Let \(\pi:\mathcal{X}\rightarrow B\) be a smooth projective morphism. As a consequence of the hard Lefschetz theorem, P. Deligne [Publ. Math., Inst. Hautes Étud. Sci. 35, 107–126 (1968; Zbl 0159.22501)] established the following decomposition (\(\ast\)) \[ R\pi_\ast\mathbb{Q} = \bigoplus_i R^i\pi_\ast \mathbb{Q}[-i] \] in the derived category of sheaves of \(\mathbb{Q}\)-vector spaces on \(B\). Such a decomposition is usually not canonical. In the decomposition (\(\ast\)), each side carries a multiplicative structure. On the right hand side, it is simply the direct sum of the cup product maps \(\mu_{i,j}:R^i\pi_\ast \mathbb{Q} \otimes R^j\pi_\ast\mathbb{Q} \rightarrow R^{i+j}\pi_\ast \mathbb{Q}\). On the left hand side, one takes a complex \(C^\ast\) representing \(R\pi_\ast\mathbb{Q}\) together with a morphism \(\mu:C^\ast\otimes C^\ast\rightarrow C^\ast\) which induces the cup product on cohomology. With the above multiplicative structures, the author of the paper under review asks if a multiplicative decomposition (\(\ast\)) (meaning a decomposition which respects the multiplicative structures) exists or not. It is shown that in general, the answer is negative, even after restricting to a Zariski open subset of \(B\). The main result (Theorem 0.7) of the paper is the existence of a decomposition (\(\ast\)) for a family of \(K3\) surfaces which becomes multiplicative after restricting to a Zariski open subfamily. To be more precise, assume that the fibers of \(\mathcal{X}\rightarrow B\) are all \(K3\) surfaces, then the author shows the following:
(i) There exists a decomposition (\(\ast\)) and a Zariski open subset \(B^\circ \) of \(B\) such that the decomposition becomes multiplicative for the restricted family \(\pi^\circ:\mathcal{X}^\circ\rightarrow B^\circ\);
(ii) The class of the relative diagonal \([\Delta_{\mathcal{X}^\circ / B^\circ}]\in \mathrm{H}^4(\mathcal{X}^\circ \times_{B^\circ} \mathcal{X}^\circ,\mathbb{Q})\) belongs to the direct summand \(\mathrm{H}^0(B^\circ, R^4(\pi, \pi)_\ast \mathbb{Q})\) of \(\mathrm{H}^4(\mathcal{X}^\circ \times_{B^\circ} \mathcal{X}^\circ,\mathbb{Q})\) under the induced decomposition of \(R(\pi,\pi)_\ast \mathbb{Q}\);
(iii) The topological first Chern class of a line bundle, restricted to \(\mathcal{X}^\circ\), belongs to the direct summand \(\mathrm{H}^0(B^\circ, R^2\pi_\ast \mathbb{Q})\) of \(\mathrm{H}^0(\mathcal{X}^\circ, \mathbb{Q})\) induced by this decomposition.
The author also discusses a geometric consequence of the main theorem which can be viewed as an evidence of the Conjecture 1.3 of [C. Voisin, Pure Appl. Math. Q. 4, No. 3, 613–649 (2008; Zbl 1165.14012)]. She gives two proofs of the main theorem. The first proof uses the concept of \(K\)-correspondences of C. Voisin [in: The Fano conference. Papers of the conference organized to commemorate the 50th anniversary of the death of Gino Fano (1871–1952), Torino, Italy, September 29–October 5, 2002. Torino: Università di Torino, Dipartimento di Matematica. 761–792 (2004; Zbl 1177.14040)]. The second proof heavily uses the result of [A. Beauville and C. Voisin, J. Algebr. Geom. 13, No. 3, 417–426 (2004; Zbl 1069.14006)]. In the last part, the author explores the possibility of generalizing the main theorem above to families of Calabi-Yau hypersurfaces. In particular, she proves an analogue for Calabi-Yau hypersurfaces of the result of Beauville and Voisin [loc. cit.]. A second ingredient needed for the generalization is stated as a conjecture in the end of the paper.