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1. On the degeneration of asymptotically conical Calabi–Yau metrics

   https://doi.org/10.1007/s00208-021-02229-z  

   Collins, T. (July 2021, Mathematische Annalen)

   null (Ed.)

   We study the degenerations of asymptotically conical Ricci-flat Kahler metrics as the Kahler class degenerates to a semi-positive class. We show that under appropriate assumptions, the Ricci-flat Kahler metrics converge to a incomplete smooth Ricci-flat Kahler metric away from a compact subvariety. As a consequence, we construct singular Calabi–Yau metrics with asymptotically conical behaviour at infinity on certain quasi-projective varieties and we show that the metric geometry of these singular metrics are homeomorphic to the topology of the singular variety. Finally, we will apply our results to study several classes of examples of geometric transitions between Calabi–Yau manifolds. 

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2. 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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3. NC-smooth algebroid thickenings for families of vector bundles and quiver representations

   https://doi.org/10.1112/S0010437X19007115  

   Dyer, Ben; Polishchuk, Alexander (April 2019, Compositio Mathematica)

   In his work on deformation quantization of algebraic varieties Kontsevich introduced the notion of algebroid as a certain generalization of a sheaf of algebras. We construct algebroids which are given locally by NC-smooth thickenings in the sense of Kapranov, over two classes of smooth varieties: the bases of miniversal families of vector bundles on projective curves, and the bases of miniversal families of quiver representations. 

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4. Spectral Data for Spin Higgs Bundles

   https://doi.org/10.1093/imrn/rny296  

   Mukhopadhyay, Swarnava; Wentworth, Richard (January 2019, International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper we determine the spectral data parametrizing Higgs bundles in a generic fiber of the Hitchin map for the case where the structure group is the special Clifford group with fixed Clifford norm. These are spin and “twisted” spin Higgs bundles. The method used relates variations in spectral data with respect to the Hecke transformations for orthogonal bundles introduced by Abe. The explicit description also recovers a result from the geometric Langlands program, which states that the fibers of the Hitchin map are the dual abelian varieties to the corresponding fibers of the moduli spaces of projective orthogonal Higgs bundles (in the even case) and projective symplectic Higgs bundles (in the odd case). 

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5. Numerical homotopies from Khovanskii bases

   https://doi.org/10.1090/mcom/3689  

   Burr, M; Sottile, F; Walker, E (September 2023, Mathematics of Computation)

   We present numerical homotopy continuation algorithms for solving systems of equations on a variety in the presence of a finite Khovanskii basis. These homotopies take advantage of Anderson’s flat degeneration to a toric variety. When Anderson’s degeneration embeds into projective space, our algorithm is a special case of a general toric two-step homotopy algorithm. When Anderson’s degeneration is embedded in a weighted projective space, we explain how to lift to a projective space and construct an appropriate modification of the toric homotopy. Our algorithms are illustrated on several examples using Macaulay2. 

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