Geometric transitions between Calabi–Yau threefolds related to Kustin–Miller unprojections

Abstract

We study Kustin-Miller unprojections between Calabi-Yau threefolds, or more precisely the geometric transitions they induce. We use them to connect many families of Calabi-Yau threefolds with Picard number one to the web of Calabi-Yau complete intersections. This result enables us to find explicit description of a few known families of Calabi-Yau threefolds in terms of equations. Moreover, we find two new examples of Calabi-Yau threefolds with Picard group of rank one, which are described by Pfaffian equations in weighted projective spaces.shall call, to avoid confusion, a geometric bitransition. This class of bitransitions, thanks to its naturality, may be much better understood, and hence gives us a new variety of tools that may be used for explicit descriptions of examples of Calabi-Yau threefolds. The theory of unprojections has already proved its efficiency in classical algebraic geometry. Thanks to works of Reid and Papadakis [12,13,14,15,16], unprojections became a powerful tool to describe and construct new varieties. They are used in descriptions of singular K3 surfaces and Fano threefolds. In this paper we would like to show how these tools work in the context of Calabi-Yau threefolds. Differently from the approach to K3 surfaces or Fano 3 folds, we will not be interested in the singular varieties arising by unprojection, but in the families of smooth Calabi-Yau threefolds that might degenerate to such. The construction shall give rise to geometric bitransitions between these families. Throughout the paper we study examples of such given geometric bitransitions. In this way we find connections (consisting of an even number of geometric transitions) between many known families of Calabi-Yau threefolds with Picard number one. In particular we connect to the web the 5 families of Calabi-Yau threefolds introduced by Borcea [17] and described as complete intersections in homogeneous spaces.The main advantage of our constructions is that at least in low codimension they are well understood in terms of equations. Thanks to that, for some known examples of Calabi-Yau threefolds we find descriptions in terms of equations in weighted projective space. This fact enables us to understand much better the geometry of those examples of Calabi-Yau varieties, which were known only as smoothings of some degenerate varieties.Moreover, in two of the cases the construction leads us to new Calabi-Yau threefolds. They are described by the vanishing of Pfaffians of some 5 × 5 matrix with polynomial entries in some weighted projective spaces. These two families, together with a few others explicitly described in this paper, have conjectured Picard-Fuchs equations of their mirror (see [18]). We hope one might use the results of this paper to find these mirrors. There are at least two possibilities to proceed. The first is to use directly the constructed geometric transitions to find candidates for the mirrors. This idea is based on the conjecture by Morrisson (see [19]) which states that any geometric transition should admit a m...

Polarized Calabi-Yau 3-folds in codimension 4

“…This complements existing approaches in the literature. In a series of papers Kapustka and Kapustka , , , , unproject anticanonically polarised del Pezzo surfaces inside Calabi–Yau 3‐folds; the resulting contraction is to a Gorenstein singularity, which they then prove can be smoothed. Neves and Papadakis apply ‘parallel’ unprojection, contracting large configurations of subvarieties to build high‐dimensional varieties, inside which they describe Calabi–Yau 3‐folds (also with orbifold points) as complete intersections.…”

A counterexample to the birational Torelli problem for Calabi-Yau threefolds

“…The threefolds studied in this paper have received some attention in recent years, partly from the perspective of mirror symmetry, see [18,[20][21][22]. In particular, the question of whether the X g have non-trivial Fourier-Mukai partners was raised in [14,Remark,p.…”

Projections of Mukai varieties

“…LG (3,6). For a chosen vector space V 2n of dimension 2n and a generic symplectic form ω ∈ 2 V 2n the variety LG ω (n, V 2n ) is the subvariety of the Grassmannian G(n, V 2n ) parametrizing n-spaces isotropic with respect to the form ω.…”

“…Proof. Fix a point p on the Lagrangian Grassmannian LG (3,6). The stabilizer of p contains GL(3) as a Levi subgroup.…”

“…In the example corresponding to the general case we project from h = 1 and the projection is given by the Pfaffians of M G2 not involving h and the quadric a(a − n − d) + gk − e(i + e + b) = 0. We can rearrange the equations to be defined by the following data a − d, For the other direction we observed that all maximal dimensional quadrics in LG (3,6) are equivalent by the action of the symplectic group. We can also observe that two general codimension 2 sections containing a fixed quadric are linearly isomorphic.…”