The Frobenius morphism on a toric variety

Multiplication maps and vanishing theorems for Toric varieties

“…) (see [BTLM,Remark 1]). We do not pursue this topic since it was already treated in [BTLM] and [B].…”

“…It was first claimed in [D,7.5.2 Theorem] without proof. See [BTLM,Theorem 5]. The readers can find that this famous vanishing theorem is stated in the standard reference [O, p.130] without proof.…”

“…See, for example, [D, 7.5.2. Theorem], [BC,Section 7], [BTLM,Theorem 5], and [M,Section 2]. Theorem 1.1 (Main theorem I).…”

Frobenius splittings of toric varieties

“…Remark 1.6. Frobenius morphisms and their lifts to characteristic zero have been used powerfully in several other contexts related to the geometry of toric varieties, including by Buch, Lauritzen, Mehta and Thomsen [1997] to prove Bott vanishing and degeneration of the Hodge to de Rham spectral sequence, by Totaro [199?] to give a splitting of the weight filtration on Borel-Moore homology, by Smith [2000] to prove global F-regularity, by Brylinski and Zhang [2003] to prove degeneration of a spectral sequence computing equivariant cohomology with rational coefficients, and by Fujino [2007] to prove vanishing theorems for vector bundles and reflexive sheaves. Frobenius splittings have also played a role in unsuccessful attempts to show that section rings of ample line bundles on smooth toric varieties are normally presented [Bøgvad 1995].…”

Cartier Isomorphism for Toric Varieties

“…Using a description of the Zariski-de Rham complex due to Danilov [Dan78], we show that this newly defined Cartier operator is an isomorphism for toric varieties. Moreover, it is induced by a split injection 0 −→ a X −→ F * a X As described in [BT97] such a result yields the Bott vanishing theorem and the degeneration of the Hodge to de Rham spectral sequence for projective toric varieties.…”