Benchmarks

杜承勇: Gromov-Witten 不变量沿曲线与曲面的加权吹胀公式

2.1 $(\backslash underline\{X\}\_a,\; Z\_a)$ 的相对轨形 GW- 不变量

类似于绝对轨形 GW- 不变量, $(\backslash underline\{X\}\_a,\; Z\_a)$ 的相对轨形 GW- 理论中的赋值映射的像落在 $\backslash underline\{X\}\_a$ 与 $Z\_a$ 的惯性轨形中. 我们前面已经分析过它们的惯性轨形, 即:

$$(\backslash mathbf\{I\}\backslash underline\{X\}\_a,\; \backslash mathbf\{I\}Z\_a)\; =\; (\backslash underline\{X\}\_a\; \backslash sqcup\; \backslash bigsqcup\_\{s\; \backslash in\; D\_a,\; s\; >\; 0\}\; Z\_a(s),\; \backslash bigsqcup\_\{s\; \backslash in\; D\_a\}\; Z\_a(s)).$$

相对模空间 $\backslash overline\{\backslash mathcal\{M\}\}\_\{g,n,\backslash vec\{\backslash mu\},A\}(\backslash underline\{X\}\_a,\; Z\_a)$ 参数化了到 $(\backslash underline\{X\}\_a,\; Z\_a)$ 的如下的相对的轨形稳定映射:

(1) 映射的原像曲线的亏格为 $g$ ;

(2) 此映射代表的下同调类为 $A\; \backslash in\; H\_2(|\backslash underline\{X\}\_a|;\; \backslash mathbb\{Z\})$ , 且 $\backslash vec\{\backslash mu\}\; =\; (\backslash mu\_1,\; \backslash dots,\; \backslash mu\_\{\backslash ell(\backslash vec\{\backslash mu\})\})$ 为相交数 $A\; \backslash cdot\; [Z\_a]$ 的一个由正有理数给出的划分, 表示映射与除子 $Z\_a$ 相交于后 $\backslash ell(\backslash vec\{\backslash mu\})$ 个标识点, 并以 $\backslash vec\{\backslash mu\}$ 为相应的分数阶相交重数 (fractional contact order). 令 $|\backslash vec\{\backslash mu\}|\; =\; \backslash sum\_\{i=1\}^\{\backslash ell(\backslash vec\{\backslash mu\})\}\; \backslash mu\_i$ , 则 $|\backslash vec\{\backslash mu\}|\; =\; A\; \backslash cdot\; [Z\_a]$ .

前 $n$ 个标识点是非轨形标识点, 称为绝对标识点, 它们决定了如下的赋值映射

$$\backslash mathrm\{ev\}\_i\; :\; \backslash overline\{\backslash mathcal\{M\}\}\_\{g,n,\backslash vec\{\backslash mu\},A\}(\backslash underline\{X\}\_a,\; Z\_a)\; \backslash to\; \backslash mathbf\{I\}\backslash underline\{X\}\_a,\; \backslash quad\; 1\; \backslash le\; i\; \backslash le\; n.$$

同时, 它们还决定了 $n$ 个余切线丛 $L\_i$ . 这些余切线丛的第一陈类也记为 $\backslash bar\{\backslash psi\}\_i$ . 后 $\backslash ell(\backslash vec\{\backslash mu\})$ 个标识点为相对标识点, 它们决定了 $\backslash ell(\backslash vec\{\backslash mu\})$ 个赋值映射

$$\backslash mathrm\{rev\}\_i\; :\; \backslash overline\{\backslash mathcal\{M\}\}\_\{g,n,\backslash vec\{\backslash mu\},A\}(\backslash underline\{X\}\_a,\; Z\_a)\; \backslash to\; Z\_a(\backslash \{-\backslash mu\_i\backslash \})\; \backslash subseteq\; \backslash mathbf\{I\}Z\_a,\; \backslash quad\; 1\; \backslash le\; i\; \backslash le\; \backslash ell(\backslash vec\{\backslash mu\}).$$

定义2.1 取 $\backslash underline\{\backslash alpha\}\; =\; (\backslash bar\{\backslash psi\}^\{a\_1\}\; \backslash alpha\_1,\; \backslash dots,\; \backslash bar\{\backslash psi\}^\{a\_n\}\; \backslash alpha\_n)\; \backslash in\; (\backslash mathbb\{C\}[\backslash bar\{\backslash psi\}]\; \backslash otimes\; H^*(\backslash underline\{X\}\_a))^\{\backslash otimes\; n\}$ 与 $\backslash underline\{\backslash epsilon\}\; =\; (\backslash epsilon\_1,\; \backslash dots,\; \backslash epsilon\_\{\backslash ell(\backslash vec\{\backslash mu\})\})\; \backslash in\; H^*(\backslash mathbf\{I\}Z\_a)^\{\backslash otimes\; \backslash ell(\backslash vec\{\backslash mu\})\}$ , 其中 $\backslash epsilon\_j\; \backslash in\; H^*(Z\_a(\backslash \{-\backslash mu\_j\backslash \}))$ . $(\backslash underline\{X\}\_a,\; Z\_a)$ 的相对轨形 GW- 不变量定义为

