Volume 211, Nº 2 (2020)

Let $k$ be an uncountable algebraically closed field of characteristic $0$, and let $X$ be a smooth projective connected variety of dimension $2p$, embedded into $\mathbb P^m$ over $k$. Let $Y$ be a hyperplane section of $X$, and let $A^p(Y)$ and $A^{p+1}(X)$ be the groups of algebraically trivial algebraic cycles of codimension $p$ and $p+1$ modulo rational equivalence on $Y$ and $X$, respectively. Assume that, whenever $Y$ is smooth, the group $A^p(Y)$ is regularly parametrized by an abelian variety $A$ and coincides with the subgroup of degree $0$ classes in the Chow group $\operatorname{CH}^p(Y)$. We prove that the kernel of the push-forward homomorphism from $A^p(Y)$ to $A^{p+1}(X)$ is the union of a countable collection of shifts of a certain abelian subvariety $A_0$ inside $A$. For a very general hyperplane section $Y$ either $A_0=0$ or $A_0$ coincides with an abelian subvariety $A_1$ in $A$ whose tangent space is the group of vanishing cycles $H^{2p-1}(Y)_\mathrm{van}$. Then we apply these general results to sections of a smooth cubic fourfold in $\mathbb P^5$. Bibliography: 33 titles.

A class of integro-differential aggregation equations with nonlinear parabolic term $b(x,u)_t$ is considered on a compact Riemannian manifold $\mathscr M$. The divergence term in the equations can degenerate with loss of coercivity and may contain nonlinearities of variable order. The impermeability boundary condition on the boundary $\partial\mathscr M\times[0,T]$ of the cylinder $Q^T=\mathscr M\times[0,T]$ is satisfied if there are no external sources of ‘mass’ conservation, $\int_\mathscr Mb(x,u(x,t)) d\nu=\mathrm{const}$. In a cylinder $Q^T$ for a sufficiently small $T$, the mixed problem for the aggregation equation is shown to have a bounded solution. The existence of a bounded solution of the problem in the cylinder $Q^\infty=\mathscr M\times[0,\infty)$ is proved under additional conditions. For equations of the form $b(x,u)_t=\Delta A(x,u)-\operatorname{div}(b(x,u)\mathscr G(u))+f(x,u)$ with the Laplace-Beltrami operator $\Delta$ and an integral operator $\mathscr G(u)$, the mixed problem is shown to have a unique bounded solution. Bibliography: 26 titles.

Left-invariant optimal control problems on Lie groups are considered. When studying the optimality of extreme trajectories, the crucial role is played by symmetries of the exponential map that are induced by symmetries of the conjugate subsystem of the Hamiltonian system of the Pontryagin maximum principle. A general construction is obtained for these symmetries of the exponential map for connected Lie groups with generic coadjoint orbits of codimension not exceeding one and with a connected stabilizer. Bibliography: 32 titles.