For each contact diffeomorphism ϕ : (Q, ξ) → (Q, ξ) of (Q, ξ), we equip its mapping torus Mϕ with a locally conformal symplectic form of Banyaga’s type, which we call the lcs mapping torus of the contact diffeomorphism ϕ. In the present paper, we consider the product Q × S 1 = M id (corresponding to ϕ = id) and develop basic analysis of the associated J-holomorphic curve equation, which has the form ∂ ˉ π w = 0 , w ∗ λ ∘ j = f ∗ d θ $$ \bar{\partial}^{\pi} w=0, \quad w^{*} \lambda \circ j=f^{*} d \theta $$ for the map u = (w, f) : Σ ˙ → Q × S 1 $\dot{\Sigma} \rightarrow Q \times S^{1}$ for a λ-compatible almost complex structure J and a punctured Riemann surface ( Σ ˙ , j ) . $(\dot{\Sigma}, j).$ In particular, w is a contact instanton in the sense of [31], [32].We develop a scheme of treating the non-vanishing charge by introducing the notion of charge class in H 1 ( Σ ˙ , Z ) $H^{1}(\dot{\Sigma}, \mathbb{Z})$ and develop the geometric framework for the study of pseudoholomorphic curves, a correct choice of energy and the definition of moduli spaces towards the construction of a compactification of the moduli space on the lcs-fication of (Q, λ) (more generally on arbitrary locally conformal symplectic manifolds).

中文翻译：

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接触流形 LCS 化的伪全纯曲线

对于每个接触微分同胚φ: (问,ξ) → (问,ξ） 的 （问,ξ), 我们装备它的映射圆环米φ 与局部共形辛Banyaga 类型的形式，我们称之为 lcs映射圆环接触微分同胚φ. 在本文中，我们考虑产品问×小号 1个=米 ID（对应于φ= id) 并开发相关的基本分析杰-全纯曲线方程，具有以下形式 ∂ ˉ π w = 0 , w ＊ λ ∘ j = F ＊ d θ $$ \bar{\partial}^{\pi} w=0, \quad w^{*} \lambda \circ j=f^{*} d \theta $$ 对于地图你= (w,F）： Σ ˙ → 问 × 小号 1个 $\dot{\Sigma} \rightarrow Q \times S^{1}$ 为一个λ-兼容几乎复杂的结构杰和一个穿孔的黎曼曲面 ( Σ ˙ , j ) . $(\dot{\Sigma}, j).$ 尤其，w是一个接触瞬间在[31]、[32]的意义上。我们通过引入以下概念来制定处理非消失电荷的方案充电类在 H 1个 ( Σ ˙ , Z ) $H^{1}(\dot{\Sigma}, \mathbb{Z})$ 并开发用于研究伪全纯曲线的几何框架，正确选择能量和模空间的定义，以在 lcs-fication 上构建模空间的紧化（问,λ)（更一般地在任意局部共形辛流形上）。