Classical algebraic geometry is full of ingenious constructions and miraculous coincidences: 27 lines on a cubic surface are related to Weyl lattice of type $E_6,$ lines on an intersection of four-dimensional quadrics are parametrized by a Jacobian of genus two curve and so on.

Existence of some constructions of this type may be predicted by simple cohomological computations. For instance, consider a smooth intersection $X$ of two quadrics $Q_1$ and $Q_2$ in $\mathbb{P}^5.$ This example is explained in detail in the beautiful paper The complete intersection of two or more quadrics of Miles Reid. It is easy to see that $h^{p,q}(X)$ equals to $1$ for $$(p,q)\in \{(0,0),\ (1,1),\ (2,2),\ (3,3)\},$$ equals to 2 for $$(p,q)\in \{(1,2),\ (2,1)\}$$ and equals to $0$ in other cases. Generalized Hodge Conjecture predicts that there exists a certain curve $C$ and a correspondence $D\subset C\times X$ which induces a surjective morphism $$H^1(C,\mathbb{Z})\longrightarrow H^3(X,\mathbb{Z})(-3).$$ Explicit construction of this correspondence is based on an identification of the Hilbert scheme of lines on $X$ with Jacobian $J(C).$

**Question 1:** What other classical constructions in algebraic geometry can be a posteriori motivated by simple computations with (mixed) Hodge structures?

**Question 2:** It will be very interesting to see some concrete examples, when such a construction is expected to exist, but has not been worked out yet.

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