Is it known whether siegel upper half plane is dense in the space of nonsingular matrices of same dimension .$.http://en.wikipedia.org/wiki/Siegel_upper_half-space. Actually the question i have in my mind is to show that almost all 2 diml complex tori are not abelian varieties.I think I have solved the problem by using the answer given below.
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1$\begingroup$ When g=1 it seems like you are asking whether the upper half plane is dense in the punctured plane. Perhaps you mean something else? $\endgroup$– Mike JuryCommented Aug 19, 2013 at 17:54
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$\begingroup$ i mean for g greater than 2. I suspect it will be even nowhere dense $\endgroup$– KoushikCommented Aug 19, 2013 at 18:20
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$\begingroup$ This question appears to be off-topic because it should be answerable by some more thought $\endgroup$– Yemon ChoiCommented Aug 19, 2013 at 19:31
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1 Answer
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There are symmetric matrices whose imaginary part is negative definite, like $-I$ for example, and these form an open set which the Siegel upper half space does not enter. On the other hand, since the Siegel upper half space is open in the symmetric matrices, it is somewhere dense, for example near $I$. But if you look inside the space of nonsingular matrices, the symmetric ones form a proper subvariety, so are nowhere dense.
answered Aug 19, 2013 at 19:09