The intersection-theoretic formulas for Siegel–Veech constants and Lyapunov exponents

Let $(\mu,\nu)$ describe a stratum ${\mathcal Q}_g(\mu,\nu)$ of quadratic differentials, with $\mu=(2m_i)_{i=1}^r$ and $\nu=(2n_j-1)_{j=1}^s$. Let $\mathbb{P}{\overline{\mathcal Q}}_g(\mu,\nu)$ be its compactified projectivization, let $\zeta$ be the first Chern class of the universal line bundle, let $\delta$ be the total boundary divisor class, let $\lambda_1$ be the first Hodge class, and let $\psi_i$ be associated with the even-order zeros.

The Siegel–Veech and Lyapunov intersection formulas. The area Siegel–Veech constant and the sum of Lyapunov exponents are

$$c_{{\rm area}}(\mu, \nu) = -\frac{1}{2\pi^2} \frac{ \int_{\mathbb{P}{\overline{\mathcal Q}}_{g}(\mu,\nu)}\zeta^{2g+s-4} \psi_{1} \cdots \psi_{r} \delta }{ \int_{\mathbb{P}{\overline{\mathcal Q}}_{g}(\mu,\nu)}\zeta^{2g+s-3} \psi_{1} \cdots \psi_{r}},$$ $$L^+(\mu, \nu) = - 2\frac{ \int_{\mathbb{P}{\overline{\mathcal Q}}_{g}(\mu,\nu)}\zeta^{2g+s-4} \psi_{1} \cdots \psi_{r} \lambda_1}{ \int_{\mathbb{P}{\overline{\mathcal Q}}_{g}(\mu,\nu)}\zeta^{2g+s-3} \psi_{1} \cdots \psi_{r}}.$$

These formulas generalize the principal-stratum identities and aim to express dynamical invariants of every quadratic-differential stratum through intersection theory; their general validity remains open in the source.