View source for Regularized moduli spaces

[[table of contents]]

Given the [[moduli spaces of pseudoholomorphic polygons]] <math>\overline\mathcal{M}(x_0;x_1,\ldots,x_d)</math> for each tuple of Lagrangians <math>L_0,\ldots,L_d\subset M</math>, generators <math>x_0\in\text{Crit}(L_0,L_d), x_1\in\text{Crit}(L_0,L_1), \ldots, x_d\in\text{Crit}(L_{d-1},L_d)</math>, and a fixed compatible almost complex structure <math>J</math>, we need to explain how to obtain regularizations <math>\overline\mathcal{M}^k(x_0;x_1,\ldots,x_d;\nu)</math> for expected dimensions <math>k=0,1</math> by a choice of perturbations <math>\nu</math>.
Moreover, we need to choose these perturbations ''coherently'' to ensure that the boundary of each 1-dimensional part is given by Cartesian products of 0-dimensional parts, 
<center>
<math>
\partial\overline\mathcal{M}^1(x_0;x_1,\ldots,x_d;\nu) = 
\bigsqcup_{m,n\geq 0} \bigsqcup_{y\in\text{Crit}(L_n,L_{m+n})}
\overline\mathcal{M}^0(y;\underline{x}';\nu) \times
\overline\mathcal{M}^0(x_0;\underline{x},y,\underline{x}'';\nu) .
</math>
</center>





'''TODO''' 


Finally, we need to check that for each pair <math>(b,b')\in\overline\mathcal{M}^0(x_0;\underline{x},y,\underline{x}'';\nu)\times\overline\mathcal{M}^0(y;\underline{x}';\nu)</math>, 
when considered as boundary point <math>(b,b')\in\partial \overline\mathcal{M}^1(x_0;x_1,\ldots,x_d;\nu)</math>, has symplectic area <math>\omega((b,b')) = \omega(b) + \omega(b')</math> and weight function <math>\text{w}((b,b')) = (-1)^{\|\underline x\|}  \text{w}(b) \text{w}(b')</math>.