# Vanishing cycles exact sequence for degeneration of curves

Let $$f : X\rightarrow D$$ be a family of smooth projective curves over the complex unit disk $$D$$, degenerating to a nodal curve $$X_0$$ above $$0\in D$$.

Let $$\eta\in D - \{0\}$$ be a general point, and let $$\gamma\subset X_\eta$$ be the vanishing cycle, which for me is a disjoint union of simple closed curves. Let $$T\supset\gamma$$ denote a tubular neighborhood of $$\gamma$$ in $$X_\eta$$, so $$T$$ is a disjoint union of annuli.

I believe there should be an exact sequence (in betti cohomology)

$$0\rightarrow H^1(X_0)\rightarrow H^1(X_\eta)\rightarrow H^1(T)\stackrel{\delta}{\rightarrow }H^2(X_0)\rightarrow H^2(X_\eta)\rightarrow H^2(T) = 0$$

The map $$H^*(X_0)\rightarrow H^*(X_\eta)$$ comes from the inclusion $$X_\eta\rightarrow X$$ using the isomorphism $$H^*(X) = H^*(X_0)$$.

1. Is there a geometric description of the connecting map $$\delta$$?
2. Does anyone have a good reference for this exact sequence?
3. Am I correct in saying that this sequence is equivariant for the monodromy action of $$\pi_1(D-\{0\},\eta)$$, where $$H^1(X_\eta)$$ is the only term where the action is nontrivial?

To set some context, while I'd eventually want a deeper understanding of vanishing cycles, I'd like to begin by understanding this simple example, and it's hard to find a reference that doesn't immediately dive into perverse sheaves and such.

This is from standard Lefschetz's theory, which works for a one-parameter family with special fiber acquiring a node. The fiber can have arbitrary dimensions, not only for curves.

To start with, I don't see where the $$H^1(T)$$ in your sequence is coming from. To me, there is a long exact sequence of the cohomology of the pair $$(X,X_{\eta})$$: $$\cdots \to H^1(X)\to H^1(X_{\eta})\xrightarrow{\alpha} H^2(X,X_{\eta})\to H^2(X)\to \cdots.\tag{1}\label{1}$$

First $$X$$ deformation retracts onto $$X_0$$, so one has $$H^*(X)\cong H^*(X_0)$$. Second, by excision, $$H^*(X,X_{\eta})\cong H^*(U,U\cap X_{\eta})$$, where $$U$$ is an open ball in $$X$$ around the nodal point on $$X_0$$. The pair $$(U,U\cap X_{\eta})$$ is homotopic equivalent to $$(C\gamma,\gamma)\simeq (B^2,S^1)$$, where $$C\gamma$$ is the cone over the vanishing cycle $$\gamma$$. So $$H^2(X,X_{\eta})\cong H^2(B^2,S^1)$$ is generated by the single class $$[C\gamma]$$. Since $$\alpha$$ is not zero map, we conclude that the long exact sequence \eqref{1} reduces to

$$0\to H^1(X_0)\to H^1(X_{\eta})\xrightarrow{\alpha} \mathbb Z\to 0, \ \text{and}$$

$$0\to H^2(X_0)\to H^2(X_{\eta})\to 0.$$

$$\alpha$$ sends the vanishing cycle to its cone, and $$\ker(\alpha)$$ is called the module of invariant cycles.

Sometimes, people work for $$X$$ projective and the base being $$\mathbb P^1$$, e.g., a Lefschetz pencil. In that case, $$\alpha$$ in \eqref{1} is still surjective, but there can be many vanishing cycles.

One can refer to [Voisin, vol II, Chapter 2 and 3], [Nicolaescu's notes], and [Lamotke, 81],