Let $X$ be a smooth projective variety and $\overline{M} = \overline{M}_{g,k}(X,\beta)$ be the moduli stack of stable maps.

Step 1: Obstruction Theory

The deformation-obstruction theory of stable maps gives a two-term complex $E^\bullet = [E^{-1} \to E^0]$ in the derived category of $\overline{M}$: $E^{-1} = R^1\pi_*f^*TX, \quad E^0 = R^0\pi_*f^*TX$

where $\pi: \mathcal{C} \to \overline{M}$ is the universal curve and $f: \mathcal{C} \to X$ is the universal map.

The obstruction space is $H^1(E^\bullet) = \text{Ext}^1(\mathbb{L}_{\overline{M}}, \mathcal{O}_{\overline{M}})$ where $\mathbb{L}_{\overline{M}}$ is the cotangent complex.

Step 2: Perfect Obstruction Theory

The complex $E^\bullet$ is a perfect obstruction theory, meaning:

1. $E^\bullet$ is perfect of amplitude $[-1,0]$
2. The morphism $E^\bullet \to \mathbb{L}_{\overline{M}}$ is a quasi-isomorphism on $h^0$ and surjection on $h^{-1}$

This data determines a cone $C \subset E^0$ called the intrinsic normal cone.

Step 3: Virtual Dimension

The virtual dimension is: $\text{vdim} = \text{rank}(E^0) - \text{rank}(E^{-1}) = \chi(f^*TX)$

By Riemann-Roch: $\chi(f^*TX) = \deg(f^*TX) + (1-g)\text{rank}(TX) = \int_\beta c_1(TX) + (1-g)\dim X$

Adding $k$ marked points gives the formula stated earlier.

Step 4: Gysin Pullback

The intrinsic normal cone $C$ sits in a vector bundle stack $E^0$. The virtual class is defined as the Gysin pullback: $[\overline{M}]^{\text{vir}} = 0_E^! [C] \in A_{\text{vdim}}(\overline{M})$

where $0_E: \overline{M} \to E^0$ is the zero section and $0_E^!$ is the refined Gysin homomorphism.

Step 5: Localized Euler Class

When $E^\bullet$ is represented by a two-term complex of vector bundles, the virtual class can be expressed using a localized Euler class: $[\overline{M}]^{\text{vir}} = e(E^0)/e(E^{-1}) \cap [\overline{M}]$

where the division is interpreted in the Chow ring via localization.

Step 6: Functoriality

For a morphism $f: \overline{M} \to \overline{M}'$ compatible with obstruction theories $E^\bullet \to (f^*E')^\bullet$, the virtual classes satisfy: $f_*[\overline{M}]^{\text{vir}} = [\overline{M}']^{\text{vir}}$

This functoriality ensures that virtual classes respect the geometry of families and gluing.

Step 7: Degeneration Formula

Under normal crossings degenerations $X \to \Delta$, the virtual class of the general fiber equals the sum of contributions from components of special fibers: $[\overline{M}(X_t)]^{\text{vir}} = \sum_{\Gamma} \frac{1}{|\text{Aut}(\Gamma)|} [\overline{M}_\Gamma(X_0)]^{\text{vir}}$

where $\Gamma$ runs over decorated dual graphs describing how curves can degenerate.