no. 5
Du Bois invariants of isolated complete intersection singularities
Annales de l'Institut Fourier, Tome 47 (1997) no. 5, pp. 1367-1377.

On définit les invariants de Du Bois pour les singularités isolées d’espaces complexes. On établit leur relation avec les nombres de Hodge des groupes de cohomologie locale et évanescente. Nos résultats principaux relient le nombre de Tjurina de certaines singularités de Gorenstein avec les invariants de Du Bois et les nombres de Hodge du bord de la singularité, et expriment les nombres de Hodge de la fibre de Milnor de certaines intersections complètes de dimension trois en des termes similaires. On discute aussi la semicontinuité des invariants de du Bois sous déformation de la singularité.

We define Du Bois invariants for isolated singularities of complex spaces. We relate them to the Hodge numbers of the local and vanishing cohomology groups. Our main results express the Tjurina number of certain Gorenstein singularities in terms of Du Bois invariants and Hodge numbers of the link, and express the Hodge numbers of the Milnor fibre of certain three-dimensional complete intersections in similar terms. We also address the question of the semicontinuity of the Du Bois invariants under deformation of the singularity.

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     author = {Steenbrink, Joseph H. M.},
     title = {Du {Bois} invariants of isolated complete intersection singularities},
     journal = {Annales de l'Institut Fourier},
     pages = {1367--1377},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {5},
     year = {1997},
     doi = {10.5802/aif.1603},
     mrnumber = {99f:32058},
     zbl = {0889.32035},
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     url = {http://www.numdam.org/articles/10.5802/aif.1603/}
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Steenbrink, Joseph H. M. Du Bois invariants of isolated complete intersection singularities. Annales de l'Institut Fourier, Tome 47 (1997) no. 5, pp. 1367-1377. doi : 10.5802/aif.1603. http://www.numdam.org/articles/10.5802/aif.1603/

[1] Ph. Du Bois, Complexe de de Rham filtré d'une variété singulière, Bull. Soc. Math. France, 109 (1981), 41-81. | Numdam | Zbl

[2] G-M. Greuel, Dualität in der lokalen Kohomologie isolierter Singularitäten, Math. Ann., 250 (1980), 157-173. | MR | Zbl

[3] G.-M. Greuel und Lê Dŭng Tràng, Spitzen, Doppelpunkte und vertikale Tangenten in der Diskriminante verseller Deformationen von vollständigen Durchschnitten, Math. Ann., 222 (1976), 71-88. | Zbl

[4] A. Grothendieck, Local cohomology, Lecture Notes in Math.41, Springer-Verlag, Berlin etc., 1967. | Zbl

[5] F. Guillén, V. Navarro Aznar, P. Pascual-Gainza, F. Puerta, Hyperrésolutions cubiques et descente cohomologique, Lecture Notes in Math. 1335, Springer-Verlag, Berlin etc., 1988. | Zbl

[6] S. Ishii, On isolated Gorenstein singularities, Math. Ann., 270 (1985), 541-554. | MR | Zbl

[7] S. Ishii, Small deformations of normal singularities, Math. Ann., 275 (1986), 139-148. | MR | Zbl

[8] E.J.N. Looijenga, J.H.M. Steenbrink, Milnor number and Tjurina number for complete intersections, Math. Ann., 251 (1985), 121-124. | Zbl

[9] Y. Namikawa, Deformation theory of Calabi-Yau threefolds and certain invariants of singularities, Preprint November 1995.

[10] Y. Namikawa, J.H.M. Steenbrink, Global smoothing of Calabi-Yau threefolds, Invent. Math., 122 (1995), 403-419. | MR | Zbl

[11] J.H.M. Steenbrink, Mixed Hodge structures associated with isolated singularities, Proc. Symp. Pure Math., 40, Part 2 (1983), 513-536. | MR | Zbl

[12] J.H.M. Steenbrink, Semicontinuity of the singularity spectrum, Invent. Math., 79 (1985), 557-565. | MR | Zbl

[13] J.H.M. Steenbrink, Vanishing theorems on singular spaces, Astérisque, 130 (1985), 330-341. | Numdam | MR | Zbl

[14] D. Van Straten, J. Steenbrink, Extendability of holomorphic differential forms near isolated hypersurface singularities, Abh. Math. Sem. Univ. Hamburg, 55 (1985), 97-110. | MR | Zbl

[15] J.M. Wahl, A characterization of quasi-homogeneous Gorenstein surface singularities, Compos. Math., 55 (1985), 269-288. | Numdam | MR | Zbl

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