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We investigate the Chern–Ricci flow, an evolution equation of Hermitian metrics generalizing the Kähler–Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a Kähler–Einstein metric from the base. Some of our estimates are new even for the Kähler–Ricci flow. A consequence of our result is that, on every minimal non-Kähler surface of Kodaira dimension one, the Chern–Ricci flow converges in the sense of Gromov–Hausdorff to an orbifold Kähler–Einstein metric on a Riemann surface.

In this paper, using connections between Clifford-Wolf isometries and Killing vector fields of constant length on a given Riemannian manifold, we classify simply connected Clifford-Wolf homogeneous Riemannian manifolds. We also get the classification of complete simply connected Riemannian manifolds with the Killing property defined and studied previously by J.E. D’Atri and H.K. Nickerson. In the last part of the paper we study properties of Clifford-Killing spaces, that is, real vector spaces of Killing vector fields of constant length, on odd-dimensional round spheres,and discuss numerous connections between these spaces and various classical objects.

In this paper, we introduce a motivic version of To¨en’s derived Hall algebra. Then we point out that the two kinds of Hall algebras in the sense of To¨en and Kontsevich–Soibelman, respectively, are Drinfeld dual pairs, not only in the classical case (by counting over finite fields) but also in the motivic version. Consequently they are canonically isomorphic. All proofs, including that for the most important associative property, are deduced in a self-contained way by analyzing the symmetry properties around the octahedral axiom, a method we used previously

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