We define the vanishing Euler characteristic of a normal toric surface , we give a formula to compute it, and we relate this number with the second polar multiplicity of . We also present a formula for the Euler obstruction of a function and for the difference between the Euler obstruction of the space and the Euler obstruction of a function . As an application of this result we compute the Euler obstruction of a type of polynomial on a family of determinantal surfaces.
Nous définissons la caractéristique d’Euler évanescente d’une surface torique normale , nous donnons une formule pour la calculer, et nous associons ce nombre avec la seconde multiplicité polaire de . Nous présentons aussi une formule pour l’obstruction d’Euler d’une fonction et pour la différence entre l’obstruction d’Euler de l’espace et l’obstruction d’Euler d’une fonction . Comme application de ce résultat nous calculons l’obstruction d’Euler des polynômes d’un certain type sur une famille de surfaces déterminantales.
@article{AFST_2015_6_24_1_1_0, author = {Tha{\'\i}s Maria Dalbelo and Nivaldo de G\'oes Grulha Jr. and Miriam Silva Pereira}, title = {Toric surfaces, vanishing {Euler} characteristic and {Euler} obstruction of a function}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {1--20}, publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques}, address = {Toulouse}, volume = {Ser. 6, 24}, number = {1}, year = {2015}, doi = {10.5802/afst.1439}, mrnumber = {3325948}, language = {en}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1439/} }
TY - JOUR AU - Thaís Maria Dalbelo AU - Nivaldo de Góes Grulha Jr. AU - Miriam Silva Pereira TI - Toric surfaces, vanishing Euler characteristic and Euler obstruction of a function JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2015 SP - 1 EP - 20 VL - 24 IS - 1 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1439/ DO - 10.5802/afst.1439 LA - en ID - AFST_2015_6_24_1_1_0 ER -
%0 Journal Article %A Thaís Maria Dalbelo %A Nivaldo de Góes Grulha Jr. %A Miriam Silva Pereira %T Toric surfaces, vanishing Euler characteristic and Euler obstruction of a function %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2015 %P 1-20 %V 24 %N 1 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1439/ %R 10.5802/afst.1439 %G en %F AFST_2015_6_24_1_1_0
Thaís Maria Dalbelo; Nivaldo de Góes Grulha Jr.; Miriam Silva Pereira. Toric surfaces, vanishing Euler characteristic and Euler obstruction of a function. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 1, pp. 1-20. doi : 10.5802/afst.1439. https://afst.centre-mersenne.org/articles/10.5802/afst.1439/
[1] Barthel (G.), Brasselet (J.-P.), Fieseler (K.-H.), and Kaupr (L.).— Diviseurs invariants et homomorphisme de Poincaré de variétés toriques complexes. Tohoku Math. J. (2), 48(3), p. 363-390, 1996. | MR | Zbl
[2] Brasselet (J.-P.).— Local Euler obstruction, old and new. In XI Brazilian Topology Meeting (Rio Claro, 1998), pages 140-147. World Sci. Publ., River Edge, NJ (2000). | MR | Zbl
[3] Brasselet (J.-P.) and Grulha Jr. (N. G.).— Local Euler obstruction, old and new, II. In Real and complex singularities, volume 380 of London Math. Soc. Lecture Note Ser., pages 23-45. Cambridge Univ. Press, Cambridge (2010). | MR | Zbl
[4] Brasselet (J.-P.), Lê (D. T.), and Seade (J.).— Euler obstruction and indices of vector fields. Topology, 39(6) p. 1193-1208 (2000). | MR | Zbl
[5] Brasselet (J.-P.), Massey (D.), Parameswaran (A.), and Seade (J.).— Euler obstruction and defects of functions on singular varieties. J. London Math. Soc. (2), 70(1) p. 59-76 (2004). | MR | Zbl
[6] Brasselet (J.-P.) and Schwartz (M.-H.).— Sur les classes de Chern d’un ensemble analytique complexe. In The Euler-Poincaré characteristic (French), volume 82 of Astérisque, pages 93-147. Soc. Math. France, Paris (1981). | MR | Zbl
