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Tropical theta functions and log Calabi–Yau surfaces

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Abstract

We generalize the standard combinatorial techniques of toric geometry to the study of log Calabi–Yau surfaces. The character and cocharacter lattices are replaced by certain integral linear manifolds described in Gross et al. (Math. Inst. Hautes Etudes Sci. 122, 65–168, 2015), and monomials on toric varieties are replaced with the canonical theta functions defined in Gross et al. (2015) using ideas from mirror symmetry. We describe the tropicalizations of theta functions and use them to generalize the dual pairing between the character and cocharacter lattices. We use this to describe generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and finite Fourier series expansions. We hope that these techniques will generalize to higher-rank cluster varieties.

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Notes

  1. Convexity along broken lines is a notion from [5] which we show is, in our situation, equivalent to [3]’s notion of “convex with respect to every seed.”

  2. Choosing a cyclic ordering for the components of D (assuming D has at least three components) is equivalent to choosing an orientation for \(N_{\mathbb {R}}\) or \(U^{{{\mathrm{trop}}}}\). It is also equivalent to fixing the sign for the holomorphic volume form \(\Omega \) on U, which we will use in Sect. 6. We assume throughout the paper that such a choice has been fixed.

  3. We assume here that there are more than three rays in \(\Sigma \), so that \(\sigma _{i-1,i}\cup \sigma _{i,i+1}\) is not all of \(N_{\mathbb {R}}\). This assumption can always be achieved by taking toric blowups of (Y, D). Alternatively, it is easy to avoid this assumption, but the notation and exposition become more complicated. We will therefore continue to implicitly assume that there are enough rays for whatever we are trying to do, without further comment.

  4. When working without the positivity assumption, [8] choose some strictly convex rational polyhedral cone containing \({{\mathrm{NE}}}(Y)_{\mathbb {R}}\) to be \(P_{\mathbb {R}}\) (\(\sigma _P\) in their notation).

  5. For \(\varphi =\eta _W\circ \varphi _{{{\mathrm{NE}}}(Y)}\) as in the second example above, we can take \(\mathbb {P}_{\varphi }\) to be the trivial bundle with \(\varphi =\varphi _W\). Since we are really only interested in tropicalizations of theta functions in this paper, restricting to this situation would be sufficient, and one can in this way avoid worrying about multi-valued functions.

  6. For details on relative Gromov–Witten invariants, see [13] or see [9] for a treatment of this particular situation.

  7. See Theorem 4.2 of [9] or Lemma 3.2 of [8].

  8. When we multiply d by a positive scalar c, we map \(L_v^d(t)\) to \(cL_v^d(t)\). That way the times \(t_1,\ldots ,t_k\) are unchanged.

  9. Consider \((\overline{Y}=\mathbb {P}^2,\overline{D}=\overline{D}_1+\overline{D}_2+\overline{D}_3)\). The \(E_7\) (resp. \(E_8\)) case refers to the Looijenga interiors \(U=Y{\setminus } D\) which can be obtained by blowing up 2 non-nodal points on \(\overline{D}_1\), 3 on \(\overline{D}_2\), and 4 (resp. 5) on \(\overline{D}_3\). \(E_7\) and \(E_8\) refer to the intersection forms on the lattices \(D^{\perp }:=\{C\in {{\mathrm{Pic\,}}}(Y)|C \cdot D_i=0 \text{ for } \text{ all } i\}\) in these cases.

  10. A previous version of this paper incorrectly claimed that \({{\mathrm{Spec\,}}}A_{\sigma } = Z_{\sigma }\) when \(\sigma ^{\vee }\) is two-dimensional, neglecting to contract the \((-1)\)-curves.

  11. It follows from Theorem 4.13 that this is equivalent to the version of convex hull used in [15]. Similarly for the Minkowski sums of Sect. 5.3. Also, Proposition 5.8 below implies that our notion of strongly convex is equivalent to [1]’s notion of convex.

  12. Here we must include broken lines through the origin, by which we mean limits of sequence of broken lines which are all equivalent to each other in the sense of Sect. 3.4. In other words, in addition to the usual broken lines, we allow sets of the form \(\langle \cdot ,v\rangle = 0\) for \(v\in V^{{{\mathrm{trop}}}}(\mathbb {Z})\). These are the lines \(L_q^0\) from Definition 2.8.

  13. This more general definition for the Minkowski sum was previously used in [1] and [15].

  14. If some \(q_1+_i\ldots +_i q_s\) is not defined in \(U^{{{\mathrm{trop}}}}\), we simply do not include it in the set. Alternatively, we could only include \(q_1+_i \ldots +_i q_s\) in the set if there is some convex cone \(\sigma \) in the complement of \(\rho _i\) containing each \(q_k\)—addition in a convex cone is always well defined.

  15. Recall that if we take the cyclic ordering of \(D=D_1+\cdots +D_n\) as part of our data, then we can use this to orient \(U^{{{\mathrm{trop}}}}\), and \(V^{{{\mathrm{trop}}}}\) gets the opposite orientation. This can be used to orient \(\gamma \) (by ordering \(z_i\) and \(z_{i+1}\)). Alternatively, we can take the sign of \(\Omega \) as part of our data and say that \(\gamma \) is oriented to make \(\int _{\gamma }\Omega >0\).

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Acknowledgments

This paper is based on part of the author’s thesis, which was written while in graduate school at the University of Texas at Austin. I would like to thank my advisor, Sean Keel, for introducing me to this topic and for all his suggestions, insights, and support and also for allowing me access to preliminary versions of his papers with Paul Hacking and Mark Gross. I also want to thank Andy Neitzke, James Pascaleff, Yuan Yao, and Yuecheng Zhu for many valuable conversations, as well as the referee for numerous helpful suggestions. The author was partially supported by the Center of Excellence Grant “Centre for Quantum Geometry of Moduli Spaces” from the Danish National Research Foundation (DNRF95).

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  1. Department of Mathematics, University of Utah, Salt Lake City, UT, USA

    Travis Mandel

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Mandel, T. Tropical theta functions and log Calabi–Yau surfaces. Sel. Math. New Ser. 22, 1289–1335 (2016). https://doi.org/10.1007/s00029-015-0221-y

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