The Evaluation Space of Logarithmic Stable Maps

Abstract

The evaluation stack ∧X for minimal logarithmic stable maps is constructed, parameterizing families of standard log points in the target log scheme. This construction provides the ingredients necessary to define appropriate evaluation maps for minimal log stable maps and establish the logarithmic Gromov-Witten theory of a log-smooth Deligne-Faltings log scheme.

Tropical refined curve counting from higher genera and lambda classes

“…Let ∆ be a balanced collection of vectors in Z 2 and let n be a nonnegative integer 1 . This determines a complex toric surface X ∆ and a counting problem of virtual dimension zero for complex algebraic curves in X ∆ of some genus g ∆,n , of some class β ∆ , satisfying some tangency conditions with respect to the toric boundary divisor, and passing through n points of X ∆ in general position.…”

Exploded manifolds

“…The tropical part of any holomorphic curve in Expl M is a tropical curve in Expl M . More generally, if M is a complex manifold with a collection N i of transversely intersecting complex codimension 1 submanifolds, then the tropical part of Expl(M ) has one vertex for each connected component of M , a copy of [0, ∞) for each submanifold, a face [0, ∞) 2 for each intersection and an n dimensional quadrant [0, ∞) n for each n-fold intersection. (This is sometimes called the dual intersection complex of M .)…”

Log geometry and exploded manifolds

“…For there to be any tropical curves of the given type, P v must be a face of P e for each edge e connected to v, and if e is an external edge, there must exist an infinite ray in P e in the direction of u e . If these conditions are satisfied, the condition that there is a tropical curve with data (p v , l q ) is equivalent to the following integral linear equations (2) p v(q,2) − p v(q,1) = l q u q for each internal edge q…”

“…Around the turn of the century, Siebert suggested defining log Gromov-Witten invariants. Siebert's program has recently been carried out by Gross and Siebert [6], and separately by Abramovich and Chen in [1], [5], [4] and [2]. Each of these groups built on work of Olsson [13] which defined an appropriate deformation theory of log schemes, which has been used by Kim in [10] to define an obstruction theory for a stack of log curves.…”