Cluster Structures on Algebraic Varieties

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At the border of Algebraic Geometry and Representation Theory, cluster algebras and their associated cluster varieties play a more and more important role in mirror symmetry and the construction of toric degenerations of projective varieties. However, their definition by Fomin-Zelevinsky in 2002 and Fock-Goncharov in 2009 still looks bizarre to Algebraic Geometers and makes it difficult to answer even basic algebraic-geometric questions on these structures:1 It is widely assumed that cluster varieties are generalizations of toric varieties, but it is not known how to produce a cluster structure for a given toric varieties besides the algebraic torus itself. The first objective of the project is to devise such a construction.2 The construction of a cluster variety can easily lead to non-Noetherian varieties, and there are no general criteria to decide when a cluster variety is Noetherian or of finite type. Another objective of the project is to provide such criteria.3 Results in the directions of both objectives above will shed new light on the construction of a cluster variety. A third objective of the project is to use this new information to descibe more algebraic-geometric ways to construct cluster varieties.A starting point for these objectives is given by Gross-Hacking-Keel's description of 2-dimensional cluster varieties as gluings of affine planes by elementary transformations. Elementary transformations also exist in higher dimension, and it is plausible to try to use them to generalize Groos-Hacking-Keel's construction. Also, using this construction to describe all 2-dimensional toric varieties as cluster varieties is an important first step towards the first objective.