Abstract. In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalisations of holomorphic discs).Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock-Goncharov dual basis conjecture, [FG06], Conjecture 4.3. In particular, under suitable hypotheses, for each Y the partial compactification of an affine cluster variety U given by allowing some frozen variables to vanish, we obtain canonical bases for H 0 (Y, O Y ) extending to a basis of H 0 (U, O U ). Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell U in the basic affine space Y we obtain a canonical basis of each irreducible representation of SL r , parameterized by a set which each choice of seed identifies with integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation theoretic considerations. Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type.

We give a canonical synthetic construction of the mirror family to pairs (Y, D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family is constructed as the spectrum of an explicit algebra structure on a vector space with canonical basis and multiplication rule defined in terms of counts of rational curves on Y meeting D in a single point. The elements of the canonical basis are called theta functions. Their construction depends crucially on the Gromov-Witten theory of the pair (Y, D). Contents1 2 MARK GROSS, PAUL HACKING, AND SEAN KEEL 6. Extending the family over boundary strata 74 6.1. Theorem 0.2 in the case that (Y, D) has a toric model 74 6.2. Proof of Theorems 0.1 and 0.2 in general 78 6.3. The case that (Y, D) is positive 81 7. Looijenga's conjecture 84 7.1. Duality of cusp singularities 84 7.2. Cusp family 86 7.3. Thickening of the cusp family 92 7.4. Smoothness 102 References 105and let X i J /S i J be the family over S i J defined by O X i J = p * OXi. Arguing as above we find that R 1 p * OXi ≃ O S i J . Pushing forward the exact sequence 0 → OX i ·t i+1

An important advance in algebraic geometry in the last ten years is the theory of variation of geometric invariant theory quotient (VGIT), see [BP], [H], [DH], [T]. Several authors, have observed that VGIT has implications for birational geometry, e.g. it gives natural examples of Mori flips and contractions, [R2], [DH], [T]. In this paper we observe that the connection is quite fundamental -Mori theory is, at an almost tautological level, an instance of VGIT, see (2.14). Here are more details:Given a projective variety X a natural problem is to understand the collection of all morphisms (with connected fibres) from X to other projective varieties. Ideally one would like to decompose each map into simple steps, and parameterize the possibilities, both for the maps, and for the factorizations of each map. An important insight, principally of Reid and Mori, is that the picture is often simplified if one allows in addition to morphisms, small modifications, i.e. rational maps that are isomorphisms in codimension one. With this extension a natural framework is the category of rational contractions. In many cases there is a nice combinatorial parameterization, given by a decomposition of a convex polyhedral cone, the cone of effective divisors N E 1 (X), into convex polyhedral chambers, which we call Mori chambers. Instances of this structure have been studied in various circumstances: The existence of such a parameterizing decomposition for Calabi-Yau manifolds was conjectured by Morrison, motivated by ideas in mirror symmetry [M]. The conjecture was proven in dimension three by Kawamata, [Ka]. Oda and Park study the decomposition for toric varieties, motivated by questions in combinatorics, [OP]. Shokurov studies such a decomposition for parameterizing log minimal models, [Sh]. In Geometric Invariant Theory there is a similar combinatorial structure, a decomposition of the G-ample cone into GIT chambers parameterizing GIT quotients, [DH]. The main observation of this paper is that whenever a good Mori chamber decomposition exists, it is in a natural way a GIT decomposition.1The main goal of this paper is to study varieties X with a good Mori chamber decomposition (see §1 for the meaning of good). We call such varieties Mori DreamSpaces. There turn out to be many examples, including quasi-smooth projective toric (or more generally, spherical) varieties, many GIT quotients, and log Fano 3-folds. We will show that a Mori dream space is in a natural way a GIT quotients of affine variety by a torus in a manner generalizing Cox's construction of toric varieties as quotients of affine space, [C]. Via the quotient description, the chamber decomposition of the cone of divisors is naturally identified with the decomposition of the G-ample cone from VGIT, See (2.9). In particular every rational contraction of a Mori dream space comes from GIT, and all possible factorizations of a rational contraction (into other contractions) can be read off from the chamber decomposition. See (2.3), (2.9) and (2.11).Content overview: In §1 we...

No abstract

In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalisations of holomorphic discs).Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock-Goncharov dual basis conjecture, [FG06], Conjecture 4.3. In particular, under suitable hypotheses, for each Y the partial compactification of an affine cluster variety U given by allowing some frozen variables to vanish, we obtain canonical bases for H 0 (Y, O Y ) extending to a basis of H 0 (U, O U ). Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell U in the basic affine space Y we obtain a canonical basis of each irreducible representation of SL r , parameterized by a set which each choice of seed identifies with integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation theoretic considerations.Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type. ContentsMARK GROSS, PAUL HACKING, SEAN KEEL, AND MAXIM KONTSEVICH 3. Broken lines 39 4. Building A from the scattering diagram and positivity of the Laurent phenomenon 45 5. Sign coherence of c-and g-vectors 50 6. The formal Fock-Goncharov Conjecture 57 7. The middle cluster algebra 66 7.1. The middle algebra for A prin 66 7.2. From A prin to A t and X . 69 8. Convexity in the tropical space 76 8.1. Convexity conditions 77 8.2. Convexity criteria 81 8.3. The canonical algebra 84 8.4. Conditions implying A prin has EGM and the full Fock-Goncharov conjecture 88 8.5. Compactifications from positive polytopes 96 9. Partial compactications and represention-theoretic results 9.1. Partial minimal models 9.2. Cones cut out by the tropicalized potential Appendix A. Review of notation and Langlands duality Appendix B. The A and X -varieties with principal coefficients Appendix C. Construction of scattering diagrams C.1. An algorithmic construction of scattering diagrams C.2. The proof of Theorem 1.28 C.3. The proof of Theorem 1.13 References