$$\backslash langle\; \backslash underline\{\backslash alpha\}\; |\; \backslash underline\{\backslash epsilon\}\; \backslash rangle\_\{g,n,\backslash vec\{\backslash mu\},A\}^\{(\backslash underline\{X\}\_a,\; Z\_a)\}\; :=\; \backslash int\_\{[\backslash overline\{\backslash mathcal\{M\}\}\_\{g,n,\backslash vec\{\backslash mu\},A\}(\backslash underline\{X\}\_a,\; Z\_a)]^\{\backslash mathrm\{vir\}\}\}\; \backslash prod\_\{i=1\}^n\; \backslash bar\{\backslash psi\}\_i^\{a\_i\}\; \backslash mathrm\{ev\}\_i^*\; \backslash alpha\_i\; \backslash wedge\; \backslash prod\_\{j=1\}^\{\backslash ell(\backslash vec\{\backslash mu\})\}\; \backslash mathrm\{rev\}\_j^*\; \backslash epsilon\_j.\; \backslash quad\; (2.1)$$

模空间 $\backslash overline\{\backslash mathcal\{M\}\}\_\{g,n,\backslash vec\{\backslash mu\},A\}(\backslash underline\{X\}\_a,\; Z\_a)$ 的期望维数 (参见 [20, Lemma 3.1]) 为

$$\backslash mathrm\{vdim\}\_\{\backslash mathbb\{C\}\}\; \backslash overline\{\backslash mathcal\{M\}\}\_\{g,n,\backslash vec\{\backslash mu\},A\}(\backslash underline\{X\}\_a,\; Z\_a)\; =\; c\_1^\{\backslash underline\{X\}\_a\}(A)\; +\; (\backslash mathrm\{dim\}\_\{\backslash mathbb\{C\}\}\; X\; -\; 3)(1\; -\; g)\; +\; n\; +\; \backslash ell(\backslash vec\{\backslash mu\})\; -\; \backslash iota^\{Z\_a\}(\backslash underline\{\backslash epsilon\})\; -\; |\backslash vec\{\backslash mu\}|,$$

类似于绝对 GW- 不变量, 当 $\backslash frac\{1\}\{2\}(\backslash mathrm\{deg\}\; \backslash underline\{\backslash alpha\}\; +\; \backslash mathrm\{deg\}\; \backslash underline\{\backslash epsilon\})\; \backslash neq\; \backslash mathrm\{vdim\}\_\{\backslash mathbb\{C\}\}\; \backslash overline\{\backslash mathcal\{M\}\}\_\{g,n,\backslash vec\{\backslash mu\},A\}(\backslash underline\{X\}\_a,\; Z\_a)$ 时, 不变量定义为 0. 有关相对轨形 GW- 理论的构造与退化公式可以参见 [1, 10].

为了后面使用方便, 我们引入两个记号:

$$\backslash iota^\{Z\_a\}(\backslash epsilon\_j)\; :=\; \backslash iota^\{Z\_a\}(\backslash \{-\backslash mu\_j\backslash \})\; =\; \backslash sum\_\{u=1\}^\{\backslash kappa\}\; \backslash \{-a\_u\; \backslash mu\_j\backslash \},\; \backslash quad\; \backslash text\{和\}\; \backslash quad\; \backslash iota^\{Z\_a\}(\backslash underline\{\backslash epsilon\})\; :=\; \backslash sum\_\{j=1\}^\{\backslash ell(\backslash vec\{\backslash mu\})\}\; \backslash iota^\{Z\_a\}(\backslash epsilon\_j).\; \backslash quad\; (2.2)$$

2.2 $\backslash kappa\; =\; 1$ 时 $(X,\; S)$ 的相对 GW- 不变量

当 $\backslash kappa\; =\; 1$ 时, 我们也需要用到 $(X,\; S)$ 的相对不变量. 令 $\backslash vec\{\backslash mu\}\; =\; (\backslash mu\_1,\; \backslash dots,\; \backslash mu\_\{\backslash ell(\backslash vec\{\backslash mu\})\})$ 为相交数 $A\; \backslash cdot\; [S]$ 的由正整数给出的划分. 如此有 $|\backslash vec\{\backslash mu\}|\; :=\; \backslash sum\_i\; \backslash mu\_i\; =\; A\; \backslash cdot\; [S]$ .

模空间 $\backslash overline\{\backslash mathcal\{M\}\}\_\{g,n,\backslash vec\{\backslash mu\},A\}(X,\; S)$ 参数化了到 $(X,\; S)$ 的如下的相对稳定映射:

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