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The answer is "no" because in general there does not even exist a neighbourhood $U$ of your submanifold that holomorphically retracts to the submanifold. In fact Finnur Lárusson has proved the followi …

Let $M$ be a differentiable manifold, $A=C^\infty (M)$ its ring of global differentiable functions and $\Omega^1 (M)$ the A-module of global differential forms of class $C^\infty$. The A-module of …

Consider the real line $\mathbb R$ and $C^1_0$ , the ring of germs of continuously differentiable functions at zero. Now take the ideal $M$ of germs vanishing at zero. The Zariski cotangent space $M/M …

Claim: The complement $U=\mathbb C^h\setminus \{F=0\}$ is path-connected and thus connected. Proof: Given $a,b\in U$ consider the affine complex line $L_{a,b}=L$ joining $a$ to $b$. The polynomial $ …

If $X$ is a $ C^\infty$ differential manifold of dimension $n$, then there exists an embedding $f:X\to \mathbb R^{2n+1}$. This is a not too difficult theorem due to Whitney, proved in many textbooks. …

1) There is a great book From Holomorphic Functions to Complex Manifolds by Fritzsche-Grauert. It is very geometric and gives you the fundamentals on complex manifolds, including specialized topics, f …

In his remarkable thesis Douady proved that, given a compact complex analytic space $X$, the set $H(X)$ of analytic subspaces of $X$ has itself a natural structure of analytic space . If $X=\mathbb P …

I interpret your line bundle $L$ to be differentiable rather than holomorphic, else the statement that line bundles are parametrized by their Chern class would be completely false. There always e …

Dear Benjamin, the statement that holomorphic sections are dense in the smooth sections is false, already for the trivial bundle of rank one $E_1=X\times \mathbb C$ over $X=\mathbb C$. Indeed on any …

Dear Timothy, here is a theorem which, according to your wish, "could be understood, and seen to be interesting, by someone who had not already studied the material in that course": Brouwer's celebrat …

An interesting application of De Rham's theorem is to show that certain differential manifolds are not diffeomorphic. Here are two examples. 1) For $n$ even the sphere $S^n$ and real projective spac …

Dear David: yes! In one direction this is just the functoriality of tangent maps. Let $f:X\to Y$ be the morphism, $x$ a point in $X$ with image $y\in Y$ and $g:V\to X$ a local section. From $f \circ …

I) The term exterior multiplication ("äussere Multiplication") is due to Grassmann, who introduced the term in his book (written in 1844) Die Wissenschaft der extensiven Grösse oder die Ausdehnungsl …

Douady introduced Banach analytic manifolds in order to solve a conjecture of Grothendieck's on the existence of a moduli space for compact analytic subspaces of a given fixed analytic space. These Ba …

Here are a few points to guide you into the beautiful subject you had the good taste to choose. 1) Hartogs extension phenomenon :given two concentric balls in $ \mathbb C^n$, any holomorphic funct …