Tropical Moduli of Curves, Gromov–Witten, and the Correspondence Principle¶
Job: develop the tropical moduli \(\mathcal{M}_{g,n}^{\mathrm{trop}}\) (cone complex / stable metric graphs), the tropical Deligne–Mumford and the comparison to the algebraic (the skeleton), the tropical Gromov–Witten / Mikhalkin / Siebert–Nishinou correspondence, the floor diagrams / log GW, and the tropical Severi degrees — with \(\mathcal{M}_{0,4}^{\mathrm{trop}}\) the interval / the skeleton of \(\overline{\mathcal{M}}_{0,4}\) as the worked example.
Core¶
The tropical moduli of curves \(\mathcal{M}_{g,n}^{\mathrm{trop}}\)¶
The moduli space of tropical curves \(\mathcal{M}_{g,n}^{\mathrm{trop}}\) (the moduli of abstract tropical curves: the stable metric graphs of genus \(g\) with \(n\) marked legs). The \(\mathcal{M}_{g,n}^{\mathrm{trop}}\) is a generalized cone complex (the cone over each combinatorial type, the moduli of the edge lengths / the cone of the graph, the face relations / the edge contractions).
| \(\mathcal{M}_{g,n}^{\mathrm{trop}}\) | Description |
|---|---|
| Moduli of tropical curves | the stable metric graphs \(\Gamma\) of genus \(g\) with \(n\) legs, edge lengths \(\ell_e\in\mathbb{R}_{>0}\) |
| Cone complex | the cone over each combinatorial type (the graph), the face relations / the edge contractions |
| Stacky \(\mathfrak{M}_{g,n}^{\mathrm{trop}}\) | the moduli stack of tropical curves (the automorphisms / the stacky structure, \(\mathcal{M}_{g,n}^{\mathrm{trop}}\) the coarse moduli) |
| Tropical curves | the abstract tropical curve: a stable metric graph \(\Gamma\) of genus \(g\) with \(n\) legs |
The tropical Deligne–Mumford and the comparison to the algebraic¶
The tropical Deligne–Mumford compactification \(\overline{\mathcal{M}}_{g,n}^{\mathrm{trop}}\) (the compactification by the stable tropical curves, the boundary / the edge-length-zero strata).
Theorem (Abramchenko–Caporaso–Payne–Melo). The tropical / the non-Archimedean skeleton: \(\mathrm{Sk}(\overline{\mathcal{M}}_{g,n}\setminus\Delta)\cong\overline{\mathcal{M}}_{g,n}^{\mathrm{trop}}\) (the dual complex of the boundary divisor \(\Delta\) of the algebraic \(\overline{\mathcal{M}}_{g,n}\), the tropical moduli as the skeleton of the algebraic moduli).
| Tropical DM / comparison | Description |
|---|---|
| \(\overline{\mathcal{M}}_{g,n}^{\mathrm{trop}}\) | the tropical DM compactification (the stable tropical curves, the boundary) |
| Comparison to algebraic | \(\mathrm{Sk}(\overline{\mathcal{M}}_{g,n}\setminus\Delta)\cong\overline{\mathcal{M}}_{g,n}^{\mathrm{trop}}\) (the skeleton / the dual complex of the boundary \(\Delta\)) |
| Tropicalization of the algebraic moduli | \(\mathrm{trop}(\overline{\mathcal{M}}_{g,n})\cong\overline{\mathcal{M}}_{g,n}^{\mathrm{trop}}\) (the tropical as the skeleton, the non-Archimedean / the Berkovich) |
The tropical Gromov–Witten and the Mikhalkin correspondence¶
The tropical Gromov–Witten invariants (the count of the tropical curves in \(\mathbb{R}^n\) through the point conditions, the weighted count with the vertex multiplicities \(m_v=|\det(u_1,u_2)|\) the complex / the Welschinger for the real).
Theorem (Mikhalkin correspondence). The algebraic GW of a toric surface / the tropical GW of \(\mathbb{R}^2\) are equal (the degeneration to the tropical / the non-Archimedean limit).
