A categorical invariant for cubic threefolds

We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class.     

Stable bundle extensions on elliptic Calabi–Yau threefolds

We construct stable bundle extensions on elliptically fibered Calabi–Yau threefolds. We show that these bundles can solve the topological anomaly constraint in heterotic string theory without the need for invoking background five-branes.     

On the existence of stable bundles with prescribed Chern classes on Calabi–Yau threefolds

We prove a case of the conjecture of Douglas, Reinbacher and Yau about the existence of stable vector bundles with prescribed Chern classes on a Calabi–Yau threefold. For this purpose we prove the existence of certain stable vector bundle extensions over elliptically fibered Calabi–Yau threefolds.     

Spectral bundles and the DRY-Conjecture

Supersymmetric heterotic string models, built from a Calabi-Yau threefold X endowed with a stable vector bundle V, usually start from a phenomenologically motivated choice of a bundle V_v in the visible sector, the spectral cover construction on an elliptically fibered X being a prominent example. The ensuing anomaly mismatch between c₂(V_v) and c₂(X), or rather the corresponding differential forms, is often ‘solved’, on the cohomological level, by including a fivebrane. This leads to the question whether the difference can be alternatively realized by a further stable bundle. The ‘DRY’-conjecture of Douglas, Reinbacher and Yau in math.AG/0604597 gives a sufficient condition on cohomology classes on X to be realized as the Chern classes of a stable sheaf. In 1010.1644 [hep-th], we showed that infinitely many classes on X exist for which the conjecture is true. In this note, we give the sufficient condition for the mentioned fivebrane classes to be realized by a further stable bundle in the hidden sector. Using a result obtained in 1011.6246 [hep-th], we show that corresponding bundles exist, thereby confirming this version of the DRY-Conjecture.     

On the number of minimal models of a log smooth threefold

We give a topological bound on the number of minimal models of a class of three-dimensional log smooth pairs of log general type.     

Rigid Curves in Complete Intersection Calabi–Yau Threefolds

Working over the complex numbers, we study curves lying in a complete intersection K3 surface contained in a (nodal) complete intersection Calabi–Yau threefold. Under certain generality assumptions, we show that the linear system of curves in the surface is a connected componend of the the Hilbert scheme of the threefold. In the case of genus one, we deduce the existence of infinitesimally rigid embeddings of elliptic curves of arbitrary degree in the general complete intersection Calabi–Yau threefold.