For the quintic $X_5 \subset \mathbb{P}^4$, the degree $d$ rational curve counts are:

Low degrees (computed via mirror symmetry, 1991):

• $n_1 = 2875$ (lines)
• $n_2 = 609250$ (conics)
• $n_3 = 317206375$ (cubics)
• $n_4 = 242467530000$ (quartics)

Method:

1. Mirror quintic has Picard-Fuchs operator $\mathcal{L} = \theta^4 - 5z(5\theta+1)(5\theta+2)(5\theta+3)(5\theta+4)$
2. Fundamental period: $\omega_0(z) = \sum_{n=0}^\infty \frac{(5n)!}{(n!)^5} z^n$
3. Mirror map: $q(z) = \exp(2\pi i \omega_1/\omega_0)$ where $\omega_1$ has logarithmic term
4. Invert to get $z(q)$ and expand $\omega_0(z(q))$ in $q$
5. Extract coefficients from $F(q) = \sum n_d \text{Li}_3(q^d)$

This calculation, originally done by Candelas-de la Ossa-Green-Parkes, was later rigorously verified by mathematicians.

The total space $X = K_{\mathbb{P}^2} = \mathcal{O}_{\mathbb{P}^2}(-3)$ counts curves in $\mathbb{P}^2$ with weights from the normal bundle. The generating function is: $F(q) = \sum_{d=1}^\infty N_d \text{Li}_3(q^d)$

where $N_d = \frac{1}{d^3}\sum_{k|d} \mu(d/k) k^2 n_k^{\mathbb{P}^2}$.

For $\mathbb{P}^2$, the counts $n_d^{\mathbb{P}^2} = (3d-1)!/(d!)^3$ are classical (Kontsevich's formula), allowing explicit computation of $N_d$.

For toric surfaces, mirror symmetry relates:

• A-side: Curve counts on toric surface $S_P$ for polytope $P$
• B-side: Oscillatory integrals on Landau-Ginzburg mirror

The generating function: $F(q) = \int_\gamma e^{W/z} \Omega$

where $W$ is the superpotential encoding toric data and $\gamma$ is a Lefschetz thimble.

A degree 8 hypersurface in weighted projective space $\mathbb{P}(1,1,2,2,2)$ is Calabi-Yau. Mirror symmetry gives:

• Picard-Fuchs operator determined by weights
• Periods computable via hypergeometric functions
• Curve counts extracted from period expansions

Low degree counts have been computed and verified against localiz ation calculations.

The bi-cubic defined by two degree $(3,3)$ polynomials has:

• Two Kähler parameters (one for each $\mathbb{P}^2$ factor)
• Mirror with two complex structure parameters
• Two-variable generating function: $F(q_1, q_2) = \sum_{d_1, d_2} n_{d_1,d_2} q_1^{d_1} q_2^{d_2}$

Computed via periods solving a GKZ hypergeometric system.

The genus 1 Gromov-Witten invariants $n_{1,d}$ are computed from: $F_1(q) = -\frac{1}{24}\log(\Delta(q))$

where $\Delta$ is a modular form related to periods. The first few values:

• $n_{1,1} = 0$
• $n_{1,2} = -5$
• $n_{1,3} = -1695$

Negative values reflect virtual contributions from obstructed curves.