I am attempting to understand and use the degeneration formula for GW invariants as stated in 'Gromov-Witten theory and Donaldson-Thomas theory II' (Maulik, Nekrasov, Okounkov, Pandharipande). The formula is found in section 3.4 of the aforementioned paper, but I will recall it here.

Let $\lambda:\chi\to C$ be a nonsingular 4-fold fibered over a nonsingular irreducible curve $C$ (I'm perfectly happy letting C = $\mathbb P^1$.) Let $X$ be a nonsingular fiber of $\lambda$ and $X_1\cup_S X_2$ be a reducible special fiber, consisting of two nonsingular 3-folds intersecting transversely along the the nonsingular surface (the relative divisor) $S$. The degeneration formula for the absolute invariants of $X$ in terms of the relative invariants of $X_1/S$ and $X_2/S$, is given as follows:

$$ Z'( X\ \left | \right. \prod _{i=1}^r \tau_0(\gamma_{l_i})_\beta = \sum Z'(\frac{X_1}{S}|\prod_{i\in P_1} \tau_0(\gamma_{l_i}))_{\beta_1,\eta} \xi(\eta) u^{2l(\eta)} Z'(\frac{X_2}{S}|\prod_{i\in P_2}\tau_0(\gamma_{l_i})_{\beta_2, \eta^{V}}$$.

In the above formula, $\eta$ is a weighted cohomology partition, and $\xi(\eta) = \prod_i \eta_i|Aut(\eta_i)$. $Z'$ is of course the GW partition function.

My Question: MNOP state that the sum is taken over curve splittings $\beta = \beta_1+\beta_2$. However, $\beta$ is a class on the nonsingular fiber $X$ and $\beta_1$ and $\beta_2$ are classes on the two components of the special fiber. I have tried to look at J. Li's paper that introduced the degeneration formula but I am unable to reconcile the two. The manner in which it is stated in MNOP II seems to indicate that the $\beta_i$ are classes on the components of the special fiber, but in that case I am not certain I understand how $\beta$ splits into these pieces. Are these classes implicitly being pulled back to classes on $X$ in some natural way?

Thanks in advance.


1 Answer 1


I have found a rough answer to a special case of my question in YP Lee's notes (roughly page 21 here http://www.math.utah.edu/~yplee/research/grenoble.pdf). If anyone has any further insight though I'd be eager to hear it

I will state it here for anyone who is interested.

The special case is deformation to the normal cone. I'm going to leave out how the insertions and genus conditions play out, but its fairly intuitive.

Let $Z\subset X$ be a smooth subvariety of a smooth variety $X$. Let $W$ be the blowup of $Z\times${0}$\hookrightarrow X\times \mathbb C$. Let $t$ be the deformation parameter. For $t\neq 0$ $W_t$ is $X$, and at $W_0$ we have a singular fiber consisting of two smooth pieces, $\hat X$, the blowup of $X$ at $Z$ and the second being $P = \mathbb P_Z(N_{Z/X}+\mathcal{O})$.

So with this the set up, let $p:\mathbb P_Z(N_{Z/X}+\mathcal{O})\to Z\subset X$ and $\phi:\hat X\to X$. In the degeneration formula, the sum $\beta_1+\beta_2=\beta$ is taken over $$ \beta = \phi_*\beta_1+p_*\beta_2. $$ (The set up is a little different in YP Lee's presentation, as we're doing this for invariants and not generating functions. There's also an additional sum over the basis of the cohomology of the relative divisor, but if the point is to understand how the curve splitting works, I needn't record that here.)

So all in all it works like you would want it to.


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