Area Siegel–Veech constants of Abelian strata

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Numbers

$g$

$\mu$

$c$

$c_{\mathrm{area}}(g,\mu,c)$

hyp:

0.3377372788077925714795982106990921296811959155741551628186967729805770698741181373040309130854061254

comment: The value is $10/(3\pi^2)$ [1], Corollary 2.

1, 1

hyp:

0.3799544386587666429145479870364786458913454050209245581710338696031492036083829044670347772210818911

comment: The value is $15/(4\pi^2)$ [1], Corollary 2.

hyp:

0.4255489712978186400642937454808560833983068536234355051515579339555271080413888530030789504876117180

odd:

0.3647562611124159771979660675550195000556915888200875758441925148190232354640475882883533861322386154

3, 1

connected:

0.3989521605917049750602753863883025781859126752719707860795855630833066637888020496903865160821359856

2, 2

hyp:

0.4728321903309096000714374949787289815536742818038172279461754821728078978237653922256432783195685755

2, 2

odd:

0.3715110066885718286275580317690013426493155071315706791005664502786347768615299510344340043939467379

2, 1, 1

connected:

0.4137281665395459000625078081063878588594649965783400744529035469012069105957947181974378685296225036

1, 1, 1, 1

connected:

0.4233778030769114021047820426977904911360705941661730791048663118435091125921980935489816089034912500

hyp:

0.5210803730177371102828086679357421429367022697429822512059893068843189078057822689833619801889123077

even:

0.4342336441814475919023405566131184524472518914524852093383244224035990898381518908194683168240935898

odd:

0.3908102797633028327121065009518066072025267023072366884044919801632391808543367017375214851416842308

5, 1

connected:

0.4221715985097407143494977633738651621014948944676939535233709662257213373426476716300386413567576567

4, 2

even:

0.4593226991785978972122535665507652963664264451808510214334276112535848150288006667334820417961523305

4, 2

odd:

0.3985299889931952343459258886249287130238111803775030921260621921170809424514594020187564774407792279

4, 1, 1

connected:

0.4307992508138306882209311358471783337778963537655428263190122201536797152185228713193507037728484859

3, 3

hyp:

0.5699316579881499643718219805547179688370181075313868372565508044047238054125743567005521658316228366

3, 3

nonhyp:

0.4179498825246433072060027857401265104804799455230170139881372565634641239692211949137382549431900802

3, 2, 1

connected:

0.4327258884724842322082352074582117911540322668293863023614552403813643707762138634207896073906765981

3, 1, 1, 1

connected:

0.4381732639371260478772609850501326319553418783709049340198213173649220654516028656353707511501186324

2, 2, 2

even:

0.4700937799621977684107921040811687750967997203261889428422401029324248405004617316529078925377950124

2, 2, 2

odd:

0.4052847345693510857755178528389105556174350986889861953824361275766924838489417647648370957024873505

2, 2, 1, 1

connected:

0.4392518234876157203498972912050215388888072833278372966811867381417238036626173865891165405872218978

2, 1, 1, 1, 1

connected:

0.4441245216322472314956716470693061505307726289800140391065862564694588468844653505548006507073090549

1, 1, 1, 1, 1, 1

connected:

0.4484873347563691143155805152022095422324899343085183776819895808313034366199745158828394585235582003

Definition

The area Siegel–Veech constant $c_{\mathrm{area}}(g,\mu,c)$ is the mean total area of maximal cylinders of circumference less than $L$, divided by $\pi L^2$, for the Masur–Veech probability measure on the area-one component $c$ of the stratum $\mathcal H(\mu)$ of holomorphic Abelian differentials [1].

Parameters

$g$

— genus ($g\geq2$)

$\mu$

— zero profile ($\mu=(m_1,\ldots,m_n)$ is a non-increasing partition of $2g-2$ into positive integers)

$c$

— connected component (A connected component allowed by the classification in [3], with the labels defined in Comments)

Formulas

(1)

Let $L_{g,\mu,c}=\sum_{j=1}^g\lambda_j$ be the sum of the nonnegative Lyapunov exponents of the real Hodge bundle for the Teichmüller flow $\operatorname{diag}(e^t,e^{-t})$, with $\lambda_1=1$ and the component probability measure used in the definition. Then $\displaystyle L_{g,\mu,c}=\frac1{12}\sum_{i=1}^n\frac{m_i(m_i+2)}{m_i+1}+\frac{\pi^2}{3}c_{\mathrm{area}}(g,\mu,c)$ [1].

(2)

For $g\geq2$, $\displaystyle c_{\mathrm{area}}(g,(2g-2),\mathrm{hyp})=\frac{g(2g+1)}{(2g-1)\pi^2}$ [1].

(3)

For $g\geq2$, $\displaystyle c_{\mathrm{area}}(g,(g-1,g-1),\mathrm{hyp})=\frac{(g+1)(2g+1)}{2g\pi^2}$ [1].

Comments

(4)

Each cylinder is counted once, without choosing an orientation; its circumference is the length of a primitive closed core geodesic. The quotient in the definition is independent of $L>0$. The zeros of the differential have positive orders $\mu=(m_1,\ldots,m_n)$, and there are no poles or added regular marked points [1].

