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{"Title": "Masur\u2013Veech volumes of Abelian strata", "Definition": "The Masur\u2013Veech volume $V_{g,\\mu,c}$ is the hypersurface measure of the area-one locus in the connected component $c$ of the stratum $\\mathcal H(\\mu)$ of holomorphic Abelian differentials on compact connected genus-$g$ Riemann surfaces with labelled zeros of orders $\\mu$ CITE{EMZ}.", "Parameters": {"g": {"type": "Z", "display": "$g$", "title": "genus", "constraints": "$g\\geq2$"}, "mu": {"type": "Symbolic", "display": "$\\mu$", "title": "zero profile", "constraints": "$\\mu=(m_1,\\ldots,m_n)$ is a non-increasing partition of $2g-2$ into positive integers"}, "component": {"type": "Symbolic", "display": "$c$", "title": "connected component", "constraints": "A component allowed by the classification in CITE{KZ}, with the labels defined in Comments"}}, "Comments": {"measure": "The relative-period lattice $H^1(X,\\{z_i\\};\\mathbb Z\\oplus i\\mathbb Z)$ has covolume one. All zeros are labelled, including zeros of equal order; sorting their orders does not divide by permutations of labels. There are no poles or extra regular marked points. The area is $A=(i/2)\\int_X\\omega\\wedge\\bar\\omega$, and the phase of $\\omega$ is retained. For $D=2g+n-1$ and $\\omega=r\\omega_1$ with $A(\\omega_1)=1$, the measure is disintegrated as $d\\nu=r^{2D-1}dr\\,d\\nu_1$. Thus the volume used here is $2D$ times the measure of the area-at-most-one cone CITE{EMZ}.", "components": "The label hyp denotes the hyperelliptic component; in $\\mathcal H(g-1,g-1)$ its hyperelliptic involution interchanges the two zeros. The labels even and odd denote the nonhyperelliptic components with even and odd spin parity, respectively. They exclude the hyperelliptic component even when it has the same spin parity. The label nonhyp denotes the nonhyperelliptic component of $\\mathcal H(g-1,g-1)$ for even $g\\geq4$, and connected denotes the sole component of any other connected stratum. Both genus-two strata use hyp CITE{KZ}.", "zeta": "The genus-two volume $V_{2,(2),\\mathrm{hyp}}=\\pi^4/120$ equals HREF{Multiple_zeta_values_of_length_2#2,2}[$\\zeta(2,2)$].", "hyperelliptic-labels": "For a hyperelliptic component with two zeros, forgetting the two labels divides the volume by $2!$. In particular, the labelled volume of $\\mathcal H(1,1)$ is $\\pi^4/135$ CITE{AEZ}. The hyperelliptic formulas use the labelled convention of CITE{CMSZ}, Section 6.5."}, "Tags": ["volume", "period"], "Keywords": ["Masur-Veech", "translation surfaces", "Abelian differentials", "Teichmuller", "Teichm\u00fcller", "flat surfaces"], "Links": {"EMZ-text": {"title": "Eskin, Masur and Zorich: full text", "url": "https://arxiv.org/pdf/math/0202134"}}, "References": {"EMZ": {"bib": "A. Eskin, H. Masur and A. Zorich, Moduli spaces of Abelian differentials: the principal boundary, counting problems, and the Siegel\u2013Veech constants. Publications Math\u00e9matiques de l\u2019IH\u00c9S 97 (2003), 61\u2013179. Sections 2.1\u20132.2 and 5.2; Table 1.", "arxiv": "math/0202134"}, "KZ": {"bib": "M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Inventiones Mathematicae 153 (2003), 631\u2013678.", "arxiv": "math/0201292", "doi": "10.1007/s00222-003-0303-x"}, "AEZ": {"bib": "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\u2019\u00c9cole Normale Sup\u00e9rieure 49 (2016), 1311\u20131386. Theorem 1.1 and Remark 1.2.", "arxiv": "1212.1660"}, "CMSZ": {"bib": "D. Chen, M. M\u00f6ller, A. Sauvaget and D. Zagier, Masur\u2013Veech volumes and intersection theory on moduli spaces of Abelian differentials. Inventiones Mathematicae 222 (2020), 283\u2013373. Sections 3.1, 6.3 and 6.5.", "arxiv": "1901.01785", "doi": "10.1007/s00222-020-00969-4"}}, "Data properties": {"type": "R", "rigour": "proven", "digits": "100", "complete": "no", "complete-note": "it