Difference between revisions of "Regularized moduli spaces"

From Polyfolds.org
Jump to: navigation, search
(Created page with "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</ma...")
 
Line 1: Line 1:
−
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>.
+
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,  
 
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>
 
<center>
Line 14: Line 14:
  
  
−
'''TODO:'''  
+
'''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>,  
 
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>.
 
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>.

Revision as of 20:21, 20 May 2017

Given the moduli spaces of pseudoholomorphic polygons \overline {\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d}) for each tuple of Lagrangians L_{0},\ldots ,L_{d}\subset M, generators 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}), and a fixed compatible almost complex structure J, we need to explain how to obtain regularizations \overline {\mathcal {M}}^{k}(x_{0};x_{1},\ldots ,x_{d};\nu ) for expected dimensions k=0,1 by a choice of perturbations \nu . 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,

\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 ).



TODO


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

Retrieved from "http://www.polyfolds.org/index.php?title=Regularized_moduli_spaces&oldid=139"