The minimal exponent and $k$-rationality for local complete intersections

Chen, Qianyu; Dirks, Bradley; Mustaţă, Mircea

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 849-873

We show that if $Z$ is a local complete intersection subvariety of a smooth complex variety $X$, of pure codimension $r$, then $Z$ has $k$-rational singularities if and only if $\tilde{\alpha }\left(Z\right)>k+r$, where $\tilde{\alpha }\left(Z\right)$ is the minimal exponent of $Z$. We also characterize this condition in terms of the Hodge filtration on the intersection complex Hodge module of $Z$. Furthermore, we show that if $Z$ has $k$-rational singularities, then the Hodge filtration on the local cohomology sheaf ${\mathscr{H}}_Z^r\left({𝒪}_X\right)$ is generated at level $dim\left(X\right)-\lceil \tilde{\alpha }\left(Z\right)\rceil -1$ and, assuming that $k\ge 1$ and $Z$ is singular, of dimension $d$, that ${\mathscr{H}}^k\left({\underline{\Omega }}_Z^{d-k}\right)\ne 0$. All these results have been known for hypersurfaces in smooth varieties.

Nous montrons que si $Z$ est une sous-variété localement intersection complète d’une variété complexe lisse $X$, de codimension pure $r$, alors $Z$ possède des singularités $k$-rationnelles si et seulement si $\tilde{\alpha }\left(Z\right)>k+r$, où $\tilde{\alpha }\left(Z\right)$ est l’exposant minimal de $Z$. Nous caractérisons également cette condition en termes de filtration de Hodge sur le module de Hodge associé au complexe d’intersection de $Z$. De plus, nous montrons que si $Z$ est à singularités $k$-rationnelles, alors la filtration de Hodge sur le faisceau de cohomologie locale ${\mathscr{H}}_Z^r\left({𝒪}_X\right)$ est engendré au niveau $dim\left(X\right)-\lceil \tilde{\alpha }\left(Z\right)\rceil -1$ et, si de plus $k\ge 1$ et $Z$ est singulière, de dimension $d$, que ${\mathscr{H}}^k\left({\underline{\Omega }}_Z^{d-k}\right)\ne 0$. Tous ces résultats sont connus pour les hypersurfaces dans les variétés lisses.

Classification : 14F10, 14B05, 14J17, 32S35
Keywords: Minimal exponent, higher rational singularities, higher Du Bois singularities, Hodge modules, V-filtration
Mots-clés : Exposant minimal, singularités rationnelles supérieures, singularités de Du Bois supérieures, modules de Hodge, V-filtration

[BK81] Brylinski, J.-L.; Kashiwara, M. Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math., Volume 64 (1981), pp. 387-410 | Zbl | DOI | MR

[BMS06] Budur, N.; Mustaţă, M.; Saito, M. Bernstein-Sato polynomials of arbitrary varieties, Compositio Math., Volume 142 (2006) no. 3, pp. 779-797 | DOI | Zbl | MR

[CD23] Chen, Q.; Dirks, B. On $V$-filtration, Hodge filtration and Fourier transform, Selecta Math. (N.S.), Volume 29 (2023) no. 4, 50, 76 pages | MR

[CDM24] Chen, Q.; Dirks, B.; Mustaţă, M. The minimal exponent of cones over smooth complete intersection projective varieties, 2024 | arXiv

[CDMO24] Chen, Q.; Dirks, B.; Mustaţă, M.; Olano, S. $V$-filtrations and minimal exponents for local complete intersections, J. Reine Angew. Math., Volume 811 (2024), pp. 219-256 | Zbl | MR

[DB81] Du Bois, P. Complexe de de Rham filtré d’une variété singulière, Bull. Soc. math. France, Volume 109 (1981) no. 1, pp. 41-81 | DOI | Zbl | MR | Numdam

[Eis95] Eisenbud, D. Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Math., 150, Springer-Verlag, New York, 1995 | MR

[FL22a] Friedman, R.; Laza, R. Deformations of singular Fano and Calabi-Yau varieties, 2022 | arXiv

[FL22b] Friedman, R.; Laza, R. Higher Du Bois and higher rational singularities, 2022 (with an appendix by M. Saito, to appear in Duke. Math. J.) | arXiv

