Embedding Complex Tori in Projective Space When we talk about projectively embedding complex tori $\mathbb{C}^{g}/\Lambda$ (i.e in Lefshetz Embedding Theorem), what exactly do we mean by an embedding. Is it in the differential geometry sense of needing to be immersive (injective on tangent spaces), or is less required ?
I suppose what I'm really asking is, what do we actually need the embedding to do to call our complex tori an abelian variety?
 A: An embedding $i: \mathbb{C}^{g}/\Lambda\to \mathbb P^N(\mathbb C)$ is first of all a holomorphic map, which has moreover to be injective and to be an immersion of complex holomorphic manifolds.
These conditions are impossible to  fulfill simustaneously  if $\Lambda$ is a general lattice in $\mathbb{C}^{g}$.
If however the lattice satisfies certain linear algebra conditions, known as Riemann's bilinear relations, then such an embeding $i$ exists.
A remarkable theorem due to Chow (and immensely generalized by Serre in his ground-breaking paper GAGA) ensures that $i(\mathbb{C}^{g}/\Lambda)\subset  \mathbb P^N(\mathbb C)$ is then, like all holomorphic submanifolds of  $\mathbb P^N(\mathbb C)$, automatically  an algebraic variety, i.e. is the common zero locus of finitely many homogeneous polynomials $P_i(z_0,...,z_N)$.
This is astonishing because the man in the street would expect $i(\mathbb{C}^{g}/\Lambda)$ to be only a holomorphic submanifold of $\mathbb P^N(\mathbb C)$, i.e. that it  can only be defined locally by using holomorphic  functions $f_i(z_0,...,z_N)$.
But Chow was not your typical man in the street... 
