Let P(x,y, z, w)= 0 be a complex surface . How can we know that this surface is a sub-variety of an abelian variety or not?
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9$\begingroup$ If this is a smooth surface in $\mathbb{P}^3$, then the answer is never: by weak Lefschetz, it is simply connected, but a subvariety of an abelian variety is not. $\endgroup$– Donu ArapuraCommented Oct 23, 2012 at 15:14
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1$\begingroup$ @Jafar: Do you have a specific surface in mind? $\endgroup$– Daniel LoughranCommented Oct 23, 2012 at 15:55
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2$\begingroup$ If your surface contains a rational curve then the answer is again never, as abelian varieties never contain any rational curves. $\endgroup$– Daniel LoughranCommented Oct 23, 2012 at 15:56
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5$\begingroup$ Could this question, please, be expanded somehow, including a more concise specification of P and an explantion what 'how can we know' should mean in detail. True, one can somehow infer/guess what might be meant, but still I think this will be a much better question with a bit of effort. Finally, a more informative title seems also desirable. $\endgroup$– user9072Commented Oct 23, 2012 at 16:38
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2$\begingroup$ @Jafar: Yes but it depends on the surface. If you have a specific surface that you are interested in, it might be possible to say something. I concur with quid that you should clarify the question. $\endgroup$– Daniel LoughranCommented Oct 23, 2012 at 22:16
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