Hodge classes associated to 1-parameter families of Calabi-Yau 3-folds (Q977135)

The \(L^2\)-Higgs cohomology is used to compute the Hodge numbers of the parabolic cohomology \(H^1(\bar{S}, j_*{\mathbf{V}})\), where the local system \(V\) arises from the third primitive cohomology of family of Calabi-Yau threefolds over a curve \(\bar{S}\). Here, \(\bar{S}={\mathbb{P}}^1\) and \(f: X\to S={\mathbb{P}}^1\setminus D\) is a smooth family of Calabi-Yau threefolds whose Kodaira-Spencer map is not identically zero. Hence \(h^{1,2}=1\) and the local system \({\mathbf V}:=R^3f_*{\mathbb{C}}_X\) of (primitive) middle cohomology is irreducible and has the Hodge numbers \((1,1,1,1)\). The cohomology group \(H:=H^1(\bar{S},j_*{\mathbf{V}})\) carries a pure Hodge structure of weight \(4\). The method presented here gives a way to predict the presence of algebraic \(2\)-cycles in the total space of the family, in particular, in \(H^{2,2}\). The algebraic \(2\)-cycles constructed by \textit{J. Walcher} [``Open mirror symmetry on the quintic'', \url{arXiv:hep-th/0605162} and ``Calculations for mirror symmetry with \(D\)-branes'', \url{arXiv:0904.4905}] are discussed to illustrate the method.