On the topology of projective subspaces in complex Fermat varieties. (English) Zbl 1354.14032

This work translates into combinatorial terms the question of whether, for a complex Fermat variety \(X\) of a certain kind, a specific submodule of its middle \(\mathbb{Z}\)-homology is primitive or not. The Fermat varieties in question have (complex) even dimension \(n=2d\) and degree \(m > 2\), and are given by \(z_0^m + \cdots + z_{n+1}^m = 0\) in a complex projective space \(\mathbb{P}^{n+1}\) with homogeneous coordinates \((z_0 : \cdots z_{n+1})\).
The submodule mentioned above is the one generated by the fundamental classes of all standard \(d\)-spaces (which are subspaces of the Fermat variety \(X\)): the standard \(d\)-space \(\mathcal{L}_{J, \beta}\) is the projective subspace of \(\mathbb{P}^{n+1}\) given by the equations \(z_{k_i} = \beta_i z_{j_i}\) (for \(i = 0, \cdots , d)\), where \(\beta = (\beta_0 , \cdots , \beta_d)\) is a \((d+1)\)-tuple of \(m\)-roots of \(-1\) and \(J = [[j_0,k_0], \cdots , [j_d,k_d]]\) is an unordered partition of the index set \(\{ 0, 1, \cdots , n+1\}\) into unordered pairs. Call \(\mathcal{I}\) the set of all such partitions and, given a non-empty subset \(\mathcal{K}\) of \(\mathcal{I}\), denote by \(\mathcal{L}_{\mathcal{K}}(X)\) the \(\mathbb{Z}\)-submodule of \(H_n(X)\) generated by the classes of all standard \(d\)-spaces \(\mathcal{L}_{J, \beta}\), for all \(J \in \mathcal{K}\) and all partitions \(\beta\) as above.
The main result in this work is Theorem 1.1., which describes the \(\mathbb{Z}\)-torsion of the quotient modules \(H_n(X)/\mathcal{L}_{\mathcal{K}}(X)\) in terms of the combinatorial properties of some subsets of elements in \(\mathbb{Z}[t_1, \cdots , t_{n+1}]\), the polynomial ring in \(n+1\) variables. This result states that this torsion is isomorphic to the torsion of any of four possible modules, all with isomorphic torsion.
The proof of the theorem, unraveled in section 4, is based upon Theorem 1.4., from section 4.3., presenting the rank of the groups \(\mathcal{L}_{\mathcal{K}}(X)\), for \(\mathcal{K}\) non-empty. Both results depend directly on Theorem 2.2., from section 3, which calculates the bilinear pairing (given from Poincaré duality) of the fundamental class of each \(\mathcal{L}_{J, \beta}\) and the cycle \(S = (1 - \gamma_1^{-1}) \cdots (1 - \gamma_{n+1}^{-1})D\), where the \(\gamma_i\) generate the Galois group of a specific covering \(\pi : X \rightarrow \Pi\), with \(\Pi\) a hyperplane, and \(D\) is a topological \(n\)-simplex. These bilinear pairings can be viewed as signed intersections of \(n\)-cycles brought to a general position.
Conjecture 1.2. declares that \(H_n(X)/\mathcal{L}_{\mathcal{K}}(X)\) has no torsion when \(\mathcal{K}\) is the full \(\mathcal{I}\). Some evidence for the plausibility of this conjecture is given in section 5, where the torsion of these groups is computed (for some values of \(m\) and \(n\)) using Corollary 1.5. of Theorem 1.4.
As explained by the authors, the results in this work on the primitivity of the submodules \(\mathcal{L}_{\mathcal{K}}(X)\) expand upon those previously known, e.g. for the case \(n=2\) (Aoki/Shioda) and whenever \(m\) is prime to 6 (Schütt/Shioda/van Luijk [M. Schütt et al., J. Number Theory 130, No. 9, 1939–1963 (2010; Zbl 1194.14057)] and A. Degtyarev [J. Number Theory 147, 454–477 (2015; Zbl 1370.14032)]).


14F25 Classical real and complex (co)homology in algebraic geometry
14J70 Hypersurfaces and algebraic geometry
13D30 Torsion theory for commutative rings
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