Universal generating function for all cohomologies on the bicubic hypersurface in P^2×P^2

Establish that for a general Calabi–Yau hypersurface X of bidegree (3,3) in P^2 × P^2, the function G(x,y) prescribed by the authors generates all line bundle cohomology series by suitable expansions and constants—recovering CS^0(X, O_X), CS^1(X, O_X), CS^2(X, O_X), and CS^3(X, O_X) as stated.

Background

Although the zeroth cohomology on the bicubic is fully captured by the Hilbert–Poincaré series of the homogeneous coordinate ring, the authors find that a more intricate function G(x,y) is required to simultaneously encode higher cohomologies via expansions at different points and the addition of constants. The conjecture posits that this G(x,y) yields all cohomology dimensions.

A proof would show that even when the nef and effective cones coincide and h0 is polynomially generated, higher cohomologies can still be recovered from a single rational generating function with appropriate expansion rules.

References

Conjecture 6. Let $X$ be a general hypersurface of bidegree $(3,3)$ in $P2\times P{2}$. Let $H_1 = \mathcal O_{P2\times P2}(1,0)|_X$ and $H_2 = \mathcal O_{P2\times P2}(0,1)|_X$. Defining $G(x,y) = %\frac{y3}{(1 - y)3} - \frac{3 (x{-1} y)3 (x (1 - x) + y (1 - y) - 3 x (1 - y))}{(1 - x{-1} y)3 (1 - y)3} \frac{(x{-1} y)3 ((1 + x - y)3 - 1 + 3 x (1 - y))}{(1 - x{-1} y)3 (1 - y)3}$, all line bundle cohomology dimensions on $X$ are encoded in the following generating functions, written in the basis ${H_1,H_2}$: ...

— Generating Functions for Line Bundle Cohomology Dimensions on Complex Projective Varieties (2401.14463 - Constantin, 25 Jan 2024) in Conjecture 6, Section 4 (General hypersurfaces of bidegree (3,3) in P^2×P^2); also previewed as Conjecture in Introduction and Overview