The generalized Hodge conjecture for the quadratic complex of lines in projective four-space
Let \(G\) be the Grassmann variety of lines in \(\mathbb{P}^4\), and embed it in \(\mathbb{P}^9= \mathbb{P}(\bigwedge^2V)\) by the Plücker embedding. Let \(X\) be a quadratic complex of lines in \(\mathbb{P}^4\), namely the intersection of \(G\) with a quadric \(Q\subset\mathbb{P}^9\). Then \(X\) is a Fano fivefold of index 3 whose cohomology group \(H^5(X)\) carries a Hodge structure of level one with \(h^{2,3} (X)=10\). The main theorem of this paper states that the generalized Hodge conjecture GHC\((X,5,2)\) holds if \(X\) is general. For the proof, the author employs the Fano variety \(F_X\) of two-planes contained in \(X\), which is shown to be a smooth connected curve of genus 161 when \(X\) is general, and shows that the cylinder homomorphism \(H_1(F_X,\mathbb{Z})\to H_5(X,\mathbb{Z})\) associated to the family \(F_X\) is surjective.
scientific article; zbMATH DE number 1221891