Geometry of nondegenerate polynomials: Motivic nearby cycles and Cohomology of contact loci

Quy Thuong Lê; Tat Thang Nguyen

doi:10.5802/crmath.492

Algebraic geometry, Geometry and Topology

Geometry of nondegenerate polynomials: Motivic nearby cycles and Cohomology of contact loci

Quy Thuong Lê ^1,²; Tat Thang Nguyen ³

¹ University of Science, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Thanh Xuan District, Hanoi, Vietnam
² Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan
³ Institute of Mathematics, Vietnam Academy of Science and Technology 18 Hoang Quoc Viet Road, Cau Giay District, Hanoi, Vietnam

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1249-1266.

Abstract

We study polynomials with complex coefficients which are nondegenerate in two senses, one of Kouchnirenko and the other with respect to its Newton polyhedron, through data on contact loci and motivic nearby cycles. Introducing an explicit description of these quantities we can answer in part the question concerning the motivic nearby cycles of restriction functions in the context of Newton nondegenerate polynomials. Furthermore, in the nondegeneracy in the sense of Kouchnirenko, we give calculations on cohomology groups of the contact loci.

Received: 2021-10-11
Revised: 2021-11-29
Accepted: 2023-03-06
Published online: 2023-10-31

DOI: 10.5802/crmath.492

Classification: 14B05, 14B07, 14J17, 32S05, 32S30, 32S55
Keywords: arc spaces, contact loci, motivic zeta function, motivic Milnor fiber, motivic nearby cycles, Newton polyhedron, nondegeneracy, sheaf cohomology with compact support

Author's affiliations:

Quy Thuong Lê ^{1, 2}; Tat Thang Nguyen ³

¹ University of Science, Vietnam National University, Hanoi, 334 Nguyen Trai Street, Thanh Xuan District, Hanoi, Vietnam
² Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan
³ Institute of Mathematics, Vietnam Academy of Science and Technology 18 Hoang Quoc Viet Road, Cau Giay District, Hanoi, Vietnam

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Quy Thuong L\^e and Tat Thang Nguyen},
     title = {Geometry of nondegenerate polynomials: {Motivic} nearby cycles and {Cohomology} of contact loci},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1249--1266},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2023},
     doi = {10.5802/crmath.492},
     language = {en},
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Quy Thuong Lê; Tat Thang Nguyen. Geometry of nondegenerate polynomials: Motivic nearby cycles and Cohomology of contact loci. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1249-1266. doi : 10.5802/crmath.492. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.492/

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