3
votes
2answers
250 views

Varieties that do not extend to flat families

Is it easy to give an example of a function field $K$ and a smooth proper variety $X$ over $K$ that does not extend to a flat scheme over $B$, where $B$ is a smooth proper variety with function field ...
0
votes
0answers
135 views

Quicker way to show that the restriction to a open subvariety is again proper?

Dear all, Let $f: X \rightarrow Y$ be a morphism of projective varieties over $\mathbb{C}$. Also let $V \subset Y$ be a nontrivial open subvariety and set $U:= f^{-1}(V)$. I would like to show that ...
7
votes
1answer
407 views

Complex analytic space with no (positive dim.) subscheme ?

Is there an example of a complex analytic space $X$ that doesn't have any (not necessarily open or closed) positive dimensional subspace $Y$ which is analytically isomorphic to (the complex ...
4
votes
2answers
417 views

Relationship between Line Bundles with isomorphic ring of sections

Suppose two positive holomorphic line bundles $L_1 \to X_1, L_2\to X_2$ over two projective complex manifold $X_1, X_2$ have isomorphic ring of sections $R=R_1=R_2$ where ...
9
votes
4answers
1k views

Nonalgebraic complex varieties

I'm turning here (a variation of) a question asked by a friend of mine. For the purposes of this question I will say that a compact complex manifold is projective if it is isomorphic to a subvariety ...