Mixing
Kähler Potential on Blowup of $mathbb{C}/{pm 1}$
$begingroup$
The book Joyce: Riemannian Holonomy Groups and Calibrated Geometry contains on page 205 the Eguchi-Hanson space as an example:
Consider $mathbb{C}^2$ with complex coordinates $(z_1,z_2)$, acted upon by the involution $-1:(z_1,z_2) mapsto (-z_1,-z_2)$.
Let $(X,pi)$ be the blow-up of $mathbb{C}^2/{pm 1}$ at $0$.
Then $X$ is a crepant resolution of $mathbb{C}^2/{pm 1}$.
(...)
Define $f:X setminus pi^{-1}(0) rightarrow mathbb{R}$ by
$$
f=
sqrt{r^4+1}+2 log r - log left( sqrt{r^4+1} +1 right),
$$
where $r=left( |z_1|^2 + |z_2 |^2 right)^{1/2}$ is the radius function on $X$.
Define a $2$-form $omega_1$ on $X setminus pi^{-1}(0)$ by $omega_1=i partial overline{partial} f$.
Then $omega_1$ extends smoothly and uniquely to $X$.
Question:
How can I see that $omega_1$ extends to $X$?
Writing $r= sqrt{z_1 overline{z_1}+z_2 overline{z_2}}$ I can compute $omega_1$ explicitly.
The expressions are long, so I used Mathematica.
All partial derivatives $frac{partial}{partial overline{z_i}}frac{partial}{partial z_k}f$ tend to $infty$ as $r rightarrow 0$.
So it seems to me as if $f$ cannot be extended.
What am I missing?
differential-geometry complex-geometry kahler-manifolds k3-surfaces
asked Jan 25 at 21:50
user505117user505117
233
$endgroup$
add a comment |
$begingroup$
The book Joyce: Riemannian Holonomy Groups and Calibrated Geometry contains on page 205 the Eguchi-Hanson space as an example:
Consider $mathbb{C}^2$ with complex coordinates $(z_1,z_2)$, acted upon by the involution $-1:(z_1,z_2) mapsto (-z_1,-z_2)$.
Let $(X,pi)$ be the blow-up of $mathbb{C}^2/{pm 1}$ at $0$.
Then $X$ is a crepant resolution of $mathbb{C}^2/{pm 1}$.
(...)
Define $f:X setminus pi^{-1}(0) rightarrow mathbb{R}$ by
$$
f=
sqrt{r^4+1}+2 log r - log left( sqrt{r^4+1} +1 right),
$$
where $r=left( |z_1|^2 + |z_2 |^2 right)^{1/2}$ is the radius function on $X$.
Define a $2$-form $omega_1$ on $X setminus pi^{-1}(0)$ by $omega_1=i partial overline{partial} f$.
Then $omega_1$ extends smoothly and uniquely to $X$.
Question:
How can I see that $omega_1$ extends to $X$?
Writing $r= sqrt{z_1 overline{z_1}+z_2 overline{z_2}}$ I can compute $omega_1$ explicitly.
The expressions are long, so I used Mathematica.
All partial derivatives $frac{partial}{partial overline{z_i}}frac{partial}{partial z_k}f$ tend to $infty$ as $r rightarrow 0$.
So it seems to me as if $f$ cannot be extended.
What am I missing?
differential-geometry complex-geometry kahler-manifolds k3-surfaces
asked Jan 25 at 21:50
user505117user505117
233
$endgroup$
add a comment |
$begingroup$
The book Joyce: Riemannian Holonomy Groups and Calibrated Geometry contains on page 205 the Eguchi-Hanson space as an example:
Consider $mathbb{C}^2$ with complex coordinates $(z_1,z_2)$, acted upon by the involution $-1:(z_1,z_2) mapsto (-z_1,-z_2)$.
Let $(X,pi)$ be the blow-up of $mathbb{C}^2/{pm 1}$ at $0$.
Then $X$ is a crepant resolution of $mathbb{C}^2/{pm 1}$.
(...)
Define $f:X setminus pi^{-1}(0) rightarrow mathbb{R}$ by
$$
f=
sqrt{r^4+1}+2 log r - log left( sqrt{r^4+1} +1 right),
$$
where $r=left( |z_1|^2 + |z_2 |^2 right)^{1/2}$ is the radius function on $X$.
Define a $2$-form $omega_1$ on $X setminus pi^{-1}(0)$ by $omega_1=i partial overline{partial} f$.
Then $omega_1$ extends smoothly and uniquely to $X$.
Question:
How can I see that $omega_1$ extends to $X$?
Writing $r= sqrt{z_1 overline{z_1}+z_2 overline{z_2}}$ I can compute $omega_1$ explicitly.
The expressions are long, so I used Mathematica.
All partial derivatives $frac{partial}{partial overline{z_i}}frac{partial}{partial z_k}f$ tend to $infty$ as $r rightarrow 0$.
So it seems to me as if $f$ cannot be extended.
What am I missing?
differential-geometry complex-geometry kahler-manifolds k3-surfaces
asked Jan 25 at 21:50
user505117user505117
233
$endgroup$
The book Joyce: Riemannian Holonomy Groups and Calibrated Geometry contains on page 205 the Eguchi-Hanson space as an example:
Consider $mathbb{C}^2$ with complex coordinates $(z_1,z_2)$, acted upon by the involution $-1:(z_1,z_2) mapsto (-z_1,-z_2)$.
Let $(X,pi)$ be the blow-up of $mathbb{C}^2/{pm 1}$ at $0$.
Then $X$ is a crepant resolution of $mathbb{C}^2/{pm 1}$.
(...)
Define $f:X setminus pi^{-1}(0) rightarrow mathbb{R}$ by
$$
f=
sqrt{r^4+1}+2 log r - log left( sqrt{r^4+1} +1 right),
$$
where $r=left( |z_1|^2 + |z_2 |^2 right)^{1/2}$ is the radius function on $X$.
Define a $2$-form $omega_1$ on $X setminus pi^{-1}(0)$ by $omega_1=i partial overline{partial} f$.
Then $omega_1$ extends smoothly and uniquely to $X$.
Question:
How can I see that $omega_1$ extends to $X$?
Writing $r= sqrt{z_1 overline{z_1}+z_2 overline{z_2}}$ I can compute $omega_1$ explicitly.
The expressions are long, so I used Mathematica.
All partial derivatives $frac{partial}{partial overline{z_i}}frac{partial}{partial z_k}f$ tend to $infty$ as $r rightarrow 0$.
So it seems to me as if $f$ cannot be extended.
What am I missing?
differential-geometry complex-geometry kahler-manifolds k3-surfaces
differential-geometry complex-geometry kahler-manifolds k3-surfaces
asked Jan 25 at 21:50
user505117user505117
233
asked Jan 25 at 21:50
user505117user505117
233
asked Jan 25 at 21:50
user505117user505117
233
asked Jan 25 at 21:50
user505117user505117
233
asked Jan 25 at 21:50
user505117user505117
233
233
add a comment |
add a comment |
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