H$1. Silverman proved that if $h_H$2 has Zariski-dense forward orbit, then

$h_H$3

(Silverman, 2015).

The proof begins by factoring the minimal polynomial of the isogeny $h_H$4 as

$h_H$5

From this one defines sub-abelian varieties

$h_H$6

A resultant argument yields $h_H$7 and shows that $h_H$8 is finite, so the addition map

$h_H$9

is an isogeny. The original map lifts to a product map

$H$0

on $H$1, and one then applies invariance of $H$2 and $H$3 under finite covers together with the product formulas

$H$4

The two factors are treated differently. On $H$5, one shows that $H$6, so $H$7 is conjugate by translation to the pure isogeny $H$8. On $H$9, the unipotent relation $P$0 leads to

$P$1

The combination yields the desired equality on $P$2.

The examples in the same work illustrate the mechanism sharply. For multiplication-by-$P$3,

$P$4

one has

$P$5

for points $P$6 of infinite order. If $P$7 is torsion, the translated map $P$8 has the same values. If $P$9, then the translated unipotent case satisfies

$X$00

5. Higher arithmetic degrees and dense-orbit applications

The higher-dimensional generalization replaces points by cycles and the first dynamical degree by the full sequence $X$01. For the graph $X$02, one defines

$X$03

and for a codimension-$X$04 subvariety $X$05, the arithmetic intersections

$X$06

lead to

$X$07

The theory parallels the usual dynamical-degree formalism: it includes birational symmetry, submultiplicativity in degree $X$08, and an arithmetic log-concavity

$X$09

A lower-bound theorem gives

$X$10

and equality is known in several special cases, including birational surface maps and polarized endomorphisms (Dang et al., 2019).

A different direction is approximation rather than exact comparison. For a dominant rational self-map of a quasi-projective variety over $X$11, Matsuzawa–Xie proved that for every $X$12, the set of points $X$13 such that $X$14 exists and

$X$15

is Zariski dense; in fact it is dense in the adelic topology. They then applied this to the Zariski dense orbit conjecture: if $X$16 is a birational map on a threefold and

$X$17

then, assuming there is no non-constant invariant rational function, $X$18 admits a point with Zariski-dense orbit. In particular, this holds for birational maps on threefolds with first dynamical degree larger than $X$19 (Matsuzawa et al., 2024).

This shifts the comparison problem from an equality for a prescribed dense orbit to a broader principle: near-maximal arithmetic growth can be produced densely, and spectral gaps among dynamical degrees can force orbit denseness.

6. Cohomological, family-theoretic, and affine-space variants

A second major usage of the term concerns the comparison between pull-back on $X$20-adic cohomology and pull-back on numerical cycle classes. For a dominant regular morphism $X$21 on a smooth projective variety over an algebraically closed field, one defines

$X$22

The corresponding Dynamical Degree Comparison Conjecture asks for

$X$23

for all $X$24. Truong proved this equality for dominant regular morphisms, and therefore also $X$25. In positive characteristic, the statement follows under the weaker assumption that either $X$26 or the Fundamental Conjecture $X$27 holds for $X$28 (Truong, 2016).

Hu–Truong developed a further framework around a quantitative Lefschetz-type conjecture $X$29. In their approach, $X$30 implies trace-degree bounds and, together with the standard conjecture $X$31, yields a norm comparison conjecture whose consequences include the cohomological–numerical DDC and a generalized semisimplicity conjecture for polarized endomorphisms. They obtained applications to abelian varieties and Kummer surfaces, and also proved a comparison result for effective finite correspondences of abelian varieties (Hu et al., 2021). In a related direction, Hu–Truong–Xie proved under Standard Conjecture $X$32 a weaker form of DDC via log-concave envelopes, together with uniform bounds on Jordan-block sizes for polarized endomorphisms over $X$33 (Hu et al., 12 Mar 2025). For abelian varieties, the equality $X$34 was also proved directly, together with a parity result on eigenvalues in prime characteristic (Hu, 2019).

There is also a family-theoretic comparison problem. For a one-parameter family $X$35 of dominant rational maps on $X$36, Silverman–Call conjectured that for every $X$37, the set

$X$38

is finite. They proved that a uniform degree-ratio bound, formulated as Conjecture 8, implies this finiteness, established that conjecture for monomial maps, and exhibited both positive and negative examples for related two-family questions (Silverman et al., 2016).

Finally, in the notation of Dang–Favre, the expression “Dynamical Degree Comparison Conjecture” also refers to the affine-space assertion that for

$X$39

the first dynamical degree

$X$40

is an algebraic integer of degree at most $X$41. In dimension four, this has been proved for affine-triangular automorphisms, and, when $X$42, for quadratic automorphisms up to affine conjugacy. The same work gives additional results for permutation-elementary maps and two special quadratic maps in dimension five (Shao et al., 18 Sep 2025).

Across these variants, several open directions recur: extending the arithmetic comparison to arbitrary rational self-maps on general smooth projective varieties beyond group varieties; treating semi-abelian varieties, holomorphic endomorphisms of Calabi–Yau manifolds, or rational maps on moduli spaces; proving the degree-ratio criterion beyond monomial maps in families; clarifying “geometric independence” for exceptional sets; and extending the affine-space algebraicity statement to higher dimensions under mild hypotheses (Silverman, 2015, Silverman et al., 2016, Shao et al., 18 Sep 2025).