Timeline for How the complex conjugation on sheaves of modules is defined?
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| Feb 11, 2022 at 11:39 | comment | added | user505117 | Does this answer imply, by choosing $\bigoplus \mathcal{O}_{\mathbb{P}}(a_i)=0$, that on $\mathbb{P}^1$ we have $\tau^* \mathcal{O}(2) \simeq \mathcal{O}(2)$, where $\tau$ is complex conjugation? That doesn't seem right, because $d \tau$ defines an isomorphism $T \mathbb{P}^1 = \overline{T \mathbb{P}^1}=(T \mathbb{P}^1)^*$, i.e. an isomorphism from $\mathcal{O}(2)$ to $\mathcal{O}(-2)$. Also, $\tau$ does not preserve $c_1(\mathcal{O}(2))$. | |
| Aug 17, 2010 at 8:27 | comment | added | Francesco Polizzi | F. Catanese wrote some papers on this subject. For instance "Moduli spaces of surfaces and real structures", Ann. Math. 158. But I do not remember whether he considers also the case of coherent sheaves. One can find all of them on arXiv. | |
| Aug 17, 2010 at 1:01 | comment | added | Donu Arapura | I wish I could recommend a reference, but I'm not sure of one. Perhaps someone else can help? | |
| Aug 17, 2010 at 0:45 | comment | added | Dmitry Kerner | Thanks for the reply. Of course I meant X is stabilized by $\tau$. (It implies that X is defined over the reals, right?) If a presentation is given one can define the complex conjugation. But the conjugation can be defined in various ways. Apparently they are all compatible, but probably this should be proved? Could you recommend some short intro/summary? | |
| Aug 17, 2010 at 0:31 | history | edited | Donu Arapura | CC BY-SA 2.5 |
added 5 characters in body
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| Aug 17, 2010 at 0:18 | history | answered | Donu Arapura | CC BY-SA 2.5 |