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The holomorphy conjecture for nondegenerate surface singularities

(2016) NAGOYA MATHEMATICAL JOURNAL. 227. p.160-188
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Abstract
The holomorphy conjecture roughly states that Igusa's zeta function associated to a hypersurface and a character is holomorphic on C whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article, we prove the holomorphy conjecture for surface singularities that are nondegenerate over C with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volumes (which appear in the formula of Varchenko for the zeta function of monodromy) of the faces in a simplex in arbitrary dimension. We then study some specific character sums that show up when dealing with false poles. In contrast to the context of the trivial character, we here need to show fakeness of certain candidate poles other than those contributed by Bi-facets.
Keywords
LOCAL ZETA-FUNCTIONS, MONODROMY, IGUSAS, CURVES

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Citation

Please use this url to cite or link to this publication:

MLA
Castryck, Wouter, Denis Ibadula, and Ann Lemahieu. “The Holomorphy Conjecture for Nondegenerate Surface Singularities.” NAGOYA MATHEMATICAL JOURNAL 227 (2016): 160–188. Print.
APA
Castryck, W., Ibadula, D., & Lemahieu, A. (2016). The holomorphy conjecture for nondegenerate surface singularities. NAGOYA MATHEMATICAL JOURNAL, 227, 160–188.
Chicago author-date
Castryck, Wouter, Denis Ibadula, and Ann Lemahieu. 2016. “The Holomorphy Conjecture for Nondegenerate Surface Singularities.” Nagoya Mathematical Journal 227: 160–188.
Chicago author-date (all authors)
Castryck, Wouter, Denis Ibadula, and Ann Lemahieu. 2016. “The Holomorphy Conjecture for Nondegenerate Surface Singularities.” Nagoya Mathematical Journal 227: 160–188.
Vancouver
1.
Castryck W, Ibadula D, Lemahieu A. The holomorphy conjecture for nondegenerate surface singularities. NAGOYA MATHEMATICAL JOURNAL. 2016;227:160–88.
IEEE
[1]
W. Castryck, D. Ibadula, and A. Lemahieu, “The holomorphy conjecture for nondegenerate surface singularities,” NAGOYA MATHEMATICAL JOURNAL, vol. 227, pp. 160–188, 2016.
@article{8568133,
  abstract     = {The holomorphy conjecture roughly states that Igusa's zeta function associated to a hypersurface and a character is holomorphic on C whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article, we prove the holomorphy conjecture for surface singularities that are nondegenerate over C with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volumes (which appear in the formula of Varchenko for the zeta function of monodromy) of the faces in a simplex in arbitrary dimension. We then study some specific character sums that show up when dealing with false poles. In contrast to the context of the trivial character, we here need to show fakeness of certain candidate poles other than those contributed by Bi-facets.},
  author       = {Castryck, Wouter and Ibadula, Denis and Lemahieu, Ann},
  issn         = {0027-7630},
  journal      = {NAGOYA MATHEMATICAL JOURNAL},
  keywords     = {LOCAL ZETA-FUNCTIONS,MONODROMY,IGUSAS,CURVES},
  language     = {eng},
  pages        = {160--188},
  title        = {The holomorphy conjecture for nondegenerate surface singularities},
  url          = {http://dx.doi.org/10.1017/nmj.2016.51},
  volume       = {227},
  year         = {2016},
}

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