Fibre de Milnor motivique à l’infini et composition avec un polynôme non dégénéré

Raibaut, Michel

Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1943-1981.

Soit $k$ un corps de caractéristique nulle, $P$ un polynôme de Laurent en $d$ variables, à coefficients dans $k$ et non dégénéré pour son polyèdre de Newton à l’infini. Soit $d$ fonctions non constantes $f_l$ à variables séparées et définies sur des variétés lisses. A la manière de Guibert, Loeser et Merle, dans le cas local, nous calculons dans cet article, la fibre de Milnor motivique à l’infini de la composée $P\left(f\right)$ en termes du polyèdre de Newton à l’infini de $P$. Pour $P$ égal à la somme $x_1+x_2$ nous obtenons une formule du type Thom-Sébastiani. Ceci permet d’introduire une notion de cycles évanescents motiviques d’une fonction $g$ pour la valeur infini notée $S_{g,\infty ,U}^{\Phi }$, qui vérifie comme dans le cas local une formule de convolution. En particulier si $g$ est le polynôme $x_1+...+x_n+1/x_1...x_n$, nous montrons que le spectre de $S_{g,\infty ,U}^{\Phi }$ vaut $1+t+..+t^n$ ce qui coïncide avec le spectre à l’infini de $g$ considéré par Douai et Sabbah.

Let $k$ be a field of characteristic zero and $P$ be a Laurent polynomial in $d$ variables, with coefficients in $k$ and non degenerate for its Newton polyhedron at infinity. Let $\left(f_l\right)$ be $d$ non constant functions with separated variables and defined on smooth varieties. As Guibert, Loeser and Merle in the local case, we compute in this article the motivic Milnor fiber at infinity of $P\left(f\right)$ in terms of the Newton polyhedron at infinity of $P$. For $P$ equal to the sum $x_1+x_2$, we obtained a Thom-Sebastiani formula. Then we can introduce a notion of motivic vanishing cycles of a function $g$ for the infinite value denoted by $S_{g\infty ,U}^{\Phi }$, and which verified, as in the local case, a convolution formula. In particular if $g$ is the polynomial $x_1+...+x_n+1/x_1...x_n$, we show that the spectrum $S_{g,\infty ,U}^{\Phi }$ is $1+t+..+t^n$ which coincides with the spectrum at infinity of $g$ considered by Douai and Sabbah.

[1] Bittner, Franziska On motivic zeta functions and the motivic nearby fiber, Math. Z., Volume 249 (2005) no. 1, pp. 63-83 | DOI | MR | Zbl

[2] Deligne, Pierre Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. (1974) no. 44, pp. 5-77 | Numdam | MR | Zbl

[3] Denef, Jan; Loeser, François Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math., Volume 135 (1999) no. 1, pp. 201-232 | DOI | MR | Zbl

[4] Denef, Jan; Loeser, François Geometry on arc spaces of algebraic varieties, European Congress of Mathematics, Vol. I (Barcelona, 2000) (Progr. Math.), Volume 201, Birkhäuser, Basel, 2001, pp. 327-348 | MR | Zbl

[5] Douai, A.; Sabbah, C. Gauss-Manin systems, Brieskorn lattices and Frobenius structures. I, Ann. Inst. Fourier (Grenoble), Volume 53 (2003) no. 4, pp. 1055-1116 | DOI | Numdam | MR | Zbl

[6] Douai, Antoine; Sabbah, Claude Gauss-Manin systems, Brieskorn lattices and Frobenius structures. II (2004), pp. 1-18

[7] Fulton, William Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 2, Springer-Verlag, Berlin, 1998 | MR | Zbl

[8] Guibert, Gil Espaces d’arcs et invariants d’Alexander, Comment. Math. Helv., Volume 77 (2002) no. 4, pp. 783-820 | DOI | MR | Zbl

[9] Guibert, Gil; Loeser, François; Merle, Michel Nearby cycles and composition with a nondegenerate polynomial, Int. Math. Res. Not. (2005) no. 31, pp. 1873-1888 | DOI | MR | Zbl

[10] Guibert, Gil; Loeser, François; Merle, Michel Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink, Duke Math. J., Volume 132 (2006) no. 3, pp. 409-457 | DOI | MR | Zbl

[11] Hironaka, Heisuke Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203 ; ibid. (2), Volume 79 (1964), pp. 205-326 | DOI | MR | Zbl

[12] Kouchnirenko, A. G. Polyèdres de Newton et nombres de Milnor, Invent. Math., Volume 32 (1976) no. 1, pp. 1-31 | DOI | MR | Zbl

[13] Loeser, François Seattle lectures on motivic integration, Algebraic geometry—Seattle 2005. Part 2 (Proc. Sympos. Pure Math.), Volume 80, Amer. Math. Soc., Providence, RI, 2009, pp. 745-784 | MR | Zbl

[14] Looijenga, Eduard Motivic measures, Astérisque (2002) no. 276, pp. 267-297 (Séminaire Bourbaki, Vol. 1999/2000) | Numdam | MR | Zbl

[15] Matsui, Yutaka; Takeuchi, Kiyoshi Monodromy at infinity of polynomial map and mixed Hodge modules, arXiv :0912.5144v5

[16] Matsui, Yutaka; Takeuchi, Kiyoshi Monodromy zeta functions at infinity, Newton polyhedra and constructible sheaves, Math. Z., Volume 268 (2011) no. 1-2, pp. 409-439 | DOI | MR

[17] Pham, Frédéric Vanishing homologies and the $n$ variable saddlepoint method, Singularities, Part 2 (Arcata, Calif., 1981) (Proc. Sympos. Pure Math.), Volume 40, Amer. Math. Soc., Providence, RI, 1983, pp. 319-333 | MR | Zbl

[18] Raibaut, Michel Fibre de Milnor motivique à l’infini, C. R. Math. Acad. Sci. Paris, Volume 348 (2010) no. 7-8, pp. 419-422 | DOI | MR | Zbl

[19] Raibaut, Michel Singularités à l’infini et intégration motivique, Thèse, Université Nice Sophia Antipolis (2010)

[20] Raibaut, Michel Singularités à l’infini et intégration motivique, Bull. Soc. Math. France, Volume 140 (2012) no. 1, pp. 51-100 | Numdam | MR

[21] Sabbah, Claude Monodromy at infinity and Fourier transform, Publ. Res. Inst. Math. Sci., Volume 33 (1997) no. 4, pp. 643-685 | DOI | MR | Zbl

[22] Saito, Morihiko Mixed Hodge modules, Publ. Res. Inst. Math. Sci., Volume 26 (1990) no. 2, pp. 221-333 | DOI | MR | Zbl

[23] Steenbrink, Joseph; Zucker, Steven Variation of mixed Hodge structure. I, Invent. Math., Volume 80 (1985) no. 3, pp. 489-542 | DOI | MR | Zbl