# Our approach

A general overview of our program, aimed at a general audience, can be found here.

At a basic level, a (projective, say) algebraic variety is given by a finite set of homogeneous polynomial equations. The possibility of writing these equations down explicitly is what lends algebraic geometry its special flavour, and also its great power in applications. The minimal model program (MMP), started by Mori, Reid and others in the 1980s, simplifies a given variety by a sequence of elementary surgery operations, called flips and extremal contractions, to reduce it to one of three classes: Fano (positively curved), Calabi–Yau (zero curvature), or general type (negatively curved). Two key open problems are the classification of $$\mathbb{Q}$$-Fano 3-folds and nonsingular Fano 4-folds. Even less is known about varieties in the other two classes; for example, we still don’t known if there are finitely many diffeomorphism types of Calabi-Yau 3-folds. Trying to understand projective algebraic varieties explicitly often leads to problems about Gorenstein graded rings. Such rings have been studied, in small codimension, from an algebraic point of view, but the geometric meaning of the resulting structures is still poorly understood. A fundamental open problem in this area is the classification of codimension-4 Gorenstein rings. Reid’s work has tackled the primary issue of constructing algebraic varieties head-on: his study of graded rings helped found the MMP and produced a wealth of new constructions of varieties. Despite much progress by Reid and his school, there is still a great deal to discover about varieties of the three classes. Direct methods typically require fine judgement, and lots of ingenuity and hard work to put in place just one variety.

The study of Calabi–Yau 3-folds was transformed in the early 1990s, when they arose in string theory–a leading candidate for a theory of quantum gravity, which postulates strings instead of particles as the elementary objects of physics. String theory is only consistent if the strings vibrate in a 10-dimensional background, with the 6 extra (real) dimensions curled up in a (complex) Calabi-Yau 3-fold. Two Calabi–Yau 3-folds X and Y are called mirrors if the type A theory on X “gives the same physics” as the type B theory on Y. Around 1990, Candelas et al. astonished the mathematical community by using  string-theoretic arguments to predict the number of rational curves of each degree on the quintic 3-fold in $$\mathbb{P}^4$$, the simplest Calabi-Yau 3-fold. This was accomplished by carrying out calculations of a completely different nature on the mirror quintic. Starting in 2001, Gross and Siebert proposed a systematic construction of mirror pairs by deforming away from a toric degeneration. Their research program has now progressed to the point that a theoretical explanation for the mirror symmetry phenomenon is within reach, but implementing even simple cases of their algorithm is fraught with difficulties.

Coates and Corti have given new insights into the classification of Fano varieties via mirror symmetry. The mirror to a Fano variety is given by a Laurent polynomial called a Landau–Ginzburg (LG) model; this version of mirror symmetry is called the Fano/LG correspondence. Laurent polynomials and their Newton polytopes are combinatorial objects adapted to computer manipulation and classification. Coates, Corti, Kasprzyk, and their collaborators classified LG models in 3 variables algorithmically. Their result recovers the Mori–Mukai classification of Fano 3-folds from the LG models. For the analogous classification of LG models in 4 variables, Coates and Kasprzyk developed a substantial computational infrastructure and software suite. This work raises the prospect of using the Fano/LG correspondence to attack the classification of Fano varieties, in particular $$\mathbb{Q}$$-Fano 3-folds and nonsingular Fano 4-folds. As it stands, though, the Fano/LG correspondence is not a rigorous mathematical theory: it predicts, but does not establish, Fano classifications.

Our joint program comes from the realization that these three research programs are fundamentally linked. Combining the explicit examples of mirror symmetry of Coates–Corti (which require a theoretical underpinning) and the theoretical advances of Gross–Siebert (which require examples to turn into a usable machine) will enable significant progress in both areas. Brown and Reid’s work on the equations of 3-dimensional flips, following Mori, contains a translation into polynomials of the structures of the Gross–Siebert algorithm. This was first noticed by Hacking, but appeared again unexpectedly in the Warwick thesis of Tom Ducat on the equations of 3-fold extremal contractions. These observations suggest the possibility of developing new graded ring methods to run important cases of the Gross–Siebert algorithm. On the other hand, it also highlights the Gross–Siebert algorithm as the source of new and powerful construction techniques in algebraic geometry. Finally, the program of Coates–Corti will populate the landscape for explicit constructions and provide challenges for Reid’s program, which in turn will allow explicit versions of the Fano/LG correspondence.

Specific objectives of the program include:

1. Construct many new varieties, writing down their equations explicitly in formats that explain their syzygies and deformation theory. Making and classifying surfaces of general type, $$\mathbb{Q}$$-Fano 3-folds, Calabi-Yau 3-folds, Fano 4-folds and key varieties controlling many of these constructions.

2. Develop a structure theory of Gorenstein rings beyond codimension 3; this includes the practicalities of using unprojection in new geometric constructions, and turning the general structure theory in codimension 4 into a set of tools that are effective in practice.

3. Give an intrinsic construction of mirror partners, in substantial generality, based on log Gromov-Witten theory. Prove that the mirror pairs meet the standard expectations in genus 0 Gromov-Witten theory.

4. Prove a general version of the Fano/LG correspondence, relating families of smooth Fano orbifolds to maximally mutable Laurent polynomials.

5. Classify Fano/LG pairs in low dimensions. This consists of a classification, up to mutation, of polytopes and maximally mutable Laurent polynomials on the B-side of mirror symmetry, and corresponding Fano orbifolds and explicit constructions on the A-side.

6. Publish our classifications of Fanos and Calabi-Yaus as online databases, together with a suite of software tools to analyse and study these objects.

7. Use the motives of pencils of Calabi-Yau 3-folds to construct L-functions of weight-3 Galois representations with h^{3,0} = 1, small rank and low conductor. Find automorphic interpretations of these L-functions, thus testing the Langlands program in global arithmetic. Study special values and congruences.