Euler-characteristic generating function for complete intersections in products of projective spaces

Establish that for a complete intersection X ↪ P^{n_1} × ⋯ × P^{n_p} cut out by q multi-homogeneous equations with multidegrees d^{(i)} = (d_{1,i}, …, d_{p,i}), the alternating sum over all 2^p choices of expanding the multivariate Hilbert–Poincaré series HS(X, t_1, …, t_p) at t_i = 0 or t_i = ∞ equals the generating function of Euler characteristics for all integer tensor powers of the restricted hyperplane line bundles H_i = O_{P^{n_1}}(0) ⊗ ⋯ ⊗ O_{P^{n_i}}(1) ⊗ ⋯ ⊗ O_{P^{n_p}}(0) restricted to X; that is, prove that ∑_{σ∈{0,∞}^p} (-1)^{sign(σ)} HS(X; t_1|_{σ(1)}, …, t_p|_{σ(p)}) = ∑_{(m_1,…,m_p)∈Z^p} χ(X, H_1^{⊗ m_1} ⊗ ⋯ ⊗ H_p^{⊗ m_p}) t_1^{m_1} ⋯ t_p^{m_p}, with the order and points of expansion as specified.

Background

The paper proposes a universal generating-function framework for encoding line bundle cohomology data. For complete intersections in products of projective spaces, the multivariate Hilbert–Poincaré series HS(X, t_1, …, t_p) captures graded dimensions of polynomial sections. The conjecture asserts that by summing expansions of HS around combinations of zero and infinity—with alternating signs—one obtains the full Euler-characteristic generating function for all line bundles descending from the ambient.

This statement, if established, would extend the well-known eventual agreement between Hilbert polynomials and Euler characteristics to all integral multi-degrees, not only sufficiently positive ones, and would provide a uniform tool for varieties of Fano, Calabi–Yau, and general type in arbitrary dimension.