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This is the exercise 2.5.5 in the book 'Complex Geometry' by Huybrechts:

Let $X$ be a K3 surface, i.e. $X$ is a compact complex surface with $K_X\cong\mathscr{O}_X$ and $h^1(X,\mathscr{O}_X)=0$. Show that $X$ is not the blow-up of any other smooth surface.

Here $\dim X=2$, so if $X=\mathrm{Bl}_{\ell}S$ where $\dim\ell=1$ in $S$, then $X=S$. So we just need to consider the case $X=\mathrm{Bl}_{x}S$ where $x$ is a point of $S$. In this case $X=\mathrm{Bl}_{x}S\cong S\#\bar{\mathbb{P}^2}$. Now we have $\mathscr{O}_X=K_X\cong\sigma^*K_S\otimes\mathscr{O}_X(E)$ where $E$ is the exceptional divisor of the blow-up $\sigma:X\to S$. I tried to show that $\sigma^*K_S\otimes\mathscr{O}_X(E)$ must has a non-trivial global section, but I can't. Moreover, I don't know how to use $h^1(X,\mathscr{O}_X)=0$.

Actually, I guess in this case $h^1(X,\mathscr{O}_X)$ means $H^1(X,\mathscr{O}_X)$ since in this book $h^1(X,\mathscr{O}_X)=\dim H^1(X,\mathscr{O}_X)$.

Thank you for your help!

Arctic Char
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DVL-WakeUp
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  • I found that in complex algebraic surfaces by Beauville, the exceptional surve $E$ should satisfies $E^2=-1$, but on K3 surface $C^2=2g-2$. So well done. But I not learned about this. – DVL-WakeUp May 10 '21 at 08:50
  • The idea is to somehow relate $H^1(X,\mathcal O_X)$ to the topology of $X$. Blowing up a smooth point increases the rank of $H^1(X,\mathbb C)$, but this vanishes for a K3 surface (can you show this with the tools you have at hand?). – Tabes Bridges May 11 '21 at 07:40
  • @DiamondVillager The comment in this question https://math.stackexchange.com/questions/348375/k3-surface-is-not-the-blow-up-of-any-other-smooth-complex-surface, you can show that this is a non trivial global section of $\sigma^*K_S\otimes \mathcal{O}_X(E)$, but $\mathcal{O}_X$ has only trivial section. – Amira Lo May 15 '21 at 16:21
  • @AmiraLo Thank you! I have known this, but I don’t know how to prove that have non-trivial global section. – DVL-WakeUp May 18 '21 at 01:59
  • @TabesBridges Thank you for your help! I will think about this! – DVL-WakeUp May 18 '21 at 02:04
  • @DiamondVillager By $X\backslash E \cong S\backslash x$, $K_{X\backslash E}\cong K_{S\backslash x}$, then $K_{S\backslash x }\cong \mathcal{O}{X\backslash E} \cong \mathcal{O}{S \backslash x} \Rightarrow K_S = \mathcal{O}_S$ only have trivial section. And $\mathcal{O}_X(E)$ has a non trivial section vanishing along $E$, therefore $\sigma^*K_S\otimes \mathcal{O}_X(E)$ has nontrivial section. – Amira Lo May 18 '21 at 09:25
  • Thank you for your help!! – DVL-WakeUp Jul 25 '21 at 12:11
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    I realise this is an old discussion, but let me note that the suggestion of Tabes Bridges here does not work. In fact the dimension of $H^1(X,C)$ does not change under blowup. The answer by ZYun gives a correct argument. More generally, there is a formula $K_Y = f^\ast K_X + cE$ where $f \colon Y \rightarrow X$ is the blowup of a smooth subvariety of codimension $c+1$. This shows that in general, a variety with trivial canonical bundle can't be a blowup of another smooth variety along a smooth subvariety. – Lazzaro Campeotti Jul 04 '24 at 10:53

1 Answers1

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I encounter the same problem, but I find you can do this without the assumption on $H^1(X,\mathcal{O}_X)$, just $K_X$ being trivial is enough to deduce $X$ is not a blow-up of a point on a surface.

Suppose $K_X$ is trivial and $X$ is the blow-up of a point of a surface $Y$, let $f:X\rightarrow Y$ be the blow-up map. Let $E$ be the exceptional divisor of the blow-up. Consider $K_E$. We know $K_E\cong K_X|_{E}\otimes \mathcal{O}_X(E)|_{E} $ by adjunction formula. $K_X$ being trivial implies $K_E\cong \mathcal{O}_X(E)|_E\cong \mathcal{O}_E(-1)$, which is a contradiction.

ZYun
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