Jonathan Wise

Birational invariance in logarithmic Gromov–Witten theory

Gromov-Witten invariants have been constructed to be deformation invariant, but their behavior under other transformations is subtle. In this note we show that logarithmic Gromov-Witten invariants are also invariant under appropriately defined logarithmic modifications.

Skeletons and Fans of Logarithmic Structures

We survey a collection of closely related methods for generalizing fans of toric varieties, include skeletons, Kato fans, Artin fans, and polyhedral cone complexes, all of which apply in the wider context of logarithmic geometry. Under appropriate assumptions these structures are equivalent, but their different realizations have provided for surprisingly disparate uses. We highlight several current applications and suggest some future possibilities.

Expanded degenerations and pairs

We provide a universal approach to the moduli of Jun Lis expanded pairs and expanded degenerations [20]. This enables us to prove algebraicity results, compare with Lis approach and with the approach of Graber and Vakil [16], and generalize to the twisted expansions used by Abramovich and Fantechi as a basis for orbifold techniques in degeneration formulas.

Moduli of morphisms of logarithmic schemes

We show that there is a logarithmic algebraic space parameterizing logarithmic morphisms between fixed logarithmic schemes when those logarithmic schemes satisfy natural hypotheses. As a corollary, we obtain the algebraicity of the stack of stable logarithmic maps without restriction on the logarithmic structure of the target.

A hyperelliptic Hodge integral

We calculate the hyperelliptic Hodge integral lambda_g lambda_{g-1} / (1 - psi) for use in arXiv:math/0702219. The proof uses the WDVV equations for the genus zero Gromov--Witten invariants of P(1,1,2).