A basis of algebraic de Rham cohomology of complete intersections over a characteristic zero field

Abstract

Let $\mathbb{k}$ be a field of characteristic 0. Let X be a smooth complete intersection over $\mathbb{k}$ of dimension n – k in the projective space ${\mathbf{P}}_{\mathbb{k}}^n,$ for given positive integers n and k. When $\mathbb{k}=\mathbb{C},$ Terasoma and Konno provided an explicit representative (in terms of differential forms) of a basis for the primitive middle-dimensional algebraic de Rham cohomology $H_{dR,\text{prim}}^{n-k}(X;\mathbb{C}).$ Later Dimca constructed another explicit representative of a basis of $H_{dR,\text{prim}}^{n-k}(X;\mathbb{C})$. Moreover, he proved that his representative gives the same cohomology class as the previous representative of Terasoma and Konno. The goal of this article is to examine the above two different approaches without assuming that $\mathbb{k}=\mathbb{C}$ and provide a similar comparison result for any field $\mathbb{k}.$ Dimca’s argument depends heavily on the condition $\mathbb{k}=\mathbb{C}$ and our idea is to find appropriate Cech-de Rham complexes and spectral sequences corresponding to those two approaches, which work without restrictions on $\mathbb{k}.$

Keywords:

• De Rham cohomology
• Gysin map
• projective smooth complete intersections
• residues

2020 Mathematics Subject Classification:

• 14M10
• 14F40 (primary)

Acknowledgements

Jeehoon Park thanks KIAS (Korea Institute for Advanced Study), where the part of work was done, for its hospitality. The authors thank the anonymous referee for useful comments to improve the article.

Notes

1 More precisely, the isomorphism is due to Griffiths in the hypersurface case [3], Terasoma in the equal degree complete intersection case [11], and Konno in the general case [8].

2 See [9, Section 1.4] for physical explanation how to understand S as an action functional of a 0-dimensional quantum field theory.

3 Here, $e^{\Gamma }=1+\Gamma +\frac{\Gamma ·\Gamma }{2}+\cdots$ and, for $x\in A,$ we think of ${\mathcal{C}}_{\gamma }^{\underset{¯}{G}}(x·e^{\Gamma })={\mathcal{C}}_{\gamma }^{\underset{¯}{G}}(x)+{\mathcal{C}}_{\gamma }^{\underset{¯}{G}}(x·\Gamma )+{\mathcal{C}}_{\gamma }^{\underset{¯}{G}}\left(x·\frac{\Gamma ·\Gamma }{2}\right)+\cdots$ as a formal expression.

4 This was already introduced in (1.1)

5 More precisely, Terasoma [11] provided such an isomorphism in the case $d_1=d_2=\cdots =d_k$ and Konno [8] extended the result of Terasoma to the general case when d_j’s are not equal. Also see [3] for the pioneering work of Griffiths in the case k = 1, hypersurface case.

6 This fact is crucially used to develop the deformation theory of period integrals of $X_{\underset{¯}{G}}$ in [7].

7 For example, $A=\mathbb{k}[\underset{¯}{y},\underset{¯}{x}],{\text{Res}}_{\underset{¯}{G}}:H_{dR}^q({\mathbf{P}}^n\setminus X_{\underset{¯}{G}};\mathbb{k})\to H_{dR}^{q-2k+1}(X_{\underset{¯}{G}};\mathbb{k}).$

Additional information

Funding

The work of Jeehoon Park was supported by the National Research Foundation of South Korea (NRF-2018R1A4A1023590 and NRF-2021R1A2C1006696). The work of Jeehoon Park was also supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.2020R1A5A1016126).