holds all 25 components of all 18 positive zero profiles in genera $2\\leq g\\leq4$, covering minimal, principal and intermediate strata and every component type occurring in this range; the genus cutoff is a practical choice supported by complete low-genus source tables, and genus one is excluded because it requires a marked regular point", "rigour details": "The exact rational coefficients of $\\pi^{2g}$ are transcribed from CITE{EMZ}, Table 1, which uses numbered zeros. Each rational is converted directly to an Arb ball and multiplied by a certified power of $\\pi$. A 64-bit working guard beyond the requested 100 significant digits gives a measured minimum of 393 bits of relative accuracy. A separate 512-bit MPFI calculation is contained in the exact uncertainty interval asserted by every written decimal.\n\nAll 25 coefficients were checked visually against the source table. Twenty-three also agree exactly with the surface-dynamics transcription in its $\\zeta(2g)$ normalization; that software cites EMZ, so this is a transcription check rather than an independent derivation. Independent rational generating-series calculations from CITE{CMSZ} recover the six minimal-component values, and the hyperelliptic formulas from CITE{AEZ} and CITE{CMSZ} recover all six hyperelliptic values, covering nine distinct entries. The principal-stratum saddle-connection ratios for genera two through four agree with EMZ Table 2. The remaining general nonhyperelliptic multizero coefficients are source transcriptions. The generator, argument-selection justification, checks, controls and numerical evidence are attached."}, "Display properties": {"number-header": "$V_{g,\\mu,c}$"}, "Numbers": {"2": {"2": {"hyp": {"number": "0.8117424252833536436370027724058759270810632139390451807622321615830904621402266349176822269562749379", "comment": "The exact volume is $\\pi^{4}/120$ CITE{EMZ}."}}, "1,1": {"hyp": {"number": "0.7215488224740921276773357976941119351831673012791512717886508102960804107913125643712730906277999448", "comment": "The exact volume is $\\pi^{4}/135$ CITE{EMZ}."}}}, "3": {"4": {"hyp": {"number": "0.1430638680915631602723540838768481992361930845358700321340887183756092132205878293897650906476065084", "comment": "The exact volume is $\\pi^{6}/6720$ CITE{EMZ}."}, "odd": {"number": "0.3956334129939524432223125282520246497395956905683319407164922582239069600174280713988565469760970108", "comment": "The exact volume is $\\pi^{6}/2430$ CITE{EMZ}."}}, "3,1": {"connected": {"number": "0.3617219775944708052318285972589939654762017742339034886550786360904292205873628081360974143781458384", "comment": "The exact volume is $(16/42525)\\pi^{6}$ CITE{EMZ}."}}, "2,2": {"hyp": {"number": "0.1017343061984449139714517929790920527901817490032853561842408664004332182901957897882773977938535171", "comment": "The exact volume is $\\pi^{6}/9450$ CITE{EMZ}."}, "odd": {"number": "0.2225437948090982493125507971417638654785225759446867166530268952509476650098032901618568076740545686", "comment": "The exact volume is $\\pi^{6}/4320$ CITE{EMZ}."}}, "2,1,1": {"connected": {"number": "0.2543357654961122849286294824477301319754543725082133904606021660010830457254894744706934944846337926", "comment": "The exact volume is $\\pi^{6}/3780$ CITE{EMZ}."}}, "1,1,1,1": {"connected": {"number": "0.1978167064969762216111562641260123248697978452841659703582461291119534800087140356994282734880485054", "comment": "The exact volume is $\\pi^{6}/4860$ CITE{EMZ}."}}}, "4": {"6": {"hyp": {"number": "0.01634240488603425031540828838718509871840756495198323600796077803796521732706296898429829476517532505", "comment": "The exact volume is $\\pi^{8}/580608$ CITE{EMZ}."}, "even": {"number": "0.1586690143127615746178663092951944636458604706212108674100322295549760715179849978303663771452637189", "comment": "The exact volume is $(32/1913625)\\pi^{8}$ CITE{EMZ}."