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1. Automorphisms and periods of cubic fourfolds

   https://doi.org/10.1007/s00209-021-02810-x  

   Laza, Radu; Zheng, Zhiwei (February 2022, Mathematische Zeitschrift)

   Abstract We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174, 960) for the automorphism group (non-necessarily symplectic) among all smooth cubic fourfolds. 

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2. Shafarevich’s Conjecture for Families of Hypersurfaces over Function Fields

   https://doi.org/10.1093/imrn/rnaf225  

   Engel, Philip; Lin, Alice; Tayou, Salim (August 2025, International Mathematics Research Notices)

   Given a smooth quasi-projective complex algebraic variety $$\mathcal{S}$$, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over $$\mathcal{S}$$ of degree $$d$$ in $$\mathbb{P}_{\mathbb C}^{n+1}$$. We prove that the finiteness is uniform in $$\mathcal{S}$$ and give examples where the result is sharp. We also prove similar results for certain complete intersections in $$\mathbb{P}_{\mathbb C}^{n+1}$$ of higher codimension and more generally for algebraic varieties whose moduli space admits a period map that satisfies the infinitesimal Torelli theorem. 

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3. Newton Polygon Stratification of the Torelli Locus in Unitary Shimura Varieties

   https://doi.org/10.1093/imrn/rnaa306  

   Li, Wanlin; Mantovan, Elena; Pries, Rachel; Tang, Yunqing (December 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $$p$$ reduction of certain Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is non-empty. This allows us to give a positive answer, under some compatibility conditions, to a question of Oort about smooth curves in characteristic $$p$$ whose Newton polygons are an amalgamate sum. As an application, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic covers of the projective line. Most of these arise in inductive systems that demonstrate unlikely intersections of the open Torelli locus with the Newton polygon stratification in Siegel modular varieties. In addition, for the 20 special Shimura varieties found in Moonen’s work, we prove that all Newton polygon strata intersect the open Torelli locus (if $p>>0$ in the supersingular cases). 

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4. The Torelli theorem for gravitational instantons

   https://doi.org/10.1017/fms.2022.67  

   Collins, Tristan; Jacob, Adam; Lin, Yu-Shen (January 2022, Forum of Mathematics, Sigma)

   Abstract We give a short proof of the Torelli theorem for $ALH^*$ gravitational instantons using the authors’ previous construction of mirror special Lagrangian fibrations in del Pezzo surfaces and rational elliptic surfaces together with recent work of Sun-Zhang. In particular, this includes an identification of 10 diffeomorphism types of $$ALH^*_b$$ gravitational instantons. 

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5. The Moduli Space of Cubic Threefolds with a Non-Eckardt Type Involution via Intermediate Jacobians

   https://doi.org/10.1093/imrn/rnad113  

   Casalaina-Martin, Sebastian; Marquand, Lisa; Zhang, Zheng (June 2023, International Mathematics Research Notices)

   Abstract There are two types of involutions on a cubic threefold: the Eckardt type (which has been studied by the first named and the third named authors) and the non-Eckardt type. Here we study cubic threefolds with a non-Eckardt type involution, whose fixed locus consists of a line and a cubic curve. Specifically, we consider the period map sending a cubic threefold with a non-Eckardt type involution to the invariant part of the intermediate Jacobian. The main result is that the global Torelli Theorem holds for the period map. To prove the theorem, we project the cubic threefold from the pointwise fixed line and exhibit the invariant part of the intermediate Jacobian as a Prym variety of a (pseudo-)double cover of stable curves. The proof relies on a result of Ikeda and Naranjo–Ortega on the injectivity of the related Prym map. We also describe the invariant part of the intermediate Jacobian via the projection from a general invariant line and show that the two descriptions are related by the bigonal construction. 

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