A mirror family is built directly from a log Calabi-Yau pair (X,D) or a maximally unipotent normal crossings degeneration, with no embedding in a toric variety required. Theta functions indexed by the integral points of the tropicalization B = Trop(X) form a free module whose structure constants count genus-zero punctured curves — logarithmic stable maps allowing a marked point of negative contact order with the boundary divisor. Taking Spf of the limit yields the mirror to the pair and Proj yields the mirror to the degeneration, recovering the Gross-Hacking-Keel log Calabi-Yau surface construction as a special case; the product is associative when K_X + D or its negative is nef.
An announcement, contributed to the 2015 AMS Summer Institute in Algebraic Geometry, of a mirror construction by Mark Gross and Bernd Siebert that requires no embedding in a toric variety. Its new ob…