[7] Brasselet (J.-P.), Seade (J.), and Suwa (T.).— Vector fields on singular varieties, volume 1987 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2009). | MR | Zbl
[8] Buchweitz (R. O.) and Greuel (G. M.).— The Milnor number and deformations of complex curve singularities. Invent. Math., 58(3) p. 241-281 (1980). | MR | Zbl
[9] Dutertre (N.) and Grulha Jr. (N. G.).— Lê-Greuel type formula for the Euler obstruction and applications. Adv. Math., 251, p. 127-146 (2014). | MR | Zbl
[10] Ebeling (W.) and Gusein-Zade (S. M.).— On the indices of 1-forms on determinantal singularities. Tr. Mat. Inst. Steklova, 267(Osobennosti i Prilozheniya), p. 119-131 (2009). | MR | Zbl
[11] Fulton (W.).— Introduction to toric varieties, volume 131 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. | MR | Zbl
[12] Gaffney (T.).— Polar multiplicities and equisingularity of map germs. Topology, 32(1), p. 185-223 (1993). | MR | Zbl
[13] Gómez-Mont (X.), Seade (J.), and Verjovsky (A.).— The index of a holomorphic flow with an isolated singularity. Math. Ann., 291(4) p. 737-751 (1991). | MR | Zbl
[14] Gonzalez-Sprinberg (G.).— Calcul de l’invariant local d’Euler pour les singularités quotient de surfaces. C. R. Acad. Sci. Paris Sér. A-B, 288(21) p. A989-A992 (1979). | MR | Zbl
[15] Greuel (G. M.) and Steenbrink (J.).— On the topology of smoothable singularities. In Singularities, Part 1 (Arcata, Calif., 1981), volume 40 of Proc. Sympos. Pure Math., pages 535-545. Amer. Math. Soc., Providence, R.I. (1983). | MR | Zbl
[16] Hamm (H.).— Lokale topologische Eigenschaften komplexer Räume. Math. Ann., 191 p. 235-252 (1971). | MR | Zbl
[17] Looijenga (E. J. N.).— Isolated singular points on complete intersections, volume 77 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1984). | MR | Zbl
[18] MacPherson (R. D.).— Chern classes for singular algebraic varieties. Ann. of Math. (2), 100 p. 423-432 (1974). | MR | Zbl
[19] Milnor (J.).— Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton, N.J. (1968). | MR | Zbl
[20] Milnor (J.) and Orlik (P.).— Isolated singularities defined by weighted homogeneous polynomials. Topology, 9 p. 385-393 (1970). | MR | Zbl
[21] Nuño-Ballesteros (J. J.), Oréfice-Okamoto (B.), and Tomazella (J. N.).— The vanishing Euler characteristic of an isolated determinantal singularity. Israel J. Math., 197(1) p. 475-495 (2013). | MR | Zbl
[22] Pereira (M. S.) and Ruas (M. A. S.).— Codimension two determinantal varieties with isolated singularities. To appear in Mathematica Scandinavica.
[23] Riemenschneider (O.).— Deformationen von Quotientensingularitäten (nach zyklischen Gruppen). Math. Ann., 209 p. 211-248 (1974). | MR | Zbl
[24] Riemenschneider (O.).— Zweidimensionale Quotientensingularitäten: Gleichungen und Syzygien. Arch. Math. (Basel), 37(5) p. 406-417 (1981). | MR | Zbl
[25] Seade (J.).— The index of a vector field on a complex analytic surface with singularities. In The Lefschetz centennial conference, Part III (Mexico City, 1984), volume 58 of Contemp. Math., pages 225-232. Amer. Math. Soc., Providence, RI (1987). | MR | Zbl
[26] Seade (J.) and Suwa (T.).— A residue formula for the index of a holomorphic flow. Math. Ann., 304(4), p. 621-634 (1996). | MR | Zbl
[27] Seade (J.), Tibăr (M.), and Verjovsky (A.).— Milnor numbers and Euler obstruction. Bull. Braz. Math. Soc. (N.S.), 36(2), p. 275-283 (2005). | MR | Zbl
[28] Teissier (B.).— Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whitney. In Algebraic geometry (La Rábida, 1981), volume 961 of Lecture Notes in Math., pages 314-491. Springer, Berlin (1982). | MR | Zbl
[29] Wahl (J.).— Smoothings of normal surface singularities. Topology, 20(3), p. 219-246, (1981). | MR | Zbl
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