The authors describe the relationships between categories of B-branes in different phases of the non-Abelian gauged linear sigma model. The relationship is described explicitly for the model proposed by Hori and Tong with non-Abelian gauge group that connects two non-birational Calabi-Yau varieties studied by Rødland. A grade restriction rule for this model is derived using the hemisphere partition function and it is used to map B-type D-branes between the two Calabi-Yau varieties.

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, we prove that if a compact Kähler manifold X has a smooth Hermitian metric ω such that (T_X,ω) is uniformly RC-positive, then X is projective and rationally connected. Conversely, we show that, if a projective manifold X is rationally connected, then there exists a uniformly RC-positive complex Finsler metric on T_X .