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1. 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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2. Symmetries of Calabi-Yau prepotentials with isomorphic flops

   https://doi.org/10.1007/JHEP02(2023)175  

   Lukas, Andre; Ruehle, Fabian (February 2023, Journal of High Energy Physics)

   A bstract Calabi-Yau threefolds with infinitely many flops to isomorphic manifolds have an extended Kähler cone made up from an infinite number of individual Kähler cones. These cones are related by reflection symmetries across flop walls. We study the implications of this cone structure for mirror symmetry, by considering the instanton part of the prepotential in Calabi-Yau threefolds. We show that such isomorphic flops across facets of the Kähler cone boundary give rise to symmetry groups isomorphic to Coxeter groups. In the dual Mori cone, non-flopping curve classes that are identified under these groups have the same Gopakumar-Vafa invariants. This leads to instanton prepotentials invariant under Coxeter groups, which we make manifest by introducing appropriate invariant functions. For some cases, these functions can be expressed in terms of theta functions whose appearance can be linked to an elliptic fibration structure of the Calabi-Yau manifold. 

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3. 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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4. Free quotients of favorable Calabi-Yau manifolds

   https://doi.org/10.1007/JHEP07(2022)116  

   Gray, James; Wang, Juntao (July 2022, Journal of High Energy Physics)

   A bstract Non-simply connected Calabi-Yau threefolds play a central role in the study of string compactifications. Such manifolds are usually described by quotienting a simply connected Calabi-Yau variety by a freely acting discrete symmetry. For the Calabi-Yau threefolds described as complete intersections in products of projective spaces, a classification of such symmetries descending from linear actions on the ambient spaces of the varieties has been given in [16]. However, which symmetries can be described in this manner depends upon the description that is being used to represent the manifold. In [24] new, favorable, descriptions were given of this data set of Calabi-Yau threefolds. In this paper, we perform a classification of cyclic symmetries that descend from linear actions on the ambient spaces of these new favorable descriptions. We present a list of 129 symmetries/non-simply connected Calabi-Yau threefolds. Of these, at least 33, and potentially many more, are topologically new varieties. 

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5. Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups

   https://doi.org/10.1016/j.aim.2025.110310  

   Pan, Jiayin (July 2025, Advances in Mathematics)

   We study the fundamental group of an open $$n$$-manifold $$M$$ of nonnegative Ricci curvature with additional stability conditions on $$\widetilde{M}$$, the Riemannian universal cover of $$M$$. We prove that if every asymptotic cone of $$\widetilde{M}$$ is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric cone, then $$\pi_1(M)$$ is finitely generated and contains a normal abelian subgroup of finite index; if in addition $$\widetilde{M}$$ has Euclidean volume growth of constant at least $$L$$, then we can bound the index of that abelian subgroup by a constant $C(n,L)$. In particular, our result implies that if $$\widetilde{M}$$ has Euclidean volume growth of constant at least $$1-\epsilon(n)$$, then $$\pi_1(M)$$ is finitely generated and $C(n)$-abelian. 

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