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1. Wonderful compactifications and rational curves with cyclic action

   https://doi.org/10.1017/fms.2023.26  

   Clader, Emily; Damiolini, Chiara; Li, Shiyue; Ramadas, Rohini (January 2023, Forum of Mathematics, Sigma)

   Abstract We prove that the moduli space of rational curves with cyclic action, constructed in our previous work, is realizable as a wonderful compactification of the complement of a hyperplane arrangement in a product of projective spaces. By proving a general result on such wonderful compactifications, we conclude that this moduli space is Chow-equivalent to an explicit toric variety (whose fan can be understood as a tropical version of the moduli space), from which a computation of its Chow ring follows. 

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2. Adic tropicalizations and cofinality of Gubler models

   https://doi.org/10.1112/blms.70001  

   Foster, Tyler; Payne, Sam (January 2025, Bulletin of the London Mathematical Society)

   Abstract We introduce adic tropicalizations for subschemes of toric varieties as limits of Gubler models associated to polyhedral covers of the ordinary tropicalization. Our main result shows that Huber's adic analytification of a subscheme of a toric variety is naturally isomorphic to the inverse limit of its adic tropicalizations, in the category of locally topologically ringed spaces. The key new technical idea underlying this theorem is cofinality of Gubler models, which we prove for projective schemes and also for more general compact analytic domains in closed subschemes of toric varieties. In addition, we introduce a ‐topology and structure sheaf on ordinary tropicalizations, and show that Berkovich analytifications are limits of ordinary tropicalizations in the category of topologically ringed topoi. 

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3. Tropical and non-Archimedean Monge–Ampère equations for a class of Calabi–Yau hypersurfaces

   https://doi.org/10.1016/j.aim.2024.109494  

   Hultgren, Jakob; Jonsson, Mattias; Mazzon, Enrica; McCleerey, Nicholas (March 2024, Advances in Mathematics)

   For a large class of maximally degenerate families of Calabi–Yau hypersurfaces of complex projective space, we study non- Archimedean and tropical Monge–Ampère equations, taking place on the associated Berkovich space, and the essential skeleton therein, respectively. For a symmetric measure on the skeleton, we prove that the tropical equation admits a unique solution, up to an additive constant. Moreover, the solution to the non-Archimedean equation can be derived from the tropical solution, and is the restriction of a continuous semipositive toric metric on projective space. Together with the work of Yang Li, this implies the weak metric SYZ conjecture on the existence of special Lagrangian fibrations in our setting. 

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4. Combinatorial aspects of virtually Cohen-Macaulay sheaves

   https://doi.org/10.1112/tlm3.12036  

   Berkesch, Christine; Klein, Patricia; Loper, Michael C.; Yang, Jay (July 2022, Seminaire lotharingien de combinatoire)

   Bauer, Tomer; Liu, Gaku (Ed.)

   There is an abundance of deep literature on the use of free resolutions to study modules and vector bundle resolutions to study coherent sheaves. When studying a module over the Cox ring of a smooth projective toric variety X, each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which are amenable to algebraic and combinatorial study and also capture desirable geometric information. In this extended abstract, we continue this program in the combinatorially-rich Stanley–Reisner setting. In particular, when X is a product of projective spaces, we produce a large new class of virtually Cohen–Macaulay Stanley–Reisner rings. After augmenting the simplicial complexes associated to these Stanley–Reisner rings with a coloring that reflects the product structure on X, our primary tool is Reisner’s criterion, whose conclusion we interpret in the virtual setting. We also provide two constructions of short virtual resolutions for use beyond the Stanley–Reisner case. 

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5. Rational homotopy type and nilpotency of mapping spaces between Quaternionic projective spaces

   https://doi.org/10.15673/pigc.v16i2.2648  

   Abebaw, Tilahun; Gatsinzi, Jean-Baptiste; Yeruk, Smegnsh Demelash (June 2024, Proceedings of the International Geometry Center)

   The rational homotopy type of a mapping space is a way to describe the structure of the space using the algebra of its homotopy groups and the differential graded algebra of its cochains. An L∞-model is a graded Lie algebra with a family of higher-order brackets satisfying the generalized Jacobi identity and antisymmetry. It can be used to study the rational homotopy type of a space. The nilpotency index of an L∞-model is useful in understanding a space's algebraic structure. In this paper, we compute the rational homotopy type of the component of some mapping spaces between projective spaces and determine the nilpotency index of corresponding L∞-models. 

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