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1. On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces

   Cifarelli, Charles ; Conlon, Ronan J. ; Deruelle, Alix ( March 2022 , ArXivorg)

   We show that the underlying complex manifold of a complete non-compact two-dimensional shrinking gradient Kähler-Ricci soliton (M,g,X) with soliton metric g with bounded scalar curvature Rg whose soliton vector field X has an integral curve along which Rg↛0 is biholomorphic to either C×P1 or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension. 

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2. A new complete two-dimensional shrinking gradient Kähler-Ricci soliton

   Bamler, Richard H. ; Cifarelli, Charles ; Conlon, Ronan J. ; Deruelle, Alix ( June 2022 , ArXivorg)

   We prove the existence of a unique complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on the blowup of C×P1 at one point. This completes the classification of such solitons in two complex dimensions. 

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3. An Aubin continuity path for shrinking gradient Kähler-Ricci solitons

   Cifarelli, Charles ; Conlon, Ronan J. ; Deruelle, Alix ( May 2022 , ArXivorg)

   Let D be a toric Kähler-Einstein Fano manifold. We show that any toric shrinking gradient Kähler-Ricci soliton on certain proper modifications of C×D satisfies a complex Monge-Ampère equation. We then set up an Aubin continuity path to solve this equation and show that it has a solution at the initial value of the path parameter. This we do by implementing another continuity method. 

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4. The existence of the Kähler–Ricci soliton degeneration

   https://doi.org/10.1017/fmp.2023.5  

   Blum, Harold ; Liu, Yuchen ; Xu, Chenyang ; Zhuang, Ziquan ( January 2023 , Forum of Mathematics, Pi)

   Abstract We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler–Ricci soliton when the ground field . 

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5. Classification of asymptotically conical Calabi-Yau manifolds

   Conlon, Ronan J. ; Hein, Hans-Joachim ( January 2022 , ArXivorg)

   A Riemannian cone (C,gC) is by definition a warped product C=R+×L with metric gC=dr2⊕r2gL, where (L,gL) is a compact Riemannian manifold without boundary. We say that C is a Calabi-Yau cone if gC is a Ricci-flat Kähler metric and if C admits a gC-parallel holomorphic volume form; this is equivalent to the cross-section (L,gL) being a Sasaki-Einstein manifold. In this paper, we give a complete classification of all smooth complete Calabi-Yau manifolds asymptotic to some given Calabi-Yau cone at a polynomial rate at infinity. As a special case, this includes a proof of Kronheimer's classification of ALE hyper-Kähler 4-manifolds without twistor theory. 

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