(5)

The Masur–Veech measure is defined by the relative-period lattice $H^1(X,\{z_i\};\mathbb Z\oplus i\mathbb Z)$ of covolume one. All zeros are labelled, including equal-order zeros. The area is $A=(i/2)\int_X\omega\wedge\bar\omega=1$, and the phase of $\omega$ is retained. For $D=2g+n-1$ and $\omega=r\omega_1$ with $A(\omega_1)=1$, the radial convention is $d\nu=r^{2D-1}dr\,d\nu_1$. The restriction to each component is divided by that component's volume to obtain its probability measure [2].

(6)

The label hyp denotes the hyperelliptic component; for $\mathcal H(g-1,g-1)$ its involution interchanges the two zeros. The labels even and odd denote the nonhyperelliptic components of even and odd spin parity; they exclude the hyperelliptic component even when it has the same parity. The label nonhyp denotes the nonhyperelliptic component of $\mathcal H(g-1,g-1)$ for even $g\geq4$. The label connected denotes the sole component of any other connected stratum. Both genus-two strata use hyp [3].

(7)

The values are the area constants themselves. In [4], the symbol $c$ denotes the scaled constant $\pi^2c_{\mathrm{area}}/3$. The exact sums in [1], Appendix A, are given separately from the experimental individual Lyapunov exponents.

Programs

(P1)

Sage

# Download generate.py from the attached files.
from generate import area_constant
area_constant(4, (4, 2), 'odd', digits=100)

(P2)

Sage

# Hyperelliptic components: zeros=1 for H(2*g-2), zeros=2 for H(g-1,g-1).
import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

def hyperelliptic_area(g, zeros=1, digits=100):
    g = ZZ(g)
    if g < 2 or zeros not in (1, 2):
        raise ValueError('Use genus g >= 2 and one or two zeros.')
    if zeros == 1:
        q = QQ(g*(2*g+1)) / QQ(2*g-1)
    else:
        q = QQ((g+1)*(2*g+1)) / QQ(2*g)
    field = RealBallField(numberdb.bits(digits, losing=64))
    return field(q) / field.pi()**2

hyperelliptic_area(4, zeros=2, digits=100)

References

[1]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow. Publications Mathématiques de l’IHÉS 120 (2014), 207–333. Section 1.6, Theorem 1, Corollaries 1–2 and Appendix A. (arXiv) (doi)

[2]

A. Eskin, H. Masur and A. Zorich, Moduli spaces of Abelian differentials: the principal boundary, counting problems, and the Siegel–Veech constants. Publications Mathématiques de l’IHÉS 97 (2003), 61–179. Sections 2.1–2.2 and 5.2, Examples 14.12–14.13 and Appendix B.2. (arXiv)

[3]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Inventiones Mathematicae 153 (2003), 631–678. Theorems 1–2. (arXiv) (doi)

[4]

D. Chen and M. Möller, Non-varying sums of Lyapunov exponents of Abelian differentials in low genus. Geometry & Topology 16 (2012), 2427–2479. Proposition 4.5 and Sections 5.1–5.2 and 6.1. (arXiv)

[5]

J. S. Athreya, A. Eskin and A. Zorich, with an appendix by J. Chaika, Right-angled billiards and volumes of moduli spaces of quadratic differentials on the complex projective line. Annales Scientifiques de l’École Normale Supérieure 49 (2016), 1311–1386. Section 4.13 of the preprint states the Vorobets relation between cylinder and area constants. (arXiv)

Links

[6]

Eskin, Kontsevich and Zorich: full text

Similar tables

T560 — the Masur–Veech volumes of the same connected components provide the normalization and boundary-volume ratios used in cylinder-counting formulas

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: proven

Table is complete: no (it holds all 25 components of all 18 positive zero profiles in genera $2\leq g\leq4$, including minimal, principal and intermediate strata and every component type in this range; the genus cutoff is a practical choice supported by complete low-genus exact source tables, favoring breadth over extending only the hyperelliptic formulas, and genus one is excluded because it requires a marked regular point)

How they were obtained:

The exact rational Lyapunov sums in [1], Corollary 2 and the exact columns of Appendix A, are substituted into Theorem 1 to obtain a positive rational coefficient of $1/\pi^2$. Each coefficient is evaluated with certified Arb arithmetic. At 100 requested significant digits, the 64-bit working guard gives at least 393 bits of relative accuracy. An independent 512-bit MPFI evaluation lies inside the exact uncertainty interval asserted by every written decimal, and 160-digit recomputations agree with all 25 enclosures.

All source sums were checked against the printed exact column. Six hyperelliptic entries also agree with the genus-zero double-cover formula. Eleven entries agree with cylinder-weighted sums of the configuration constants in [2], using the Vorobets relation stated in [5]. EKZ itself uses EMZ data, so this is a source-consistency check rather than an independent derivation. Three entries additionally agree with the geometric divisor arguments of [4]; altogether, additional mathematical checks cover 16 distinct entries. The remaining nine rely on the exact published EKZ data and theorem, with transcription and interval checks. The generator, selection rationale, controls and reproducible checks are attached.