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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:

  1. Deligne proves Hodge implies absolute Hodge for abelian varieties by deforming the general case to CM abelian varieties (using principle $B$).
  2. 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.
  3. 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).
  4. 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).
  5. Generic vanishing theorem for abelian varieties.
  6. Perverse continuation principle.
  7. Mori's bound and break techniques.
  8. Deformation of Galois representations.
  9. Gross’s proof of Chowla-Selberg formula.
  10. Tits deformation principle.
  11. Deformation to the normal cone.
  12. 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.

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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 $$\pi_1(S^n, \, x_0)=\{1\}.$$

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    $\begingroup$ But $S^n$ does not contain any straight line segment. Or do you mean a homotopy equivalent polyhedron boundary? $\endgroup$ – მამუკა ჯიბლაძე Mar 28 at 5:13
  • $\begingroup$ @მამუკაჯიბლაძე: you are obviously right. I wanted to say that every point in $S^n$ has a neighborhood that is diffeomorphic to a star-shaped set, and this implies that every path is homotopic to a piecewise smooth path. Thanks for the remark. $\endgroup$ – Francesco Polizzi Mar 28 at 10:35
  • $\begingroup$ Sorry I still do not understand - the same is true of all differentiable manifolds (including non-simply connected ones), no? $\endgroup$ – მამუკა ჯიბლაძე Mar 28 at 19:13
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    $\begingroup$ @მამუკაჯიბლაძე: yes, but in the proof I also use the fact that $S^n$ minus a point is contractible. This is not true for a general manifold. $\endgroup$ – Francesco Polizzi Mar 28 at 21:26
  • $\begingroup$ One also need that the dimension is at least $2$, in order to use Sard's argument to prove the non-surjectivity of the path. $\endgroup$ – Francesco Polizzi Mar 28 at 21:29
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Almost all of Brill-Noether theory (e.g. Eisenbud-Harris limit linear series) is based on deformation (or "degeneration") arguments.

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In Character Varieties of Free Groups are Gorenstein but not always Factorial (co-authored with C. Manon), we show a family of moduli spaces are Gorenstein using deformation techniques.

In particular, we construct a flat degeneration of our moduli spaces to a affine toric variety (that we show is Gorenstein), and then invoke the Gorenstein property is "open" (the property of the special fiber of the deformation implies the general case).

Depending on your point-of-view, this could be considered an example of deformation techniques being used in commutative algebra.

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    $\begingroup$ That's very good, thank you a lot ! $\endgroup$ – sawdada Mar 28 at 16:52
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In this paper http://www.ams.org/journals/tran/2000-352-09/S0002-9947-00-02416-8/S0002-9947-00-02416-8.pdf, Ciliberto and Miranda use a very clever degeneration of the complex projective plane $\mathbb P^2$ into a union of a $\mathbb P^2$ and a Hirzebruch surface $\mathbb F_1$, meeting transversally along a curve, to be able to apply a recursive argument to study the dimension of some linear systems of curves in $\mathbb P^2$.

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Tropical geometry emerges from a deformation procedure called Maslov dequantization.

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A classical(!) example of deformation is quantization. Here the Planck constant is the deformation parameter.

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One can prove genus-degree formula for plane curves by deforming the plane curve of degree $d$ into a configuration of $d$ lines on the plane.

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Q-analogs introduce a parameter $q$ that recovers known results as $q \rightarrow 1$.

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