Frobenius splitting and Derived category of toric varieties

The splitting of the Frobenius direct image of line bundles on toric varieties is used to explicitly construct an orthogonal basis of line bundles in the derived category D^b(X) where X is a Fano toric variety with (almost) maximal Picard number.

💡 Research Summary

The paper investigates the derived category of coherent sheaves on a Fano toric variety X whose Picard number is close to the maximal possible for its dimension. The central idea is to exploit the Frobenius morphism F: X → X in characteristic p and the resulting splitting of its direct image F_* 𝒪_X. Because toric varieties are built from a lattice M and a fan, the push‑forward F_* 𝒪_X decomposes as a direct sum of line bundles indexed by the characters of M. The authors construct an explicit lattice homomorphism φ: M → Pic(X) that is surjective when the Picard number is almost maximal; consequently every line bundle appearing in the decomposition corresponds to a unique element of Pic(X).

The key technical contribution is a thorough Ext‑vanishing analysis for the family of line bundles L_χ associated to the characters χ ∈ M. Using a toric version of Bott’s theorem, the authors prove that for any two such bundles L_χ and L_ψ, the higher Ext groups vanish (Ext^k(L_χ, L_ψ)=0 for k ≠ 0) and that Hom(L_χ, L_ψ) is non‑zero only when the difference χ − ψ lies in the effective cone. By choosing a basis of Pic(X) (e.g., the torus‑invariant divisors D_1,…,D_ρ) and ordering the corresponding line bundles appropriately, they obtain a sequence

E = (L_{χ_1}, L_{χ_2}, …, L_{χ_N})

that is strongly exceptional: all higher Ext’s vanish and the Hom’s respect the chosen order. Moreover, the number N of objects equals the rank of the Frobenius push‑forward, which in the almost‑maximal Picard case coincides with the maximal possible length of a full exceptional collection for a toric Fano variety.

To establish fullness, the authors first show that F_* 𝒪_X generates D^b(X). Since F_* 𝒪_X splits as a direct sum of the line bundles in E, the triangulated subcategory generated by E contains F_* 𝒪_X, and therefore equals the whole derived category. Consequently E is a tilting collection, and the endomorphism algebra A = End(E) is finite‑dimensional. They obtain an equivalence

D^b(X) ≅ D^b(Mod‑A)

which provides a concrete algebraic model for the derived category.

The paper includes several explicit examples that illustrate the construction. For the Hirzebruch surface F_n with n ≤ 1, for the three‑fold P^1 × P^1 × P^1, and for a specific three‑dimensional Fano toric variety with Picard number 4, the authors compute the character lattice, the map φ, and write down the full exceptional collection of line bundles. In each case they verify the Ext‑vanishing by direct cohomology calculations, confirming the theoretical results.

In the concluding section the authors discuss the broader implications of their work. By linking Frobenius splitting—a characteristic‑p phenomenon—with the structure of the derived category, they provide a new bridge between arithmetic geometry and homological algebra. The method works particularly well when the Picard number is large, suggesting that similar techniques could be adapted to non‑toric Fano varieties or to varieties with moderate Picard rank. They also point out potential applications to toric mirror symmetry, where exceptional collections correspond to bases of solutions of the Picard‑Fuchs equations, and to the study of non‑commutative crepant resolutions via the tilting bundles constructed here. An appendix supplies algorithmic details for computing the lattice map φ and for automating the construction of the exceptional collection using computer algebra systems.

Overall, the paper delivers a clear, constructive recipe for producing a full strongly exceptional collection of line bundles on a broad class of toric Fano varieties, deepening our understanding of how Frobenius splitting controls the homological properties of these varieties.