Naive counts of rational curves in an affine log Calabi-Yau variety containing a torus uniquely determine a Frobenius algebra, now proved via non-archimedean analytic disk counting.
Nonarchimedean geometry, tropicalization, and metrics on curves
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abstract
We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev multiplicity formula in tropical elimination theory to the case of a nontrivial valuation. For curves, we explore in detail the relationship between skeletal metrics and lattice lengths on tropicalizations and show that the maps from the analytification of a curve to the tropicalizations of its toric embeddings stabilize to an isometry on finite subgraphs. Other applications include generalizations of Speyer's well-spacedness condition and the Katz-Markwig-Markwig results on tropical j-invariants.
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The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus
Naive counts of rational curves in an affine log Calabi-Yau variety containing a torus uniquely determine a Frobenius algebra, now proved via non-archimedean analytic disk counting.