Tautological classes of strata of exact differentials
[Classes tautologiques des strates de différentiels exacts]
Benirschke, Frederik1
1Eckhart Hall, 5734 S University Ave, Chicago, IL 60637 (USA)
Annales de l'Institut Fourier, Online first, 53 p.

Strata of exact differentials are moduli spaces for differentials on Riemann surfaces with vanishing absolute periods. Our main result is that classes of closures of strata of exact differentials inside the moduli space of multi-scale differentials lie in the divisorial tautological ring. By relating exact differentials to rational functions we obtain a new proof that classes of Hurwitz spaces are tautological and a new method for computations

Les strates des différentielles exactes sont des espaces de modules pour les différentielles sur les surfaces de Riemann avec des périodes absolues nulles. Notre résultat principal est que les classes des adhérences des strates de différentielles exactes à l’intérieur de l’espace de modules des différentielles multi-échelles appartiennent à l’anneau tautologique divisoriel. En reliant les différentielles exactes aux fonctions rationnelles, nous obtenons une nouvelle preuve que les classes des espaces de Hurwitz sont tautologiques et une nouvelle méthode pour les calculs.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3795
Classification : 14H10, 32G15, 14C25
Keywords: tautological classes, moduli spaces of Riemann surface, exact differentials
Mots-clés : classes tautologiques, espaces de modules de surfaces de Riemann, différentielles exactes

Benirschke, Frederik  1

1 Eckhart Hall, 5734 S University Ave, Chicago, IL 60637 (USA) 
Benirschke, Frederik. Tautological classes of strata of exact differentials. Annales de l'Institut Fourier, Online first, 53 p.
@unpublished{AIF_0__0_0_A84_0,
     author = {Benirschke, Frederik},
     title = {Tautological classes of strata of exact differentials},
     journal = {Annales de l'Institut Fourier},
     year = {2026},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     doi = {10.5802/aif.3795},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Benirschke, Frederik
TI  - Tautological classes of strata of exact differentials
JO  - Annales de l'Institut Fourier
PY  - 2026
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3795
LA  - en
ID  - AIF_0__0_0_A84_0
ER  - 
%0 Unpublished Work
%A Benirschke, Frederik
%T Tautological classes of strata of exact differentials
%J Annales de l'Institut Fourier
%D 2026
%V 0
%N 0
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3795
%G en
%F AIF_0__0_0_A84_0

[1] Aluffi, Paolo Chern classes of blow-ups, Math. Proc. Camb. Philos. Soc., Volume 148 (2010) no. 2, pp. 227-242 | DOI | Zbl | MR

[2] Bae, Younghan; Holmes, David; Pandharipande, Rahul; Schmitt, Johannes; Schwarz, Rosa Pixton’s formula and Abel–Jacobi theory on the Picard stack (2021) | arXiv

[3] Bainbridge, Matt; Chen, Dawei; Gendron, Quentin; Grushevsky, Samuel; Möller, Martin The moduli space of multi-scale differentials (2019) | arXiv | Zbl

[4] Benirschke, Frederik The boundary of linear subvarieties in strata of differentials, J. Eur. Math. Soc., Volume 25 (2023) no. 11, pp. 4521-4582 | DOI | Zbl | MR

[5] Benirschke, Frederik; Dozier, Benjamin; Grushevsky, Samuel Equations of linear subvarieties of strata of differentials, Geom. Topol., Volume 26 (2022) no. 6, pp. 2773-2830 | DOI | Zbl | MR

[6] Buryak, Alexandr; Rossi, Paolo; Zvonkine, Dimitri Moduli spaces of residueless meromorphic differentials and the KP hierarchy (2021) | arXiv

[7] Chen, Dawei; Grushevsky, Samuel; Holmes, David; Möller, Martin; Schmitt, Johannes A tale of two moduli spaces: logarithmic and multi-scale differentials (2022) | arXiv

[8] Chen, Dawei; Möller, Martin; Sauvaget, Adrien Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials (2019) | arXiv

[9] Costantini, Matteo; Möller, Martin; Schwab, Johannes Chern classes of linear submanifolds with application to spaces of k-differentials and ball quotients (2023) | arXiv | Zbl

