Geodesic
Generalized Airy Functions and the Cohomological Intersection Numbers
Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Differential and Functional-Differential Equations — Satellite of International Congress of Mathematicians ICM-2002 (Moscow, MAI, 11–17 August, 2002). Part 2, Tome 2 (2003), pp. 83-94.

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We compute explicitly the cohomological intersection numbers for the basis and extend the result of Iwasaki and Matsumoto. To this end, we establish the exterior power structure for the polynomial twisted de Rham cohomology group associated with the generalized Airy functions at a point of extended Veronese variety. Using this structure, we introduce a natural basis of the twisted de Rham cohomology group coming from that of the one-dimensional case, which is considered as an analogue of a flat basis of the Jacobi ring of $A$-type simple singularity. 
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H. Kimura. Generalized Airy Functions and the Cohomological Intersection Numbers. Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Differential and Functional-Differential Equations — Satellite of International Congress of Mathematicians ICM-2002 (Moscow, MAI, 11–17 August, 2002). Part 2, Tome 2 (2003), pp. 83-94. http://geodesic.mathdoc.fr/item/CMFD_2003_2_a4/

[1] Aomoto K., “Les équation aux différences linéaires et les intégrales des fonctions multiformes”, J. Fac. Sci. Univ. Tokyo, Sec. IA, 22 (1975), 271–297 | MR | Zbl

[2] Cho K., Matsumoto K., “Intersection theory for twisted cohomologies and twisted Riemann's period relations I”, Nagoya Math. J., 139 (1995), 67–86 | MR | Zbl

[3] Gelfand I. M., “General theory of hypergeometric functions”, Soviet Math. Dokl., 33 (1986), 9–13 | MR

[4] Gelfand I. M., Retahk V. S., Serganova V. V., “Generalized Airy functions, Schubert cells, and Jordan groups”, Soviet Math. Dokl., 37 (1988), 8–12 | MR

[5] Ishiura S., Noumi M., “Calculus of the Gauss-Manin system of type $A_l$, I”, Proc. Jpn. Acad., Ser. A, 58 (1982), 13–16 | DOI | MR | Zbl

[6] Ishiura S., Noumi M., “Calculus of the Gauss-Manin system of type $A_l$, II”, Proc. Jpn. Acad., Ser. A, 58 (1982), 62–65 | DOI | MR | Zbl

[7] Iwasaki K., Isolated singularity, Witten's Laplacian and duality for twisted de Rham cohomology, Preprint | MR

[8] Iwasaki K., Matsumoto K., “Intersection matrix of a generalized Airy function in terms of skew-Schur polynomials”, Proc. Jpn. Acad., Ser. A, 76 (2000), 135–140 | DOI | MR | Zbl

[9] Iwasaki K., Kita M., “Exterior power structure on the twisted de Rham cohomology of the complements of real Veronese arrangements”, J. Math. Pures Appl., IX Sér., 75 (1995), 69–84 | MR

[10] Kimura H., “On rational de Rham cohomology associated with the generalized Airy functions”, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV Ser., 24 (1997), 351–366 | MR | Zbl

[11] Kimura H., “On the homology group associated with the general Airy integral”, Kumamoto J. Math., 10 (1997), 11–29 | MR | Zbl

[12] Kimura H., Haraoka Y., Takano K., “The generalized confluent hypergeometric functions”, Proc. Jpn. Acad., Ser. A, 68 (1992), 290–295 | DOI | MR | Zbl

[13] Kimura H., Taneda M., “Analogue of flat basis and cohomological intersection number for general hypergeometric functions”, J. Math. Sci., Tokyo, 6 (1999), 415–436 | MR | Zbl

[14] Majima H., Matsumoto K., Takayama N., “Quadratic relations for confluent hypergeometric functions”, Tohoku Math. J., II Ser., 52 (2000), 489–513 | DOI | MR | Zbl

[15] Matsumoto K., “Quadratic identities for hypergeometric series of type $(k,l)$”, Kyushu J. Math., 2 (1994), 335–345 | DOI | MR | Zbl

[16] Pham F., “Vanishing homologies and the $n$-variable saddle point method”, Proc. Symp. Pure Math., 40 (1983), 319–334 | MR

[17] Pham F., “La descente des cols par les onglets de Lefschetz, avec vues sur Gauss-Manin”, Astérisque, 130 (1985), 11–47 | MR | Zbl