Henri
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Registered User
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Phd Student in Complex Geometry
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May 3 |
revised |
Hyperbolic Riemann Surface added 91 characters in body |
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May 3 |
answered | Hyperbolic Riemann Surface |
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Apr 27 |
awarded | ● Yearling |
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Mar 20 |
comment |
Extension of pluriharmonic functions Yes you are right of course; I just had in mind the case where $M$ is a domain in $\mathbb C^n$. My point was just that the assumptions on $\Omega$ should not be too restrictive like pseudoconvexity e.g.; but on $M surely! |
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Mar 20 |
comment |
Extension of pluriharmonic functions No, not even. You just need $\Omega$ relatively compact, and $M\setminus \overline{\Omega}$ connected. (btw, you should have defined $u$ on $M\setminus \overline{\Omega}$ only if you want to make sense of pluriharmonicity) |
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Mar 19 |
comment |
Extension of pluriharmonic functions Oh, I just realized that I misunderstood your question; I thought that $u$ was only defined on $\Omega$. In fact, if $M\setminus \Omega$ has $b_1=0$, then every pluriharmonic function on $M\setminus \Omega$ is the real part of a holomorphic function; so $u=Re(f)$ for $f \in \mathcal O(M\setminus \Omega)$. Now, if $M\setminus \Omega$ is connected, Hartog's extension theorem tells you that $f$ extends (uniquely) to $M$, and then $Re(f)$ extends $u$ has a pluriharmonic function. |
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Mar 19 |
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Extension of pluriharmonic functions Yes; if you want consider $\Omega = B(0, 1/2)$; by uniqueness of the pluriharmonic extension (ph functions are real analytic), $u$ won't extend to $\mathbb C^2$ either. |
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Mar 19 |
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Extension of pluriharmonic functions How about $\Omega=B(0,1)\subset \mathbb C^2$ and $u(z,w)=Re(e^{1/(z-1)})$? (cf remark of Alexandre) |
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Feb 27 |
awarded | ● Civic Duty |
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Jan 31 |
comment |
When does $Aut(X)=Bir(X)$ hold? Yes, Y. Consider the two projection $p_1$, $p_2$ from the graph to $X$ and $Y$. We know that the exceptional locus of $p_1$ contains rational curves $C \subset X\times Y$. By assumption, $p_2$ must contract $C$, so that $C$ is contracted by $p_1$ and $p_2$, which is absurd. |
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Jan 31 |
answered | When does $Aut(X)=Bir(X)$ hold? |
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Nov 21 |
revised |
restriction of sheaf added 276 characters in body |
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Nov 21 |
comment |
restriction of sheaf In particular, as $X$ is smooth hence normal, the desired property holds as soon as $U$ has codimension at least $2$. |
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Nov 21 |
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restriction of sheaf Thanks, you are perfectly right! (I had in mind a morphism with connected fibers) So here here one can just say that $i_*i^*F= i_* \mathcal O_X \otimes F$. |
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Nov 21 |
answered | restriction of sheaf |
