Lectures on Gromov–Witten Invariants of Orbifolds

Abstract

Introduction 1 Lecture 1. Gromov-Witten theory 2 Lecture 2. Orbifolds / Stacks 16 Lecture 3. Twisted stable maps 20 Lecture 4. Gromov-Witten classes 28 Lecture 5. WDVV, grading and computations 35 Appendix A. The legend of String Cohomology: two letters of Maxim Kontsevich to Lev Borisov 42 References 45

Logarithmic Geometry, Minimal Free Resolutions and Toric Algebraic Stacks

“…Suppose (1) be a set of positive integers indexed by σ (1). Let ι : P → F be an admissible free resolution defined to be the composite P → F n → F where P → F is the minimal free resolution and n : F → F is defined by e ρ → n ρ · e ρ for each ray ρ ∈ σ (1).…”

“…§4. • A pair (Σ, n = {n ρ } ρ∈Σ (1) ) is called a simplicial fan with a level structure n if Σ is simplicial fan in N R and n = {n ρ } ρ∈Σ (1) is the set of positive integers indexed by the set of rays Σ(1).…”

“…(This free-net may be regarded as the geometric realization of the level n.) The canonical free-net Σ 0 can of Σ is the free-net associated to the level {n ρ = 1} ρ∈Σ (1) .…”

“…For a ray (1) ) be a simplicial fan with a level structure n = {n ρ }. The free-net of Σ associated to level n is the subset Σ 0 n ⊂ |Σ| ∩ N such that for each cone σ ∈ Σ the set σ ∩ Σ 0 n is the free submonoid generated by {n ρ · v ρ } ρ∈σ (1) , where v ρ denotes the first lattice point on a ray ρ ∈ Σ(1).…”

“…Therefore the data Σ 0 is determined by the set of points {w ρ } ρ∈Σ (1) . Let us denote by v ρ the first lattice point on a ray ρ ∈ Σ(1).…”