The idea of proving something by deforming the general case to some special cases is very powerful. For example, one can prove certain equalities by regarding both sides as functions/sheaves, and show they are equal in some good domains.
Also, one can study something like moduli spaces by deforming a point to formal neighborhood, and use this (plus obstruction theory) to detect dimension and other properties.
Therefore, the idea of deformation is very common and useful , and there are many famous applications:
- Deligne proves Hodge implies absolute Hodge for abelian varieties by deforming the general case to CM abelian varieties (using principle $B$).
- If $X,Y$ are two smooth projective curves of genus $>1$, then $Hom_{nonconst}(X,Y)$ is finite by studying the Hom scheme using deformation theory.
- We can use deformation to detect dimension of moduli spaces we hope to understand, or prove some smoothness/non-smoothness theorems ($\mathcal{M_g}$, $Bun_G$, Hilbert schemes, smoothness of Jacobian for smooth projective curves, modulis of Calabi-Yau by Tian–Todorov).
- Lifting theorems by considering infinitesimal thickenings and algebraic theorems, for example Grothendieck for curves, Serre-Tate for ordinary abelian varieties, Deligne for K3 surfaces (up to a finite extension).
- Generic vanishing theorem for abelian varieties.
- Perverse continuation principle.
- Mori's bound and break techniques.
- Deformation of Galois representations.
- Gross’s proof of Chowla-Selberg formula.
- Tits deformation principle.
- Deformation to the normal cone.
- The proof of standard conjectures for any smooth projective variety of $K3^{[n]}$ type (see https://arxiv.org/abs/1009.0413).
There is of course more, what are some other applications of the idea of deformation in algebraic geometry, number theory and representation theory etc? I believe some are still very important but maybe non-experts don't know those examples and ideas.
In some sense, proving something in good case and using density argument may be regarded as applying deformation as well. As a very naive example, one can prove Cayley-Hamilton theorem by first considering diagonalizable matrices.
Best Answer
I like the following proof that the sphere $S^n$ is simply connected for $n \geq 2$, that does not use the Seifert-Van Kampen theorem, but rather a deformation argument and basic tools of differential topology.
Let us consider a closed path $\alpha \colon I \to S^n$ based at $x_0$. Since every point of $S^n$ has a neighborhood that is diffeomorphic to a star-shaped subset of $\mathbb{R}^{n+1}$, the path $\alpha$ is homotopic to a piecewise smooth path $\beta \colon X \to S^n$ based at the same point, see for instance this MSE question.
By a standard application of Sard's lemma, a piecewise smooth path cannot be space-filling, in particular the image $\beta(I)$ is not the whole of $S^n$. In other words, there is a point $p \in S^n - \beta(I)$, and we can see $\beta$ as a path $$\beta \colon I \to S^n-\{p\} \simeq \mathbb{R}^n.$$
Since $\mathbb{R}^n$ is contractible, the path $\beta$ is homotopic to a constant path, and so the same holds for $\alpha$. This shows that $$\pi_1(S^n, \, x_0)=\{1\}.$$