Theorem (Siebert–Nishinou). The tropical / log GW for the toric surfaces (the log / the tropical degeneration, the correspondence theorem for the toric surfaces).
| Tropical GW / correspondence | Description |
|---|---|
| Tropical GW | the count of the tropical curves in \(\mathbb{R}^n\) through the point conditions, the vertex multiplicities |
| Mikhalkin correspondence | the algebraic GW (toric surface) = the tropical GW (\(\mathbb{R}^2\)), the degeneration / the non-Archimedean limit |
| Siebert–Nishinou | the tropical / log GW for the toric surfaces (the log / the tropical degeneration) |
| Welschinger (real) | the real enumerative invariants (the signed count, the Welschinger for the real) |
The floor diagrams and the logarithmic Gromov–Witten¶
| Floor diagrams / log GW | Description |
|---|---|
| Floor diagrams | the combinatorial / tropical tool for the enumeration (the floor diagram = the tropical curve with "floors" / the horizontal edges) |
| Logarithmic GW | the Gross–Siebert / the Chen–Ranganathan: the log GW as the tropical / the logarithmic refinement (the log structure from the tropicalization) |
| Mirror / GW connection | the tropical / the log GW as the degeneration / the tropical limit of the algebraic GW (mirror symmetry / GW) |
The tropical Severi and the node counts¶
| Tropical Severi / node counts | Description |
|---|---|
| Tropical Severi degrees | the count of the nodal curves / the tropical curves with the node conditions (the tropical / the Vakil) |
| Göttsche–Vainsencher | the generating function / the tropical Severi (the Block–Göttsche / the tropical / the multiplicity) |
| Block–Göttsche | the tropical / the relative invariants (the generating function, the Block–Göttsche) |
See it¶
\(\mathcal{M}_{0,4}^{\mathrm{trop}}\) and the tropical \(\overline{\mathcal{M}}_{0,4}\) skeleton. The moduli \(\mathcal{M}_{0,4}\) (the algebraic: the 4-pointed genus-0 curves, the \(\mathbb{P}^1\) minus the 3 special points, the cross-ratio). The tropical \(\mathcal{M}_{0,4}^{\mathrm{trop}}\): the 4-legged tropical genus-0 curves, the single internal edge of length \(\ell\in\mathbb{R}_{>0}\) (the 4 possible trivalent types, the cross-ratio the edge length). The compactification \(\overline{\mathcal{M}}_{0,4}^{\mathrm{trop}}=[0,\infty]\) (the edge length \(\ell\in[0,\infty]\), the \(\ell=0\) the 3-node / the 3-combinatorial-type boundary, the \(\ell=\infty\) the "infinite edge" the non-compact). The comparison to the algebraic: \(\mathrm{Sk}(\overline{\mathcal{M}}_{0,4}\setminus\Delta)=[0,\infty]\) (the dual complex of the boundary \(\Delta\) = the 3 special points, the interval, the skeleton). The key point: the tropical \(\overline{\mathcal{M}}_{0,4}^{\mathrm{trop}}\) is the interval (the 1-dimensional skeleton of the algebraic \(\overline{\mathcal{M}}_{0,4}\cong\mathbb{P}^1\)), the tropical moduli as the non-Archimedean skeleton — the prototypical example of the tropical / the algebraic comparison.
Check¶
Show that \(\overline{\mathcal{M}}_{0,4}^{\mathrm{trop}}=[0,\infty]\) and identify the comparison to the algebraic \(\overline{\mathcal{M}}_{0,4}\cong\mathbb{P}^1\).
\(\mathcal{M}_{0,4}\): the 4-pointed genus-0 curves, the algebraic moduli \(\mathcal{M}_{0,4}=\mathbb{P}^1\setminus\{0,1,\infty\}\) (the cross-ratio, the 3 special points / the boundary \(\Delta\)). The tropical \(\mathcal{M}_{0,4}^{\mathrm{trop}}\): the 4-legged tropical genus-0 curves, the single internal edge of length \(\ell\in\mathbb{R}_{>0}\) (the 4 possible trivalent types parametrized by \(\ell\), the cross-ratio the edge length). The compactification \(\overline{\mathcal{M}}_{0,4}^{\mathrm{trop}}=[0,\infty]\) (the \(\ell=0\) the boundary / the 3-combinatorial-type, the \(\ell=\infty\) the non-compact). The comparison: \(\mathrm{Sk}(\overline{\mathcal{M}}_{0,4}\setminus\Delta)=\mathrm{Sk}(\mathbb{P}^1\setminus\{0,1,\infty\})=[0,\infty]\) (the dual complex of \(\Delta=\{0,1,\infty\}\) is the interval / the 3 points on \(\mathbb{P}^1\) give the dual complex = the interval, the skeleton). The \(\overline{\mathcal{M}}_{0,4}\cong\mathbb{P}^1\) (the algebraic), the \(\overline{\mathcal{M}}_{0,4}^{\mathrm{trop}}=[0,\infty]\) (the tropical / the skeleton), the Abramchenko–Caporaso–Payne–Melo comparison.
