The intersection-theoretic formulas for affine invariant submanifolds

Let ${\mathcal N}$ be an affine invariant submanifold in a stratum of Abelian differentials. Suppose its tangent space projects onto an $a$-dimensional subspace of absolute periods with an $r$-dimensional kernel of relative periods, so that $\dim_{\mathbb C}{\mathcal N}=a+r$. Let $\mathbb{P}\overline{\mathcal N}$ be the closure of its projectivization in the incidence-variety compactification, let $\xi$ be the first Chern class of the universal line bundle, let $\delta$ be the boundary divisor class, let $\lambda_1$ be the first Hodge class, and let $\psi_i$ be associated with the relative-period zero $z_i$.

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

$$c_{{\rm area}}({\mathcal N}) = -\frac{1}{4\pi^2} \frac{ \int_{\mathbb{P}\overline{\mathcal{N}}}\xi^{a-2} \psi_{1} \cdots \psi_{r} \delta }{ \int_{\mathbb{P}\overline{\mathcal{N}}}\xi^{a-1} \psi_{1} \cdots \psi_{r}},$$ $$L({\mathcal N}) = - \frac{ \int_{\mathbb{P}\overline{\mathcal{N}}}\xi^{a-2} \psi_{1} \cdots \psi_{r} \lambda_1}{ \int_{\mathbb{P}\overline{\mathcal{N}}}\xi^{a-1} \psi_{1} \cdots \psi_{r}}.$$

The conjecture seeks uniform intersection-theoretic formulas for dynamical invariants of arbitrary affine invariant submanifolds; beyond the cases established in the paper, the general statement remains open.