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Results tagged with complex-geometry
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Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
188
votes
Accepted
Why is the Hodge Conjecture so important?
Let $K$ be one of the following fields: the complex numbers, a finite field, a number field (and we could amalgamate the last two into the more general case of a field finitely generated overs its pri …
- 57.6k
4
votes
Covers of the projective line over Z and arithmetic Grauert-Remmert
This is a question which is too long to put in a comment box: What exactly do you mean by a simple normal crossings divisor in $\mathbb P^1_{\mathbb Z}$?
Let me recall that an irreducible divisor in …
- 57.6k
22
votes
Accepted
what is the motivation of Shimura variety?
The theory of Shimura varieties was begun by Shimura, and further developed by Langlands (who introduced the name), and is now a central part of arithemtic geometry and of the theories
of automorphic …
- 57.6k
13
votes
Why do people think that abelian varieties are the hardest case for the Hodge conjecture?
Related to Jim Milne's answer, one might mention that Deligne proved that for abelian varieties, all Hodge cycles are "absolutely Hodge" (i.e. when you think of them embedded diagonally inside the pro …
- 57.6k
6
votes
Accepted
Principal bundles and associated vector bundles, the case of the complex projective space (1...
In $SU(n+1)/U(n)$ there is a natural basepoint, the coset of the identity, which is fixed by
the action of $U(n)$ (thought of as acting on the quotient by virtue of being a subgroup
of $SU(n1)$). Sin …
- 57.6k
14
votes
Accepted
What do intermediate Jacobians do?
Probably you know this, but just to be sure: they receive cycle class maps from
codimension $k$ cycles. More precisely, if $Z$ is a cycle on $X$ of codimension $k$
that is cohomologically trivial, th …
- 57.6k