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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.

4 votes
1 answer
3k views

First Chern class of canonical bundle ?

This is a somewhat simple question: consider a complex manifold $M$ and its canonical bundle $\omega_X$. It is clear that in $H^2(X,\mathbb{R})$, $$c_1(\omega_X) = - c_1(T_X)$$ (Obvious using Chern-W …
Youloush's user avatar
  • 365
1 vote
1 answer
808 views

Picard group of a K3 surface generated by a curve

In Lazarsfeld's article "Brill Noether Petri without degenerations" he mentions the fact that for any integer $g \geq 2$, one may find a K3 surface $X$ and a curve $C$ of genus $g$ on $X$ such that th …
Youloush's user avatar
  • 365
13 votes
1 answer
2k views

Chern classes of ideal sheaf of an analytic subset

Let $X$ be a Kähler manifold of dimension $n$, and $Z \subset X$ an analytic subset of codimension $k$. I have read in a paper the following result, a proof of which I cannot find: $$c_k(\mathscr{I}_ …
Youloush's user avatar
  • 365
4 votes
2 answers
1k views

Holomorphic bundles and maps to the Grassmannian ?

Hello, In the differentiable case it is quite easy to prove that vector bundles are equivalent to smooth maps to the Grassmannian $G_{k}(\mathbb{R}^N)$ for some integer $N>>0$. The proofs I have seen …
Youloush's user avatar
  • 365