Under broad hypotheses we derive a scalar reduction of the generalized Kähler–Ricci soliton system. We realize solutions as critical points of a functional, analogous to the classical Aubin energy, defined on an orbit of the natural Hamiltonian action of diffeomorphisms, thought of as a generalized Kähler class. This functional is convex on a large set of paths in this space, and using this we show rigidity of solitons in their generalized Kähler class. As an application we prove uniqueness of the generalized Kähler–Ricci solitons on Hopf surfaces constructed in Streets and Ustinovskiy [Commun. Pure Appl. Math. 74(9), 1896–1914 (2020)], finishing the classification in complex dimension 2.
In this article, we present the concavity of the minimal $L^2$ integrals related to multiplier ideals sheaves on Stein manifolds. As applications, we obtain a necessary condition for the concavity degenerating to linearity, a characterization for 1-dimensional case, and a characterization for the equality in 1-dimensional optimal $L^{2}$ extension problem to hold.
In this note we derive an improved no-local-collapsing theorem of Ricci flow under the scalar curvature bound condition along the worldline of the basepoint. It is a refinement of Perelman’s no-local-collapsing theorem.
Let G be a connected, complex reductive group. In this paper, we classify $G\times G$-equivariant normal ${\mathbb {R}}$-test configurations of a polarized G-compactification. Then, for ${\mathbb {Q}}$-Fano G-compactifications, we express the H-invariants of their equivariant normal ${\mathbb {R}}$-test configurations in terms of the combinatory data. Based on Han and Li “Algebraic uniqueness of Kähler–Ricci flow limits and optimal degenerations of Fano varieties”, we compute the semistable limit of a K-unstable Fano G-compactification. As an application, we show that for the two smooth K-unstable Fano SO$_4({\mathbb {C}})$-compactifications, the corresponding semistable limits are indeed the limit spaces of the normalized Kähler–Ricci flow.
Using an argument of Baldwin–Hu–Sivek, we prove that if K is a hyperbolic fibered knot with fiber F in a closed, oriented 3-manifold Y, and $\widehat{HFK}(Y,K,[F], g(F)-1)$ has rank 1, then the monodromy of K is freely isotopic to a pseudo-Anosov map with no fixed points. In particular, this shows that the monodromy of a hyperbolic L-space knot is freely isotopic to a map with no fixed points.