A compactification of the moduli space of twisted holomorphic maps

I. Mundet i Riera, G. Tian

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Given compact symplectic manifold X with a compatible almost complex structure and a Hamiltonian action of S1 with moment map μ : X → i R, and a real number K ≥ 0, we compactify the moduli space of twisted holomorphic maps to X with energy ≤K. This moduli space parameterizes equivalence classes of tuples (C, P, A, φ{symbol}), where C is a smooth compact complex curve of fixed genus g, P is a principal S1 bundle over C, A is a connection on P and φ{symbol} is a section of P ×S1 X satisfyingover(∂, -)A φ{symbol} = 0, ιv FA + μ (φ{symbol}) = c, {norm of matrix} FA {norm of matrix}L22 + {norm of matrix} dA φ{symbol} {norm of matrix}L22 + {norm of matrix} μ (φ{symbol}) - c {norm of matrix}L22 ≤ K . Here FA is the curvature of A, v is the restriction to C of a volume form on the universal curve over over(M, -)g and c is a fixed constant. Two tuples (C, P, A, φ{symbol}) and (C, P, A, φ{symbol}) are equivalent if there is a morphism of bundles ρ : P → P lifting a biholomorphism C → C such that ρ* v = v, ρ* A = A and ρ* φ{symbol} = φ{symbol}. The topology of the moduli space is the quotient topology of the topology of C convergence on the set of tuples (C, P, A, φ{symbol}). We also incorporate marked points in the picture. There are two sources of noncompactness. First, bubbling off phenomena, analogous to the one in Gromov-Witten theory. Second, degeneration of C to nodal curves. In this case, there appears a phenomenon which is not present in Gromov-Witten: near the nodes, the section φ{symbol} may degenerate to a chain of gradient flow lines of - i μ.

Original languageEnglish (US)
Pages (from-to)1117-1196
Number of pages80
JournalAdvances in Mathematics
Issue number4
StatePublished - Nov 10 2009

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Compactification of moduli spaces
  • Hamiltonian actions
  • Pseudoholomorphic curves
  • Vortex equations


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