variety是A^n / P^n -> k的一个polynomial map的kernel，即0的preimage。preimage很复杂，故ag中也经常面对singularity。

但若从整体来看，岂不可以有类似morse theory的一些手段。

Morse理论需要一个关键的形变引理，没有它，后面的理论都是不可能的。代数几何的函数或者映射具有极强的刚性，不能象光滑范畴中那样随意变形，因而也不可能有Morse理论。Morse理论可以推广到很多地方，甚至可以推广到距离空间上并用来研究PDE.但是不大可能推广到代数几何或者复几何中。

这些是尽人皆知的，办法是总有的。再复杂，都不会超出stack的能力。我倒怀疑这个已经有人做过了。

就像我以前说的推广，用一样的办法照样可以定义moduli space，但总之不简单，高维的moduli space比较抽象。need more time to find applications.

Morse理论需要一个关键的形变引理，没有它，后面的理论都是不可能的。代数几何的函数或者映射具有极强的刚性，不能象光滑范畴中那样随意变形，因而也不可能有Morse理论。Morse理论可以推广到很多地方，甚至可以推广到距离空间上并用来研究PDE.但是不大可能推广到代数几何或者复几何中。
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Mvorse 理论中的这个关键的形变引理是什么呢？
突然想到，代数几何很像量子力学。
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这些是尽人皆知的，办法是总有的。再复杂，都不会超出stack的能力。我倒怀疑这个已经有人做过了。

就像我以前说的推广，用一样的办法照样可以定义moduli space，但总之不简单，高维的moduli space比较抽象。need more time to find applications.
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Stack也不是万能的，况且现在对于Stack的一些性质仍然不是很清楚，对它的知识还很有限。
“就像我以前说的推广，用一样的办法照样可以定义moduli space”，用什么样的办法定义哪个的moduli space？高维的还是低维的?

Actually the idea is extremely boring. For example, instead of studying the affine variety f[x_1, x_2, ... , x_n] = 0, maybe we should study a parameterized family of varieties f[x_1, x_2, ... , x_n] - x_(n+1) = 0, which will always be a morse map: A^n -> k after a little perturbation.

How do we study Y-valued morse theory? We can have the definition for the critical set from geometry instincts. Take R^2 for the easiest example. A "common" projection T^2 -> R^2 will have the inner and outer S^1 as the critical set of the map [this looks similar to the Morse-Bott case, because we embedd the T^2 in R^3 which is R x R^2] --- basically the critical set is where the topology of the preimage changes. Therefore, giving a smooth map f: X->Y [so, df: TX->TY] between smooth closed manifolds, and which satisfy some similar transversality and non-degenerate conditions, we say a point p of X is in the critical set if there exist a smooth map g: Y->R defined locally at p s.t. d_p(g o f) = 0 and d_(f(p))(g) != 0. In some sense, the R^2-valued theory is the study of a family of R-valued theories, and we hope it is more powerful, and we hope we can define things such as cup product & massey product on it directly.

The flow comes from the tangent space decomposition. On critical points we have 1-d flow, while on regular points we have 2-d flow. We moduli the R^2-action to get the moduli space of flows.

Acutally have you guys heard of Picard-Lefschetz Theory? It's a bit like my idea, but different.