[FL24] Friedman, Robert; Laza, Radu The higher Du Bois and higher rational properties for isolated singularities, J. Algebraic Geom., Volume 33 (2024) no. 3, pp. 493-520 | DOI | Zbl | MR

[GNAPGP88] Guillén, F.; Navarro Aznar, V.; Pascual-Gainza, P.; Puerta, F. Hyperrésolutions cubiques et descente cohomologique, Lect. Notes in Math., 1335, Springer, 1988 | DOI

[HTT08] Hotta, R.; Takeuchi, K.; Tanisaki, T. D-modules, perverse sheaves, and representation theory, Birkhäuser, Boston, 2008 | DOI | MR

[JKSY22] Jung, S.-J.; Kim, I.-K.; Saito, M.; Yoon, Y. Higher Du Bois singularities of hypersurfaces, Proc. London Math. Soc. (3), Volume 125 (2022) no. 3, pp. 543-567 | DOI | Zbl | MR

[Kas83] Kashiwara, M. Vanishing cycle sheaves and holonomic systems of differential equations, Algebraic geometry (Tokyo/Kyoto, 1982) (Lect. Notes in Math.), Volume 1016, Springer, Berlin, 1983, pp. 134-142 | DOI | Zbl | MR

[Mal83] Malgrange, B. Polynômes de Bernstein-Sato et cohomologie évanescente, Analyse et topologie sur les espaces singuliers (Luminy, 1981) (Astérisque), Volume 101, Société Mathématique de France, Paris, 1983, pp. 243-267 | Zbl | Numdam | MR

[MOPW23] Mustaţă, M.; Olano, S.; Popa, M.; Witaszek, J. The Du Bois complex of a hypersurface and the minimal exponent, Duke Math. J., Volume 172 (2023) no. 7, pp. 1411-1436 | DOI | Zbl | MR

[MP20a] Mustaţă, M.; Popa, M. Hodge filtration, minimal exponent, and local vanishing, Invent. Math., Volume 220 (2020) no. 2, pp. 453-478 | Zbl | DOI | MR

[MP20b] Mustaţă, M.; Popa, M. Hodge ideals for $\mathbf{Q}$-divisors, $V$-filtration, and minimal exponent, Forum Math. Sigma, Volume 8 (2020), e19, 41 pages | DOI | Zbl | MR

[MP22a] Mustaţă, M.; Popa, M. Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension, Forum Math. Pi, Volume 10 (2022), e22, 58 pages | DOI | Zbl | MR

[MP22b] Mustaţă, M.; Popa, M. On $k$-rational and $k$-Du Bois local complete intersections, 2022 (to appear in Algebraic Geometry) | arXiv

[Ola23] Olano, S. Weighted Hodge ideals of reduced divisors, Forum Math. Sigma, Volume 11 (2023), 51, 28 pages | DOI | Zbl | MR

[PS08] Peters, C.; Steenbrink, J. Mixed Hodge structures, Ergeb. Math. Grenzgeb. (3), 52, Springer-Verlag, Berlin, 2008 | MR

[Sai84] Saito, M. Hodge filtrations on Gauss-Manin systems. I, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 30 (1984) no. 3, pp. 489-498 | Zbl | MR

[Sai88] Saito, M. Modules de Hodge polarisables, Publ. RIMS, Kyoto Univ., Volume 24 (1988) no. 6, pp. 849-995 | DOI | Zbl

[Sai90] Saito, M. Mixed Hodge modules, Publ. RIMS, Kyoto Univ., Volume 26 (1990) no. 2, pp. 221-333 | Zbl | DOI

[Sai93] Saito, M. On $b$-function, spectrum and rational singularity, Math. Ann., Volume 295 (1993) no. 1, pp. 51-74 | DOI | Zbl | MR

[Sai94] Saito, M. On microlocal $b$-function, Bull. Soc. math. France, Volume 122 (1994) no. 2, pp. 163-184 | DOI | Zbl | MR | Numdam

[Sai00] Saito, M. Mixed Hodge complexes on algebraic varieties, Math. Ann., Volume 316 (2000) no. 2, pp. 283-331 | DOI | Zbl | MR