This question is motivated by this one.
I would like to hear about results concerning complex projective varieties which
For example, I don't think there exists an equivalent of Mori's bend-and-break argument that avoids reduction to positive characteristic. So the existence of rational curves on Fano varieties would be an example of 2.
Here is one I am curious about : Suppose X is a proper variety over $\mathbb{C}$. Then there are only finitely etale covers of X in each degree.
This is proven in SGA 1 by comparison with the classical fundamental group, but is there a purely algebraic proof?
Concerning your example, there is definitely no analytic proof of the existence of rational curves on Fano manifolds. This is one of the dream of complex geometers... You can also consider this weaker statement: given a Fano manifold $X$, can you construct an entire curve (i.e. a non constant holomorphic image of the complex plane) in it by analytic methods? Even this is not known...
On the other hand, no algebraic proof is known for invariance of plurigenera for varieties not of general type (the analytic proof is due to Siu, later refined by Paun, and the algebraic proof for varieties of general type is due to Kawamata).
If $X$ is a proper curve of genus $g$ over an algebraically closed field $K$ of characteristic $0$, and $U$ an open subset, say obtained by removing $n$ closed points from $X$, then by comparison with the complex topology (more precisely by the Riemann Existence Theorem) one can derive that $\pi_1^{\text{ét}}(U)$ is isomorphic to the profinite completion of the group $$\langle a_1,\ldots,a_g, b_1,\ldots,b_g,c_1,\ldots,c_n \mid[a_1,b_1]\cdot\ldots\cdot[a_g,b_g]c_1\ldots c_n\rangle.$$ As far as I know, there is no algebraic proof for this fact.
I've wondered if the following is an example of 1, but I'm not expert enough in algebra to know.
The set of smooth points of an irreducible complex projective variety is connected in the classical topology.
The argument I know goes like this: Suppose it were disconnected, a disjoint union of $A$ and $B$, say. These are locally analytic sets. Then, by a theorem of Remmert and Stein, their closures $\overline{A}$ and $\overline{B}$ are analytic sets. Then, by Chow's part of the GAGA principle, $\overline{A}$ and $\overline{B}$ are varieties, and the original variety is not irreducible.
I've always wondered if you can avoid the Remmert-Stein theorem in the middle (without using Hironaka's theorem).
The Bogomolov-Tian-Todorov theorem states that deformations of Calabi-Yau manifolds (compact Kaehler manifolds with trivial canonical bundle) are unobstructed. This recent paper of Iacono and Manetti gives an algebraic proof of the theorem for algebraically closed fields of characteristic 0. As far as I know this is the only algebraic result in this direction.
1) I don't know if this qualifies, but it seems to me that there is no non-analytic proof of the statement that a surface of general type with $c_1^2=3c_2$ is a quotient of a unit two-dimensional complex ball. This is a theorem of Yau.
2) There is a notion of K-stability of complex projective (polarised) varieties (this is an algebraic notion). This K-stability holds for varieties that admit a constant scalar curvature Kähler (cscK) metric in the chosen polarisation (Donaldson). Usually it is hard to prove that a polarised variety is K-stable unless you know that is has a cscK metric.
Kevin are you aware of the Ran/Kawamata argument for this theorem? (1992 pair of articles in Journal of Algebraic Geometry) Kawamata's version is not only algebraic but shockingly cute. The only problem is the profs who told me about it couldn't really explain why having unobstructed formal deformations (the algebraic side) means also that the analytic obstruction vanishes/the Kuranishi space is smooth.
I am fairly certain that there is no complex analytic proof of the following theorem (but I would love to be proven wrong!). This is not strictly speaking an answer to the question, because the available proof is not exactly algebraic either; rather, it uses $p$-adic (analytic) methods.
Theorem. (Batyrev) Let $X$ and $Y$ be birational Calabi–Yau varieties (that is, smooth projective over $\mathbb C$ with $\Omega^n \cong \mathcal O$). Then $H^i(X,\mathbb C) \cong H^i(Y,\mathbb C)$.
The same methods were later refined to prove the following theorem:
Theorem. (Ito) Let $X$ and $Y$ be birational smooth minimal models (that is, smooth projective over $\mathbb C$ with $\Omega^n$ nef). Then $h^{p,q}(X) = h^{p,q}(Y)$ for all $p,q$.
Again, the proof goes through $p$-adic analytic methods, this time combined with $p$-adic Hodge theory (which I think counts as an algebraic method).
References.
V. V. Batyrev, Birational Calabi–Yau $n$-folds have equal Betti numbers. arXiv:alg-geom/9710020
T. Ito, Birational smooth minimal models have equal Hodge numbers in all dimensions. arXiv:math/0209269
The following statement about holomorphic vectors bundles is in the same spirit as Dmitri's answer. The tensor product of two slope-polystable bundles is again slope stable. Assuming the Hitchin-Kobayashi correspondence, this is easy. Slope-polystablility is equivalent to the existence of a Hermitian-Yang-Mills connection whilst the tensor product of two HYM connections is easily seen to be again HYM. On the other hand, a purely algebraic proof of this fact is not so straightforward. (That said, neither is the proof of the Hitchin-Kobayashi theorem.)
To my knowledge, there is no characteristic-free proof of Grauert–Riemenschneider vanishing theorem: Let $\pi \colon \widetilde{X} \to X$ a desingularization of $X$, $\mathcal{F}$ an ample locally free sheaf on $X$ such that $\pi^{\ast}(\mathcal{F})$ is also locally free, then $R^p\pi_*(\pi^{\ast}(\mathcal{F}) \otimes \omega_{\widetilde{X}}) = 0$ for all $p \geq 1$, where $\omega_{\widetilde{X}}$ denotes the canonical sheaf on $\widetilde{X}$.