}, "odd": {"number": "0.2510193390494860848446713096271631163147401976624625050822775506631457381436872035988218075930929928", "comment": "The exact volume is $\\pi^{8}/37800$ CITE{EMZ}."}}, "5,1": {"connected": {"number": "0.2581913201651856872688047756165106339237327747385328623703426235392356163763639808445024306671813640", "comment": "The exact volume is $\\pi^{8}/36750$ CITE{EMZ}."}}, "4,2": {"even": {"number": "0.08824898638458495170320475729079953307940085074070947444298820140501217356614003251521079173194675527", "comment": "The exact volume is $\\pi^{8}/107520$ CITE{EMZ}."}, "odd": {"number": "0.1195330185949933737355577664891252934832096179345059548010845479348313038779462874280103845681395204", "comment": "The exact volume is $\\pi^{8}/79380$ CITE{EMZ}."}}, "4,1,1": {"connected": {"number": "0.1597924033301126697506588197858098541355406350860583076333942741489932360868379189575833265928254005", "comment": "The exact volume is $(11/653184)\\pi^{8}$ CITE{EMZ}."}}, "3,3": {"hyp": {"number": "0.009562641487599469898844621319130023478656769434760476384086763834786504310235702994240830765451161630", "comment": "The exact volume is $\\pi^{8}/992250$ CITE{EMZ}."}, "nonhyp": {"number": "0.1859402511477674702553120812053060120849927390092314852461315190097375838101386693324605982171059206", "comment": "The exact volume is $\\pi^{8}/51030$ CITE{EMZ}."}}, "3,2,1": {"connected": {"number": "0.1338769808263925785838246984678203287011947720866466693772146936870110603432998419193716307163162628", "comment": "The exact volume is $\\pi^{8}/70875$ CITE{EMZ}."}}, "3,1,1,1": {"connected": {"number": "0.1024737384103251836073719914198130911046182206095320185356458149209220461886986444321116185729828185", "comment": "The exact volume is $(62/5740875)\\pi^{8}$ CITE{EMZ}."}}, "2,2,2": {"even": {"number": "0.05159841969350547299584910253447241835358548507506173715580149652520217950731348073975781600524689296", "comment": "The exact volume is $(37/6804000)\\pi^{8}$ CITE{EMZ}."}, "odd": {"number": "0.06754860686227490130368759200036507470275126846819737549957121589025623161852693846843295169605801021", "comment": "The exact volume is $(31/4354560)\\pi^{8}$ CITE{EMZ}."}}, "2,2,1,1": {"connected": {"number": "0.09134314837634076976292205989210657843675268303828496712716210871353358804673062130957126887415328348", "comment": "The exact volume is $(131/13608000)\\pi^{8}$ CITE{EMZ}."}}, "2,1,1,1,1": {"connected": {"number": "0.06972759418041280134574203045198975453187227712846180696729931962865159392880200099967272433141472022", "comment": "The exact volume is $\\pi^{8}/136080$ CITE{EMZ}."}}, "1,1,1,1,1,1": {"connected": {"number": "0.05310566263841540627746413228363664133033504742915171964984210808081141598213808965025579206655222126", "comment": "The exact volume is $(377/67359600)\\pi^{8}$ CITE{EMZ}."}}}}, "Formulas": {"hyperelliptic-one": "For $g\\geq2$, $\\displaystyle V_{g,(2g-2),\\mathrm{hyp}}=\\frac{2\\pi^{2g}}{(2g+1)!}\\frac{(2g-3)!!}{(2g-2)!!}$ CITE{CMSZ}. The double factorial $k!!$ is the product of positive integers at most $k$ with the same parity as $k$, and $0!!=1$.", "hyperelliptic-two": "For $g\\geq2$, $\\displaystyle V_{g,(g-1,g-1),\\mathrm{hyp}}=\\frac{8\\pi^{2g}}{(2g+2)!}\\frac{(2g-2)!!}{(2g-1)!!}$ CITE{CMSZ}."}, "Similar tables": [{"table": "HREF{Multiple_zeta_values_of_length_2}", "relation": "the entry $\\zeta(2,2)$ equals the volume of $\\mathcal H(2)$ in the labelled EMZ convention"}, {"table": "HREF{Weil-Petersson_volumes_of_the_moduli_spaces_of_hyperbolic_surfaces}", "relation": "volumes of moduli spaces of Riemann surfaces for the Weil-Petersson measure"}], "Programs": {"sage": {"language": "Sage", "code": "# Download the attached generate.py into the current directory.\nfrom generate import volume\nvolume(4, (3, 3), 'hyp', digits=100)"}}}