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1. Steady gradient Kähler-Ricci solitons on crepant resolutions of Calabi-Yau cones

   Conlon, Ronan J. ; Deruelle, Alix ( June 2020 , ArXivorg)

   We show that, up to the flow of the soliton vector field, there exists a unique complete steady gradient Kähler-Ricci soliton in every Kähler class of an equivariant crepant resolution of a Calabi-Yau cone converging at a polynomial rate to Cao's steady gradient Kähler-Ricci soliton on the cone. 

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2. 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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3. Hermitian–Yang–Mills connections on some complete non-compact Kähler manifolds

   https://doi.org/10.1007/s00208-024-02849-1  

   Zhang, Junsheng ( April 2024 , Mathematische Annalen)

   We give an algebraic criterion for the existence of projectively Hermitian–Yang–Mills metrics on a holomorphic vector bundleEover some complete non-compact Kähler manifolds$$(X,\omega )$$$(X,\omega )$, whereXis the complement of a divisor in a compact Kähler manifold and we impose some conditions on the cohomology class and the asymptotic behaviour of the Kähler form$$\omega $$$\omega$. We introduce the notion of stability with respect to a pair of (1, 1)-classes which generalizes the standard slope stability. We prove that this new stability condition is both sufficient and necessary for the existence of projectively Hermitian–Yang–Mills metrics in our setting.

    

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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. Special Lagrangian submanifolds of log Calabi-Yau manifolds

   Collins, Tristan C. ; Jacob, A. ; Lin, Y.-S. ( January 2021 , Duke mathematical journal)

   null (Ed.)

   We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D is a smooth divisor in the linear system of K_Y with D^2=d, then X=Y/D admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y=P^2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type I_d fiber appearing as a singular fiber in a rational elliptic surface. 

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