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ON A COMPARISON BETWEEN DWORK AND RIGID COHOMOLOGIES OF PROJECTIVE COMPLEMENTS
- Part of
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- Journal:
- Nagoya Mathematical Journal / Volume 254 / June 2024
- Published online by Cambridge University Press:
- 01 December 2023, pp. 420-461
- Print publication:
- June 2024
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- Article
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For homogeneous polynomials
$G_1,\ldots ,G_k$ over a finite field, their Dwork complex is defined by Adolphson and Sperber, based on Dwork’s theory. In this article, we will construct an explicit cochain map from the Dwork complex of 
$G_1,\ldots ,G_k$ to the Monsky–Washnitzer complex associated with some affine bundle over the complement 
$\mathbb {P}^n\setminus X_G$ of the common zero 
$X_G$ of 
$G_1,\ldots ,G_k$, which computes the rigid cohomology of 
$\mathbb {P}^n\setminus X_G$. We verify that this cochain map realizes the rigid cohomology of 
$\mathbb {P}^n\setminus X_G$ as a direct summand of the Dwork cohomology of 
$G_1,\ldots ,G_k$. We also verify that the comparison map is compatible with the Frobenius and the Dwork operator defined on both complexes, respectively. Consequently, we extend Katz’s comparison results in [19] for projective hypersurface complements to arbitrary projective complements.
