Some questions on the tangent bundle of manifolds

Question

For what type of manifolds $M$,

1) $TM$ does not admit a holomorphic structure?

2)$TM$ is not diffeomorphic to an algebraic variete in some $\mathbb{C}^n$?

Answer 1

1. There is always an integrable almost complex structure on $TM$, see Theorem 2.2 in

R. Szöke, Complex structures on tangent bundles of Riemannian manifolds. Mathematische Annalen, 291, (1991) 3, page 409-428.

2. I know of two types of obstructions to the existence of a structure of a smooth complex quasiprojective variety on $TM$:

a. Each quasiprojective variety is tame (admits a compactification as a manifold with boundary), while $TM$ need not be tame, e.g. it can have non-finitely generated fundamental group (for instance, take $M$ to be a surface of infinite genus) or infinite Betti numbers.

b. Assume, therefore, that $M$ is compact. There are some known restrictions on fundamental groups of smooth complex quasiprojective varieties coming from the Rational Homotopy Theory:

J. Morgan, The algebraic topology of smooth algebraic varieties. Inst. Hautes Études Sci. Publ. Math. No. 48 (1978), 137–204.

Since every finitely presented group can be realized as the fundamental group of some smooth closed 4-dimensional manifold, we obtain, therefore, more examples.

Answer 2

I would like to point out that the fact that $TM$ admits a complex structure follows from an earlier result than the one mentioned in Moishe Cohen's answer.

Landweber proved the following result in his $1974$ paper Complex Structures on Open Manifolds:

Let $X$ be an open manifold of dimension $2q$. If $H^i(X; \mathbb{Z}) = 0$ for $i > q$, then each almost complex structure on $X$ is homotopic to an integrable almost complex structure.

If $M$ is a $q$-dimensional manifold without boundary, then $TM$ is a $2q$-dimensional open manifold. Moreover, $TM$ is homotopy equivalent to $M$ and hence for $i > q$, $H^i(TM; \mathbb{Z}) \cong H^i(M; \mathbb{Z}) = 0$. As pointed out in the comments, $TM$ always admits an almost complex structure, so by the result above, it admits an integrable almost complex structure, i.e. a complex structure.