Let \(X = \mathbb P^1\) with toroidal log structure \(D = [0] + [\infty]\) and \(\beta = ([\mathbb P^1], g=0, n=3, u_1=u_2=1, u_3=0)\) be the curve class of a stable log curve. Then for any stable log map \((C,f, p_1,p_2,p_3) \in M(X, \beta)\), the prescribed contact orders require that \(f(p_1) = [0]\) and \(f(p_2) = \infty\). We are left with three cases:
- If \(f(p_3) = f(p_1)\), then \(C\) consists of two components both isomorphic to \(\mathbb P^1\), one is contracted to \(0\in X\) and contains both \(p_3\) and \(p_1\) and the other is mapped identically to \(X\).
- If \(f(p_3) = f(p_2)\), we have a similar situation except the contracted component is sent to \(\infty\) and contains both \(p_2\) instead of \(p_1\).
- If neither \(f(p_3) = f(p_1)\) nor \(f(p_3) = f(p_1)\), then stability implies \(C\cong \mathbb P^1\).
Thus \(M(X, \beta) \cong \mathbb P^1\) via the third evaluation map \(e_3: [C, f, p_1,p_2,p_3] \mapsto f(p_3)\).
Computing a curve count
Choose a cohomology class in \(X\) corresponding to a point, \([\text{pt}] = c_1(\mathcal O(1))\). Pulling back to \(M(X, \beta)\) via \(e_3\) yields \(c_1(\mathcal O(1))\) again, so \[\int_{M(X, \beta)} e_3^*\big(c_1(\mathcal O(1))\big) = \deg(c_1(\mathcal O(1)) \cap [\mathbb P^1]) = 1.\]
Computing via localization
Now let \(T = \mathbb C^*\) act on \(X=\mathbb P^1\) via distinct characters \(\chi_0, \chi_\infty \in M = \Hom(T, \mathbb C^*)\); in coordinates, \(t\cdot [x:y] = [\chi_0(t)x : \chi_\infty(t)y]\). The induced action on \(M(X, \beta)\) is given by postcomposing \(f:C\to X\) with the action of \(T\), so under the identification \(M(X, \beta) \cong \mathbb P^1\) above, \(T\) acts with the same characters on \(M(X, \beta)\). Under this action the fixed points of \(M(X, \beta)\) are \(C_0\) and \(C_\infty\) corresponding to the stable curves where \(f(p_3)\) is \(0 = [1:0]\) and \(\infty = [0:1]\) respectively.
Localization tells us that
\[\int_{M(X, \beta)} c_1(\mathcal O(1)) = \frac{c_1^T(\mathcal O(1))|_{C_0}}{c_1^T(T_{C_0}M(X,\beta))} + \frac{c_1^T(\mathcal O(1))|_{C_\infty}}{c_1^T(T_{C_\infty}M(X,\beta))}.\]
Or rather
\[\int_{\mathbb P^1} c_1(\mathcal O(1)) = \frac{c_1^T(\mathcal O(1))|_{0}}{c_1^T(T_{0}\mathbb P^1)} + \frac{c_1^T(\mathcal O(1))|_\infty}{c_1^T(T_{\infty}\mathbb P^1)}.\]
This is valued in the equivariant cohomology ring \(S^{-1}H^*_T(\mathbb P^1) = S^{-1}\Lambda_{T}[\zeta]/(\zeta+\chi_1)(\zeta+\chi_2)\). Here,
- \(\Lambda_{T} = \mathbb Z[\hbar]\) where \(\hbar = c_1(\mathcal O_{\mathbb P^{m-1}}(-1))\) (recall that \(H^*_T(\text{pt}) = H^*(\mathbb P^{m-1})\) up to degree \(m\)).
- The class \(\zeta\) is \(c_1^T(\mathcal O_{\mathbb P^1}(1))\)
- The character \(\chi_i\) is identified with \(c^T_1(\mathbb C_{\chi_i})\) under the identification \(M\to \Lambda^2_T = H^2_T(\text{pt})\). Here \(\mathbb C_{\chi}\) denotes the line bundle over \(\text{pt}\) given by the \(T\) representation \(t\cdot z = \chi(t)z\).
- The multiplicative set \(S\) contains \((\chi_\infty - \chi_0)^2\).
We now just compute:
The tangent space \(T_0\mathbb P^1\) is a line bundle over \(0 = [1:0]\) on which \(\mathbb C^*\) acts by character \(\chi_\infty - \chi_0\). This can easily be seen in coordinates, where \(T_0\mathbb P^1 = \{[1:z]\}\) and so
\[t\cdot [1:z] = [\chi_0(t) : \chi_\infty(t)z] = \left[1:\frac{\chi_\infty(t)}{\chi_0(t)}z\right].\]
The equivariant Chern class of a line bundle over a point is the ordinary Chern class of a line bundle over \(\mathbb BG\), so in this case \[c_1^T(T_0\mathbb P^1) = c_1((\mathbb C^m \setminus \{0\})\times^T \mathbb C_{\chi_\infty - \chi_0}) = \chi_\infty-\chi_0.\]
- Similarly, \[c_1^T(T_\infty\mathbb P^1) = \chi_0 - \chi_\infty.\]
- By definition \(c_1^T(\mathcal O(1)) = \zeta\). The restriction of the tautological line bundle \(\mathcal O(-1)\) to a point \(p\in \mathbb P^1\) gives the line \(L\) over that point, so \[c_1^T(\mathcal O(1))|_{\{0\}} = -c_1^T(\mathcal O(-1)|_{\{0\}}) = -c_1^T(L_0) = -\chi_0. \]
- Similarly, \[c_1^T(\mathcal O(1))|_{\{\infty\}} = -\chi_\infty\].
We now just add everything up:
\begin{align*} \int_{\mathbb P^1} c_1(\mathcal O(1)) &= \frac{c_1^T(\mathcal O(1))|_{0}}{c_1^T(T_{0}\mathbb P^1)} + \frac{c_1^T(\mathcal O(1))|_\infty}{c_1^T(T_{\infty}\mathbb P^1)} \\ &= \frac{-\chi_0}{\chi_\infty - \chi_0} + \frac{-\chi_\infty}{\chi_0 - \chi_\infty} \\ &= \frac{-\chi_0}{\chi_\infty - \chi_0} + \frac{\chi_\infty}{\chi_\infty - \chi_0} \\ &= \frac{\chi_\infty - \chi_0}{\chi_\infty - \chi_0}\\ &= 1 \end{align*}as desired.