We show that a Ricci flow in four dimensions can develop singularities modeled on the Eguchi–Hanson space. In particular, we prove that starting from a class of asymptotically cylindrical U(2)-invariant initial metrics on $TS^2$, a Type II singularity modeled on the Eguchi–Hanson space develops in finite time. Furthermore, we show that for these Ricci flows the only possible blow-up limits are (i) the Eguchi–Hanson space, (ii) the flat ${\mathbb {R}}^4 /{\mathbb {Z}}_2$ orbifold, (iii) the 4d Bryant soliton quotiented by ${\mathbb {Z}}_2$, and (iv) the shrinking cylinder ${\mathbb {R}}\times {\mathbb {R}}P^3$. As a byproduct of our work, we also prove the existence of a new family of Type II singularities caused by the collapse of a two-sphere of self-intersection $|k| \ge 3$.
Over any smooth algebraic variety over a p-adic local field k, we construct the de Rham comparison isomorphisms for the étale cohomology with partial compact support of de Rham ${\mathbb {Z}}_p$-local systems, and show that they are compatible with Poincaré duality and with the canonical morphisms among such cohomology. We deduce these results from their analogues for rigid analytic varieties that are Zariski open in some proper smooth rigid analytic varieties over k. In particular, we prove finiteness of étale cohomology with partial compact support of any ${\mathbb {Z}}_p$-local systems, and establish the Poincaré duality for such cohomology after inverting p.
Motivated by Witten’s work, we propose a K-theoretic Verlinde/Grassmannian correspondence which relates the GL Verlinde numbers to the K-theoretic quasimap invariants of the Grassmannian. We recover these two types of invariants by imposing different stability conditions on the gauged linear sigma model associated with the Grassmannian. We construct two families of stability conditions connecting the two theories and prove two wall-crossing results. We confirm the Verlinde/Grassmannian correspondence in the rank two case.