Explain the Mikhalkin correspondence: the algebraic GW (toric surface) = the tropical GW (\(\mathbb{R}^2\)), and identify the role of the degeneration.
The Mikhalkin correspondence: the algebraic GW invariants (the count of the algebraic curves on a toric surface through the point conditions) equal the tropical GW invariants (the weighted count of the tropical curves in \(\mathbb{R}^2\) through the corresponding tropical points, the vertex multiplicities \(m_v=|\det(u_1,u_2)|\)). The mechanism: the degeneration of the complex structure / the coefficients to a non-Archimedean limit (the Puiseux family, the \(t\to0\) limit), where the algebraic curves degenerate to the tropical curves (the tropicalization, the Kapranov / the Berkovich). The correspondence: the algebraic curves through the point conditions \(\to\) the tropical curves through the tropical points (the valuation of the conditions), the multiplicities matched (the vertex multiplicity \(m_v\) the complex / the Welschinger for the real). The role of the degeneration: the non-Archimedean / the tropical limit is the mechanism, the Kapranov / the Berkovich the framework, the Mikhalkin the correspondence theorem.
Pitfalls¶
- The tropical moduli \(\mathcal{M}_{g,n}^{\mathrm{trop}}\) is the cone complex (not the scheme / the stack — the generalized cone complex, the polyhedral / the combinatorial, not the algebraic).
- The Abramchenko–Caporaso–Payne–Melo comparison is the skeleton of the algebraic moduli (not the algebraic moduli itself — the \(\mathrm{Sk}(\overline{\mathcal{M}}_{g,n}\setminus\Delta)\cong\overline{\mathcal{M}}_{g,n}^{\mathrm{trop}}\) is the skeleton / the dual complex, not the algebraic moduli).
- The Mikhalkin / Siebert–Nishinou correspondence is for the toric surfaces (the non-toric / the general case is open / partial — the toric is the special / the log case, the general / the non-toric is the open problem).
- The floor diagrams are the combinatorial / the computational model (the tropical curves with the floors), not the definition of the tropical curves — the floor diagram is the tool, the tropical curve is the object.
- The log GW (the Gross–Siebert / the Chen–Ranganathan) is the tropical / the logarithmic refinement, the relation to the classical GW (mirror symmetry / GW) via the degeneration / the tropical limit, not the direct equality.
- The tropical Severi / the Göttsche–Vainsencher is the tropical / the relative count, the Block–Göttsche the generating function — the tropical enumerative, not the classical Severi (the two are related via the Mikhalkin / the correspondence, but the tropical is the combinatorial / the weighted count).
Refs¶
- Abramchenko–Caporaso–Payne–Melo, "The tropicalization of the moduli space of curves" (Ann. Sci. ENS 2013)
- Mikhalkin, "Enumerative tropical algebraic geometry in \(\mathbb{R}^2\)" (JAMS 2005); Siebert–Nishinou, "Toric degenerations and the correspondence theorem" (2012)
- Brugallé–Mikhalkin, "Floor decompositions of tropical curves" (2012); Gross–Siebert, "Logarithmic Gromov–Witten invariants" (2013)
- Göttsche–Vainsencher, "Enumeration of nodal curves" (2006); Block–Göttsche, "Refined curve counting" (2014)
Next¶
TG index · Foundations · Berkovich & Skeletons · Matroids & Linear Spaces · Scattering & Mirror · Mirror symmetry / GW · Algebraic geometry