On the deformation chirality of real cubic fourfolds

According to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Here, we show how to eliminate the projective equivalence and to obtain a pure deformation classification, that is how to respond to the chirality question: which cubics are not deformation equivalent to their image under a mirror reflection. We provide an arithmetical criterion of chirality, in terms of the eigen-sublattices of the complex conjugation involution in homology, and show how this criterion can be effectively applied taking as examples $M$-cubics (that is those for which the real locus has the richest topology) and $(M-1)$-cubics (the next case with respect to complexity of the real locus). It happens that there is one chiral class of $M$-cubics and three chiral classes of $(M-1)$-cubics, contrary to two achiral classes of $M$-cubics and three achiral classes of $(M-1)$-cubics.Comment: 25 pages, 8 figure

Explicit integration of one problem of motion of the generalized Kowalevski top

In the problem of motion of the Kowalevski top in a double force field the 4-dimensional invariant submanifold of the phase space was pointed out by M.P.Kharlamov (Mekh. Tverd. Tela, 32, 2002). We show that the equations of motion on this manifold can be separated by the appropriate change of variables, the new variables s1, s2 being elliptic functions of time. The natural phase variables (components of the angular velocity and the direction vectors of the forces with respect to the movable basis) are expressed via s1, s2 explicitly in elementary algebraic functions.Comment: 6 page