Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 042, 32 pages      arXiv:1407.6241

Classification of Rank 2 Cluster Varieties

Travis Mandel
School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, UK

Received May 09, 2018, in final form May 15, 2019; Published online May 27, 2019

We classify rank $2$ cluster varieties (those for which the span of the rows of the exchange matrix is $2$-dimensional) according to the deformation type of a generic fiber $U$ of their ${\mathcal X}$-spaces, as defined by Fock and Goncharov [Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865-930]. Our approach is based on the work of Gross, Hacking, and Keel for cluster varieties and log Calabi-Yau surfaces. Call $U$ positive if $\dim[\Gamma(U,{\mathcal O}_U)] = \dim(U)$ (which equals 2 in these rank 2 cases). This is the condition for the Gross-Hacking-Keel construction [Publ. Math. Inst. Hautes Études Sci. 122 (2015), 65-168] to produce an additive basis of theta functions on $\Gamma(U,{\mathcal O}_U)$. We find that $U$ is positive and either finite-type or non-acyclic (in the usual cluster sense) if and only if the inverse monodromy of the tropicalization $U^{\rm trop}$ of $U$ is one of Kodaira's monodromies. In these cases we prove uniqueness results about the log Calabi-Yau surfaces whose tropicalization is $U^{\rm trop}$. We also describe the action of the cluster modular group on $U^{\rm trop}$ in the positive cases.

Key words: cluster varieties; log Calabi-Yau surfaces; tropicalization; cluster modular group.

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  1. Assem I., Schiffler R., Shramchenko V., Cluster automorphisms, Proc. Lond. Math. Soc. 104 (2012), 1271-1302, arXiv:1009.0742.
  2. Cadavid C.A., Vélez J.D., Normal factorization in ${\rm SL}(2,{\mathbb Z})$ and the confluence of singular fibers in elliptic fibrations, Beiträge Algebra Geom. 50 (2009), 405-423, arXiv:0802.0005.
  3. Chan K., Ueda K., Dual torus fibrations and homological mirror symmetry for $A_n$-singlarities, Commun. Number Theory Phys. 7 (2013), 361-396, arXiv:1210.0652.
  4. Dupont G., An approach to non-simply laced cluster algebras, J. Algebra 320 (2008), 1626-1661, arXiv:math.RT/0512043.
  5. Fock V.V., Goncharov A.B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865-930, arXiv:math.AG/0311245.
  6. Fomin S., Zelevinsky A., Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), 63-121, arXiv:math.RA/0208229.
  7. Gross M., Examples of special Lagrangian fibrations, in Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci. Publ., River Edge, NJ, 2001, 81-109, arXiv:math.AG/0012002.
  8. Gross M., Hacking P., Keel S., Birational geometry of cluster algebras, Algebr. Geom. 2 (2015), 137-175, arXiv:1309.2573.
  9. Gross M., Hacking P., Keel S., Mirror symmetry for log Calabi-Yau surfaces I, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 65-168, arXiv:1106.4977.
  10. Gross M., Hacking P., Keel S., Mirror symmetry for log Calabi-Yau surfaces II, in preparation.
  11. Gross M., Hacking P., Keel S., Moduli of surfaces with an anti-canonical cycle, Compos. Math. 151 (2015), 265-291, arXiv:1211.6367.
  12. Gross M., Hacking P., Keel S., Kontsevich M., Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497-608, arXiv:1411.1394.
  13. Gross M., Pandharipande R., Quivers, curves, and the tropical vertex, Port. Math. 67 (2010), 211-259, arXiv:0909.5153.
  14. Gross M., Pandharipande R., Siebert B., The tropical vertex, Duke Math. J. 153 (2010), 297-362, arXiv:0902.0779.
  15. Gross M., Siebert B., From real affine geometry to complex geometry, Ann. of Math. 174 (2011), 1301-1428, arXiv:math.AG/0703822.
  16. Heckman G., Looijenga E., The moduli space of rational elliptic surfaces, in Algebraic Geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math., Vol. 36, Math. Soc. Japan, Tokyo, 2002, 185-248, arXiv:math.AG/0010020.
  17. Kodaira K., On compact analytic surfaces. II, Ann. of Math. 77 (1963), 563-626.
  18. Kontsevich M., Soibelman Y., Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and mirror symmetry, in Homological Mirror Symmetry and Tropical Geometry, Lect. Notes Unione Mat. Ital., Vol. 15, Springer, Cham, 2014, 197-308, arXiv:1303.3253.
  19. Mandel T., Tropical theta functions and log Calabi-Yau surfaces, Selecta Math. (N.S.) 22 (2016), 1289-1335, arXiv:1407.5901.
  20. Mandel T., Cluster algebras are Cox rings, Manuscripta Math., to appear, arXiv:1707.05819.
  21. Symington M., Four dimensions from two in symplectic topology, in Topology and Geometry of Manifolds (Athens, GA, 2001), Proc. Sympos. Pure Math., Vol. 71, Amer. Math. Soc., Providence, RI, 2003, 153-208, arXiv:math.SG/0210033.

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