Expanded Degenerations and Pairs

Abstract

Since Jun Li's original definition, several other definitions of expanded pairs and expanded degenerations have appeared in the literature. We explain how these definitions are related and introduce several new variants and perspectives. Among these are the twisted expansions used by Abramovich and Fantechi as a basis for orbifold techniques in degeneation formulas.

Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations

“…It is possible to isolate this global geometry by comparing the space of maps to X/V with the space of maps to a universal target X /V . This method was introduced in [2] and [1] and has been used in [15] as well.…”

“…However, Siebert's proposal awaited the development of logarithmic algebraic geometry, particularly the logarithmic cotangent complex, on which it relied to define virtual fundamental classes. (1) In the interim, Cadman saw that orbifold Gromov-Witten theory [17,8,7] could be used for some of the same purposes as Li's relative GromovWitten theory [14]. Abramovich and Fantechi adapted and generalized Cadman's method to apply to Li's expanded targets.…”

Skeletons and Fans of Logarithmic Structures

“…The idea is that since an Artin fan A is logarithmicallyétale, a map f : C → A X from a curve to A X is logarithmically unobstructed. Precursors to this result for specific X were obtained in [ACFW13,ACW10,AMW14,CMW12]. In [ACFW13] an approach to Jun Li's expanded degenerations was provided using what in hindsight we might call the Artin fan of the affine line A = A A 1 .…”

“…Precursors to this result for specific X were obtained in [ACFW13,ACW10,AMW14,CMW12]. In [ACFW13] an approach to Jun Li's expanded degenerations was provided using what in hindsight we might call the Artin fan of the affine line A = A A 1 . The papers [ACW10,AMW14,CMW12] use this formalism to prove comparison results in relative Gromov-Witten theory.…”

Boundedness of the space of stable logarithmic maps

“…A more complete picture of the relationship between these objects, as well as with Berkovich spaces, is given in [Uli13]. The simplest cases of Artin fans were used previously in [ACFW11,ACW10,CMW12].…”