Smoothing Pairs Over Degenerate Calabi–Yau Varieties

We apply the techniques developed in [2] to study smoothings of a pair $(X,\mathfrak{C}^*)$, where $\mathfrak{C}^*$ is a bounded perfect complex of locally free sheaves over a degenerate Calabi–Yau variety $X$. In particular, if $X$ is a projective Calabi–Yau variety admitting the structure of a toroidal crossing space and with the higher tangent sheaf $\mathcal{T}^1_X$ globally generated, and $\mathfrak{F}$ is a locally free sheaf over $X$, then we prove, using the results in [ 8], that the pair $(X,\mathfrak{F})$ is formally smoothable when $\textrm{Ext}^2(\mathfrak{F},\mathfrak{F})_0 = 0$ and $H^2(X,\mathcal{O}_X) = 0$.

THE LIPMAN–ZARISKI CONJECTURE IN GENUS ONE HIGHER

We prove the Lipman–Zariski conjecture for complex surface singularities with $p_{g}-g-b\leqslant 2$ . Here $p_{g}$ is the geometric genus, $g$ is the sum of the genera of exceptional curves and $b$ is the first Betti number of the dual graph. This improves on a previous result of the second author. As an application, we show that a compact complex surface with a locally free tangent sheaf is smooth as soon as it admits two generically linearly independent twisted vector fields and its canonical sheaf has at most two global sections.

Completed tensor products and a global approach to p-adic analytic differential operators

Ardakov-Wadsley defined the sheaf $\wideparen{\Ncal{D}}$X of p-adic analytic differential operators on a smooth rigid analytic variety by restricting to the case where X is affinoid and the tangent sheaf admits a smooth Lie lattice. We generalise their results by dropping the assumption of a smooth Lie lattice throughout, which allows us to describe the sections of $\wideparen{\Ncal{D}}$ for arbitrary affinoid subdomains and not just on a suitable base of the topology. The structural results concerning $\wideparen{\Ncal{D}}$ and coadmissible $\wideparen{\Ncal{D}}$-modules can then be generalised in a natural way.The main ingredient for our proofs is a study of completed tensor products over normed K-algebras, for K a discretely valued field of mixed characteristic. Given a normed right module U over a normed K-algebra A, we provide several exactness criteria for the functor $U\widehat{\otimes}_A$ - applied to complexes of strict morphisms, including a necessary and sufficient condition in the case of short exact sequences.

Hypersurfaces invariant by Pfaff systems

We present results expressing conditions for the existence of meromorphic first integrals for Pfaff systems of arbitrary codimension on complex manifolds. Some of the results presented improve previous ones due to Jouanolou and Ghys. We also present an enumerative result counting the number of hypersurfaces invariant by a projective holomorphic foliation with split tangent sheaf.

Flags of holomorphic foliations

A flag of holomorphic foliations on a complex manifold M is an object consisting of a finite number of singular holomorphic foliations on M of growing dimensions such that the tangent sheaf of a fixed foliation is a subsheaf of the tangent sheaf of any of the foliations of higher dimension. We study some basic properties oft hese objects and, in <img src="/img/revistas/aabc/2011nahead/aop2411pcn.jpg" align="absmiddle" />, n > 3, we establish some necessary conditions for a foliation, we find bounds of lower dimension to leave invariant foliations of codimension one. Finally, still in <img src="/img/revistas/aabc/2011nahead/aop2411pcn.jpg" align="absmiddle" /> involving the degrees of polar classes of foliations in a flag.

ON THE SINGULAR SCHEME OF CODIMENSION ONE HOLOMORPHIC FOLIATIONS IN ℙ3

In this work, we begin by showing that a holomorphic foliation with singularities is reduced if and only if its normal sheaf is torsion-free. In addition, when the codimension of the singular locus is at least two, it is shown that being reduced is equivalent to the reflexivity of the tangent sheaf. Our main results state on one hand, that the tangent sheaf of a codimension one foliation in ℙ3 is locally free if and only if the singular scheme is a curve, and that it splits if and only if it is arithmetically Cohen–Macaulay. On the other hand, we discuss when a split foliation in ℙ3 is determined by its singular scheme.