Calabi–Yau threefolds in $${\mathbb {P}}^6$$ P 6

Abstract

We study the geometry of 3-codimensional smooth subvarieties of the complex projective space. In particular, we classify all quasi-Buchsbaum Calabi-Yau threefolds in projective 6-space. Moreover, we prove that this classification includes all Calabi-Yau threefolds contained in a possibly singular 5-dimensional quadric as well as all Calabi-Yau threefolds of degree at most 14 in P 6 . Keywords Calabi-Yau threefolds · Pfaffian varieties · Canonical surfaces Mathematics Subject Classification Primary: 14J32 IntroductionIt is conjectured that, when 2n ≥ N , there is a finite number of smooth families of smooth n-dimensional subvarieties of P N that are not of general type. This conjecture was inspired by [15] where the statement was formulated and proven in the case of surfaces in P 4 . In [9], the conjecture was proven in the case of threefolds in P 5 . Moreover, Schneider in [33] proved that the statement is true when 2n ≥ N + 2. In this context, it is a natural problem to classify families of smooth n-folds of small degree in P N for chosen n, N ∈ N satisfying 2n ≥ N . In the case of codimension 2 subvarieties, this problem was addressed by many authors (see [3,6,11,17]).The next step is to study codimension 3 subvarieties in P 6 . In this case, standard tools such as the Barth-Lefschetz theorem do not apply. However, some general structure theorems were recently developed. We say that a submanifolds X ⊂ P n is subcanonical when ω X = O X (k) for some k ∈ Z. A codimension 3 submanifolds X is called Pfaffian if it is the first nonzero degeneracy locus of a skew-symmetric morphism of vector bundles of odd rank E * (−t) ϕ − → E where t ∈ Z. In this case, X is given locally by the vanishing of 2u × 2u Pfaffians of an alternating map ϕ from the vector bundle E of odd rank 2u + 1 to its twisted dual. More precisely, if X is Pfaffian, then we have:where s = c 1 (E) + ut. Moreover, from [29, Sect.3], we have in this caseSince the choice of an alternating map ϕ is equivalent to the choice of a section σ ∈ H 0 ( 2 E(t)), we will use the notation Pf(σ ) for the variety described by the Pfaffians of the map corresponding to σ . Answering a question of Okonek (see [29]), Walter, in [37], proved that if n is not divisible by 4 then a locally Gorenstein codimension 3 submanifold of P n+3 is Pfaffian if and only if it is subcanonical. In the case when n = 4k, the last statement is not true; however, there is another structure theorem (see [16]).The nongeneral type subcanonical threefolds in P 6 are either well understood Fano threefolds or threefolds with trivial canonical class. A very natural class of varieties among varieties with trivial canonical class are Calabi-Yau threefolds, i.e., smooth threefolds X with K X = 0 and H 1 (X, O X ) = 0. In the paper, we shall sometimes also consider singular Calabi-Yau threefolds by which we mean complex projective threefolds with Gorenstein singularities, ω X = 0 and with h 1 (O X ) = 0.For Calabi-Yau threefolds, the theory of Pfaffians is more specific. For instance, Schreyer,...

Tonoli's Calabi–Yau threefolds revisited

“…Observe first that from (2.3) we infer that Syz 1 (M ) has no section and c 1 (Syz 1 (M )) = −q. By analogous reasoning to [KKb,Lemma 3.5] and by Formula (1.1) for the canonical class we conclude that the number of negative a i 's, if nonzero, must be smaller than 3 − (p − q). The latter is negative by assumption so all a i ≥ 0.…”

“…Vector bundle defining projected del Pezzo surfaces in P 5 3 O P 5 (−1) ⊕ 2O Observe that a Tonoli Calabi-Yau threefold of degree 12 is just a complete intersection of type (2, 2, 3) and is naturally described by the Pfaffians of O P 6 ⊕ 2O P 6 (1). We have changed this vector bundle to an equivalent one (see the proof of [KKb,Lem. 3.4]) in order to make the analogy more transparent.…”

Arithmetically Gorenstein Calabi--Yau threefolds in $\mathbb{P}^7$

“…If a subcanonical codimension 3 manifold is additionally aG, then E can be chosen to be a sum of line bundles. Note that Tonoli in his PhD thesis [41] constructed examples of Calabi-Yau threefolds in P 6 using the Pfaffian resolution; see also [26].…”

Categorical generic fiber

“…There is another Fourier-Mukai partner called IMOU varieties [IMOU,Kuz18], consisting of derived-equivalent Calabi-Yau threefolds Y 0 , Y 0 which are deformation equivalent to X 0 , X 0 respectively [KK16,IIM19]. Extending L if necessary, we may assume that also Y 0 , Y 0 are defined over L. Then either of them fails to be isomorphic to X Q, X Q as an abstract scheme, otherwise we would have X L Y L and X L Y L .…”