Toric surfaces, vanishing Euler characteristic and Euler obstruction of a function
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 1, pp. 1-20.

Nous définissons la caractéristique d’Euler évanescente d’une surface torique normale X σ , nous donnons une formule pour la calculer, et nous associons ce nombre avec la seconde multiplicité polaire de X σ . Nous présentons aussi une formule pour l’obstruction d’Euler d’une fonction f:X σ et pour la différence entre l’obstruction d’Euler de l’espace X σ et l’obstruction d’Euler d’une fonction f. Comme application de ce résultat nous calculons l’obstruction d’Euler des polynômes d’un certain type sur une famille de surfaces déterminantales.

We define the vanishing Euler characteristic of a normal toric surface X σ , we give a formula to compute it, and we relate this number with the second polar multiplicity of X σ . We also present a formula for the Euler obstruction of a function f:X σ and for the difference between the Euler obstruction of the space X σ and the Euler obstruction of a function f. As an application of this result we compute the Euler obstruction of a type of polynomial on a family of determinantal surfaces.

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     author = {Maria Dalbelo, Tha{\'\i}s and de G\'oes Grulha Jr., Nivaldo and Silva Pereira, Miriam},
     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},
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Maria Dalbelo, Thaís; de Góes Grulha Jr., Nivaldo; Silva Pereira, Miriam. Toric surfaces, vanishing Euler characteristic and Euler obstruction of a function. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 1, pp. 1-20. doi : 10.5802/afst.1439. http://archive.numdam.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.), (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). | Numdam | 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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