an:01442160
Zbl 0980.14017
Adolphson, Alan; Sperber, Steven
Dwork cohomology, de Rham cohomology, and hypergeometric functions
EN
Am. J. Math. 122, No. 2, 319-348 (2000).
00063427
2000
j
14F40 14F43 14D05 53C05
integrable connection; Gauss-Manin connection; complete intersection; de Rham cohomology; Dwork cohomology; hypergeometric equations; Picard-Fuchs equations
Let \(S\) be a smooth affine \(\mathbb C\)-scheme, and let \(X = \text{Spec}(A)\) be an affine \(S\)-scheme. Consider a projective \({\mathcal O}_X\)-module \(\mathcal E\) of finite rank, and an integrable \(\mathbb C\)-connection \(\nabla : {\mathcal E} \rightarrow \Omega^1_{X/{\mathbb C}} \otimes_{{\mathcal O}_X}{\mathcal E}.\) Let \(f_1, \ldots, f_r \in A\) be a regular sequence defining the smooth complete intersection \(Y = \text{Spec}(A/(f_1, \ldots, f_r)),\) and let \(j : Y \rightarrow X\) be the inclusion of \(S\)-schemes. Set \({\mathbb A}_X^r = \text{Spec}(A[T_1, \ldots, T_r])\) so that \(\pi : {\mathbb A}_X^r \rightarrow X\) is the projection. It is proved that for any \(n \in {\mathbb N}\) there is an isomorphism of \({\mathcal O}_S\)-modules with \(\mathbb C\)-connection: \(H^n_{DR}(Y/S, (j^\ast({\mathcal E}), \nabla_Y)) \cong H^{n+2r}_{DR}({\mathbb A}_X^r/S, (\pi^\ast({\mathcal E}), \nabla_F)),\) where \(\nabla_Y\) is the pullback of \(\nabla\) to a connection on \(j^\ast({\mathcal E}),\) \(F = T_1f_1 + \ldots + T_rf_r,\) and \(\nabla_F\) is the corresponding twisted connection. The cohomology groups on the right hand side can be interpreted as analogues of Dwork cohomology groups in the case of characteristic zero. This result can be considered as a generalization to the complete intersections case of a result by \textit{N. Katz} [Publ. Math., Inst. Hautes ??tud. Sci. 39, 175-232 (1970; Zbl 0221.14007)] which states that for smooth projective hypersurfaces the Dwork cohomology and the primitive part of the de Rham cohomology coincide. As an application the authors show how one can compute the Picard-Fuchs equations for smooth complete intersections.
Aleksandr G.Aleksandrov (Moskva)
Zbl 0221.14007