Kodaira Type Vanishing Theorem for the Hirokado Variety

Abstract

The Hirokado variety is a Calabi-Yau threefold in characteristic 3 that is not liftable either to characteristic 0 or the ring W 2 of the second Witt vectors. Although Deligne-Illusie-Raynaud type Kodaira vanishing cannot be applied, we show that H 1 (X, L −1 ) = 0, for an ample line bundle such that L 3 has a non-trivial global section, holds for this variety. MSC Code: 14F17, 14J32, 14G17, 14M15.

On Kodaira type vanishing for Calabi–Yau threefolds in positive characteristic

“…In this paper, we give a lower bound of h 1 (X, L −1 ) = dim k H 1 (X, L −1 ) if L is an ample divisor with H 1 (X, L −1 ) = 0. Moreover, we show that a Kodaira type vanishing holds if X is a Schröer variety [21] or a Schoen variety [20], which extends the similar result given in [25] for the Hirokado variety [9]. We show that such kind of vanishing holds for Calabi-Yau manifold whose Picard variety has no p-torsion.…”

“…Theorem 13 (cf. Theorem 9 [25]). Let k be an algebraically closed field of characteristic p > 0 and X a Calabi-Yau threefold over k. If H 0 (X, Ω 1 X ) = 0, then weak H 1 -Kodaira vanishing holds for X.…”

“…Namely, for an ample divisor D we have H 1 (X, O X (−D)) = 0 if H 0 (X, O X (D)) = 0. It turns out that by the result given in [25] this condition can be weaken to H 0 (X, O X (pD)) = 0 when X has no global 1-forms. One of such varieties is the Hirokado variety [9] as observed in [25].…”

“…It turns out that by the result given in [25] this condition can be weaken to H 0 (X, O X (pD)) = 0 when X has no global 1-forms. One of such varieties is the Hirokado variety [9] as observed in [25]. In this paper, we show that a larger class of the varieties satisfies this condition such as Schröer varieties [21] and Schoen varieties [20] (see Corollary 15).…”

“…In [25], we applied Theorem 13 to the Hirokado variety [9]. Now we consider more examples which we can apply this theorem.…”