[10] Costantini, Matteo; Möller, Martin; Zachhuber, Jonathan The Chern classes and the Euler characteristic of the moduli space of Abelian differentials (2020) | arXiv | Zbl

[11] Costantini, Matteo; Möller, Martin; Zachhuber, Jonathan diffstrata – a Sage package for calculations in the tautological ring of the moduli space of Abelian differentials (2020) | arXiv

[12] Delecroix, Vincent; Schmitt, Johannes; van Zelm, Jason admcycles – a Sage package for calculations in the tautological ring of the moduli space of stable curves, J. Softw. Algebra Geom., Volume 11 (2021) no. 1, pp. 89-112 | DOI | Zbl | MR

[13] Eisenbud, David; Harris, Joe The Kodaira dimension of the moduli space of curves of genus ≥23, Invent. Math., Volume 90 (1987) no. 2, pp. 359-387 | DOI | Zbl | MR

[14] Faber, Carel; Pagani, Nicola The class of the bielliptic locus in genus 3, Int. Math. Res. Not., Volume 2015 (2015) no. 12, pp. 3943-3961 | DOI | Zbl | MR

[15] Faber, Carel; Pandharipande, Rahul Relative maps and tautological classes, J. Eur. Math. Soc., Volume 7 (2005) no. 1, pp. 13-49 | DOI | Zbl | MR

[16] Fulton, William Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 2, Springer, 1998 | Zbl | DOI

[17] Fulton, William; MacPherson, Robert A Compactification of Configuration Spaces, Ann. Math., Volume 139 (1994) no. 1, pp. 183-225 | DOI | Zbl

[18] Graber, Tom; Pandharipande, Rahul Constructions of nontautological classes on moduli spaces of curves, Mich. Math. J., Volume 51 (2003) no. 1, pp. 93-109 | DOI | Zbl | MR

[19] Klingler, Bruno; Lerer, Leonardo A. Abelian differentials and their periods: the bialgebraic point of view (2022) | arXiv

[20] Lian, Carl The ℋ-tautological ring, Sel. Math., New Ser., Volume 27 (2021), 96, 74 pages | DOI | Zbl | MR

[21] Molcho, Samouil; Pandharipande, Rahul; Schmitt, Johannes The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves, Compos. Math., Volume 159 (2023) no. 2, pp. 306-354 | DOI | Zbl | MR

[22] Mullane, Scott Strata of differentials of the second kind, positivity, and irreducibility of certain Hurwitz spaces, Ann. Inst. Fourier, Volume 72 (2022) no. 4, pp. 1379-1416 | DOI | MR | Numdam | Zbl

[23] Mumford, David Towards an enumerative geometry of the moduli space of curves, Arithmetic and geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. Vol. II: Geometry (Progress in Mathematics), Volume 36, Birkhäuser, 1983, pp. 271-328 | Zbl

[24] Pervova, Ekaterina; Petronio, Carlo On the existence of branched coverings between surfaces with prescribed branch data. I, Algebr. Geom. Topol., Volume 6 (2006) no. 4, pp. 1957-1985 | MR | DOI | Zbl

[25] Sauvaget, Adrien Théorie de l’intersection sur les espaces de différentielles holomorphes et méromorphes, Ph. D. Thesis, Université Paris 6, Paris, France (2017)

[26] Sauvaget, Adrien Cohomology classes of strata of abelian differentials, Geom. Topol., Volume 23 (2019) no. 3, pp. 1085-1171 | MR | DOI | Zbl

[27] Schmitt, Johannes; van Zelm, Jason Intersections of loci of admissible covers with tautological classes, Sel. Math., New Ser., Volume 26 (2020), 79, 69 pages | DOI | MR | Zbl

[28] The Stacks Project Authors Stacks, https://stacks.math.columbia.edu, 2023

[29] Vistoli, Angelo Alexander duality in intersection theory, Compos. Math., Volume 70 (1989) no. 3, pp. 199-225 | Zbl | MR | Numdam

[30] Vistoli, Angelo Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math., Volume 97 (1989) no. 3, pp. 613-670 | DOI | Zbl | MR

Cité par Sources :