More Like this

1. Cubic fourfolds with an involution

   https://doi.org/10.1090/tran/8811  

   Marquand, Lisa (December 2022, Transactions of the American Mathematical Society)

   There are three types of involutions on a cubic fourfold; two of anti-symplectic type, and one symplectic. Here we show that cubics with involutions exhibit the full range of behaviour in relation to rationality conjectures. Namely, we show a general cubic fourfold with a symplectic involution has no associated $K3$surface and is conjecturely irrational. In contrast, a cubic fourfold with a particular anti-symplectic involution has an associated $K3$, and is in fact rational. We show such a cubic is contained in the intersection of all non-empty Hassett divisors; we call such a cubic Hassett maximal. We study the algebraic and transcendental lattices for cubics with an involution both lattice theoretically and geometrically. 

   more » « less
2. An inverse Jacobian algorithm for Picard curves

   https://doi.org/10.1007/s40993-021-00253-1  

   Lario, Joan-C.; Somoza, Anna; Vincent, Christelle (June 2021, Research in Number Theory)

   null (Ed.)

   Abstract We study the inverse Jacobian problem for the case of Picard curves over $${\mathbb {C}}$$ C . More precisely, we elaborate on an algorithm that, given a small period matrix $$\varOmega \in {\mathbb {C}}^{3\times 3}$$ Ω ∈ C 3 × 3 corresponding to a principally polarized abelian threefold equipped with an automorphism of order 3, returns a Legendre–Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike–Weng (Math Comput 74(249):499–518, 2005) which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain equations of all the isomorphism classes of Picard curves with maximal complex multiplication by the maximal order of the sextic CM-fields with class number at most $$4$$ 4 . In particular, we obtain the complete list of maximal CM Picard curves defined over $${\mathbb {Q}}$$ Q . In the appendix, Vincent gives a correction to the generalization of Takase’s formula for the inverse Jacobian problem for hyperelliptic curves given in [Balakrishnan–Ionica–Lauter–Vincent, LMS J. Comput. Math., 19(suppl. A):283-300, 2016]. 

   more » « less
3. Quasi-projectivity of images of mixed period maps

   https://doi.org/10.1515/crelle-2023-0070  

   Bakker, Benjamin; Brunebarbe, Yohan; Tsimerman, Jacob (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We prove a mixed version of a conjecture of Griffiths: that the closure of the image of any admissible mixed period map is quasi-projective, with a natural ample bundle. Specifically, we consider the map from the image of the mixed period map to the image of the period map of the associated graded. On the one hand, we show in a precise manner that the parts of this map parametrizing extension data of non-adjacent-weight pure Hodge structures are quasi-affine. On the other hand, extensions of adjacent-weight pure polarized Hodge structures are parametrized by a compact complex torus (the intermediate Jacobian) equipped with a natural theta bundle which is ample in Griffiths transverse directions.Our proof makes heavy use of o-minimality, and recent work with B. Klingler associating an ${ℝ}_{\mathrm{an},\exp }${\mathbb{R}_{\mathrm{an},\exp}}-definable structure to mixed period domains and admissible mixed period maps. 

   more » « less
4. Orientifold Calabi-Yau threefolds with divisor involutions and string landscape

   https://doi.org/10.1007/JHEP03(2022)087  

   Altman, Ross; Carifio, Jonathan; Gao, Xin; Nelson, Brent D. (March 2022, Journal of High Energy Physics)

   A bstract We establish an orientifold Calabi-Yau threefold database for h 1 , 1 ( X ) ≤ 6 by considering non-trivial ℤ 2 divisor exchange involutions, using a toric Calabi-Yau database ( www.rossealtman.com/tcy ). We first determine the topology for each individual divisor (Hodge diamond), then identify and classify the proper involutions which are globally consistent across all disjoint phases of the Kähler cone for each unique geometry. Each of the proper involutions will result in an orientifold Calabi-Yau manifold. Then we clarify all possible fixed loci under the proper involution, thereby determining the locations of different types of O -planes. It is shown that under the proper involutions, one typically ends up with a system of O 3 /O 7-planes, and most of these will further admit naive Type IIB string vacua. The geometries with freely acting involutions are also determined. We further determine the splitting of the Hodge numbers into odd/even parity in the orbifold limit. The final result is a class of orientifold Calabi-Yau threefolds with non-trivial odd class cohomology ( $$ {h}_{-}^{1,1} $$ h − 1 , 1 ( X/σ * ) ≠ 0). 

   more » « less
5. Scaling Limit of Multi-Type Invariant Measures via the Directed Landscape

   https://doi.org/10.1093/imrn/rnae168  

   Busani, Ofer; Seppäläinen, Timo; Sorensen, Evan (August 2024, International Mathematics Research Notices)

   Abstract This paper studies the large scale limits of multi-type invariant distributions and Busemann functions of planar stochastic growth models in the Kardar–Parisi–Zhang (KPZ) class. We identify a set of sufficient hypotheses for convergence of multi-type invariant measures of last-passage percolation (LPP) models to the stationary horizon (SH), which is the unique multi-type stationary measure of the KPZ fixed point. Our limit theorem utilizes conditions that are expected to hold broadly in the KPZ class, including convergence of the scaled last-passage process to the directed landscape. We verify these conditions for the six exactly solvable models whose scaled bulk versions converge to the directed landscape, as shown by Dauvergne and Virág. We also present a second, more general, convergence theorem with future applications to polymer models and particle systems. Our paper is the first to show convergence to the SH without relying on information about the structure of the multi-type invariant measures of the prelimit models. These results are consistent with the conjecture that the SH is the universal scaling limit of multi-type invariant measures in the KPZ class. 

   more » « less