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Commit 31b8ae55 authored by Luke Naylor's avatar Luke Naylor
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Start notebook for generic choice of P

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%% Cell type:code id:2c1f46cb tags:
``` sage
from pseudowalls import *
%display latex
```
%% Cell type:markdown id:48112244 tags:
# Initialize Cherns
%% Cell type:code id:7a52d103 tags:
``` sage
var("R C D r c d A B", domain="real")
P = A, B
```
%% Cell type:code id:cfde4b23 tags:
``` sage
v = Chern_Char(R,C,D)
v
```
%% Output
$\displaystyle \text{Chern Character:} \\ \begin{array}{l} \mathrm{ch}_{0} = R \\ \mathrm{ch}_{1} = C \ell^{1} \\ \mathrm{ch}_{2} = D \ell^{2} \end{array}$
<pseudowalls.chern_character.Chern_Char object at 0x7f7bcab93ad0>
%% Cell type:code id:f015f4ab tags:
``` sage
twisted_v = Twisted_Chern_Char(B,
R,
var("twisted_v1", latex_name = r"\mathrm{ch}_1^B(v)", domain="real"),
var("twisted_v2", latex_name = r"\mathrm{ch}_2^B(v)", domain="real"),
)
twisted_v
```
%% Output
$\displaystyle \text{ Twisted Chern Character for $\beta={ B }$ } \\ \begin{array}{l} \mathrm{ch}_{0} = R \\ \mathrm{ch}_{1} = {\mathrm{ch}_1^B(v)} \ell^{1} \\ \mathrm{ch}_{2} = {\mathrm{ch}_2^B(v)} \ell^{2} \end{array}$
<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f7bbf22f110>
%% Cell type:code id:711e4205 tags:
``` sage
twisted_u = Twisted_Chern_Char(B,
r,
var("twisted_u1", latex_name = r"\mathrm{ch}_1^B(u)", domain="real"),
var("twisted_u2", latex_name = r"\mathrm{ch}_2^B(u)", domain="real"),
)
twisted_u
```
%% Output
$\displaystyle \text{ Twisted Chern Character for $\beta={ B }$ } \\ \begin{array}{l} \mathrm{ch}_{0} = r \\ \mathrm{ch}_{1} = {\mathrm{ch}_1^B(u)} \ell^{1} \\ \mathrm{ch}_{2} = {\mathrm{ch}_2^B(u)} \ell^{2} \end{array}$
<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f7bbf0399d0>
%% Cell type:markdown id:5f3e6e12 tags:
# Numerical Conditions
%% Cell type:markdown id:1206d912 tags:
Condition of $P = (A,B)$ being on $\Theta_v$ (i.e. $ch_2^{A,B}(v) = 0$) expressed in terms of twisted Chern character for $v$ at $\beta=B$:
%% Cell type:code id:17c390cd tags:
``` sage
A2_subs = solve(
stability.Tilt(*P).degree(twisted_v) == 0,
A^2)[0]
A2_subs
```
%% Output
$\displaystyle A^{2} = \frac{2 \, {\mathrm{ch}_2^B(v)}}{R}$
A^2 == 2*twisted_v2/R
%% Cell type:markdown id:11b9c67b tags:
## Condition: $ch_2^{P}(u) > 0$
%% Cell type:code id:47b34ed7 tags:
``` sage
stability.Tilt(*P).degree(twisted_u) > 0
```
%% Cell type:code id:d8abf566 tags:
``` sage
(stability.Tilt(*P).degree(twisted_u) / r > 0).expand().subs(
A2_subs
)
```
%% Cell type:markdown id:fe9fe5b8 tags:
## Condition: $\Delta(u) \geq 0$
%% Cell type:code id:f09514cb tags:
``` sage
```
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