Coates Tom, Kasprzyk Alexander M, Pitton Giuseppe, Tveiten Ketil
Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK.
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK.
Proc Math Phys Eng Sci. 2021 Oct;477(2254):20210584. doi: 10.1098/rspa.2021.0584. Epub 2021 Oct 20.
We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), which we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are expected to correspond to Fano varieties with terminal locally toric singularities. We prove that there are exactly 10 mutation classes of rigid MMLPs in two variables; under mirror symmetry these correspond one-to-one with the 10 deformation classes of smooth del Pezzo surfaces. Furthermore, we give a computer-assisted classification of rigid MMLPs in three variables with reflexive Newton polytope; under mirror symmetry these correspond one-to-one with the 98 deformation classes of three-dimensional Fano manifolds with very ample anti-canonical bundle. We compare our proposal to previous approaches to constructing mirrors to Fano varieties, and explain why mirror symmetry in higher dimensions necessarily involves varieties with terminal singularities. Every known mirror to a Fano manifold, of any dimension, is a rigid MMLP.
我们引入了一类洛朗多项式，称为最大可变洛朗多项式（MMLP），我们认为在镜像对称下它与法诺簇相对应。其中一个子类，称为刚性的，预计与具有终端局部环面奇点的法诺簇相对应。我们证明在两个变量中恰好有10个刚性MMLP的突变类；在镜像对称下，这些与光滑德尔佩佐曲面的10个变形类一一对应。此外，我们给出了具有自反牛顿多面体的三个变量的刚性MMLP的计算机辅助分类；在镜像对称下，这些与具有非常丰富反典范丛的三维法诺流形的98个变形类一一对应。我们将我们的提议与以前构造法诺簇镜像的方法进行比较，并解释为什么高维镜像对称必然涉及具有终端奇点的簇。任何维度的法诺流形的每个已知镜像都是一个刚性MMLP。
