Formality of Donaldson submanifolds

We introduce the concept of s-formal minimal model as an extension of formality. We prove that any orientable compact manifold M, of dimension 2n or (2n-1), is formal if and only if M is (n-1)-formal. The formality and the hard Lefschetz property are studied for the symplectic manifolds constructed by Donaldson with asymptotically holomorphic techniques. This study permits us to show an example of a Donaldson symplectic manifold of dimension eight which is formal simply connected and does not satisfy the hard Lefschetz theorem.Comment: 24 pages, no figures, Latex2e; v3. statement of Lemma 2.7 correcte

An 8-dimensional non-formal simply connected symplectic manifold

A non-formal simply connected compact symplectic manifold of dimension 8 is constructed.Comment: 8 pages, 1 figure; v2. exposition greatly improved; v3. final version. To appear in Annals of Mathematic

Formality and the Lefschetz property in symplectic and cosymplectic geometry

We review topological properties of K\"ahler and symplectic manifolds, and of their odd-dimensional counterparts, coK\"ahler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the K\"ahler/symplectic situation) and the $b_1=1$ case (in the coK\"ahler/cosymplectic situation).Comment: 27 pages, no figures. Comments are welcome

Non-formal co-symplectic manifolds

We study the formality of the mapping torus of an orientation-preserving diffeomorphism of a manifold. In particular, we give conditions under which a mapping torus has a non-zero Massey product. As an application we prove that there are non-formal compact co-symplectic manifolds of dimension $m$ and with first Betti number $b$ if and only if $m=3$ and $b \geq 2$, or $m \geq 5$ and $b \geq 1$. Explicit examples for each one of these cases are given.Comment: Only minor changes with respect to version 1 (some terminology clarified). 21 pages, no figures. To appear in Trans. Am. Math. So

On formality of Sasakian manifolds

We investigate some topological properties, in particular formality, of compact Sasakian manifolds. Answering some questions raised by Boyer and Galicki, we prove that all higher (than three) Massey products on any compact Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using this we produce a method of constructing simply connected K-contact non-Sasakian manifolds. On the other hand, for every $n \geq 3$, we exhibit the first examples of simply connected compact Sasakian manifolds of dimension $2n + 1$ which are non-formal. They are non-formal because they have a non-zero triple Massey product. We also prove that arithmetic lattices in some simple Lie groups cannot be the fundamental group of a compact Sasakian manifold.Comment: 22 pages, no figures; v2. some corrections; v3. Accepted in J. Topolog

Resistance to the SDHI fungicides boscalid and fluopyram in Podosphaera xanthii from commercial cucurbit fields in Spain

Powdery mildew elicited by Podosphaera xanthii is a devastating disease of cucurbits worldwide and one of the most important diseases affecting these crops in Spain. Application of fungicides is the main control practice for managing P. xanthii; however, isolates resistant to multiple classes of site-specific fungicides have been recently reported in the Spanish cucurbit powdery mildew population. Succinate dehydrogenase inhibitors (SDHIs) constitute a relatively novel class of fungicides registered for powdery mildew control representing new alternatives for cucurbit growers. In the present study, 30 P. xanthii isolates were used to determine the effective concentration that reduces mycelial growth by 50% (EC50) to boscalid and fluopyram. The present study was also conducted to obtain discriminatory doses to monitor SDHI fungicide resistance in 180 P. xanthii isolates collected from several commercial cucurbit fields in Spain during 2017-2018. Three SDHI resistance patterns were observed in our population, which include patterns I (resistance to boscalid), II (resistance to fluopyram), and III (resistance to boscalid and fluopyram). The amino acid changes associated with these resistance patterns in the Sdh protein were also examined. Based on our results, SDHI fungicides are good alternatives for cucurbit powdery mildew control, although they should be applied with caution.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tec

Nearly hypo structures and compact Nearly K\"ahler 6-manifolds with conical singularities

We prove that any totally geodesic hypersurface $N^5$ of a 6-dimensional nearly K\"ahler manifold $M^6$ is a Sasaki-Einstein manifold, and so it has a hypo structure in the sense of \cite{ConS}. We show that any Sasaki-Einstein 5-manifold defines a nearly K\"ahler structure on the sin-cone $N^5\times\mathbb R$, and a compact nearly K\"ahler structure with conical singularities on $N^5\times [0,\pi]$ when $N^5$ is compact thus providing a link between Calabi-Yau structure on the cone $N^5\times [0,\pi]$ and the nearly K\"ahler structure on the sin-cone $N^5\times [0,\pi]$. We define the notion of {\it nearly hypo} structure that leads to a general construction of nearly K\"ahler structure on $N^5\times\mathbb R$. We determine {\it double hypo} structure as the intersection of hypo and nearly hypo structures and classify double hypo structures on 5-dimensional Lie algebras with non-zero first Betti number. An extension of the concept of nearly K\"ahler structure is introduced, which we refer to as {\it nearly half flat} SU(3)-structure, that leads us to generalize the construction of nearly parallel $G_2$-structures on $M^6\times\mathbb R$ given in \cite{BM}. For $N^5=S^5\subset S^6$ and for $N^5=S^2 \times S^3\subset S^3 \times S^3$, we describe explicitly a Sasaki-Einstein hypo structure as well as the corresponding nearly K\"ahler structures on $N^5\times\mathbb R$ and $N^5\times [0,\pi]$, and the nearly parallel $G_2$-structures on $N^5\times\mathbb R^2$ and $(N^5\times [0,\pi])\times [0,\pi]$.Comment: 28 pages, new four figures, references added, final version to appear in the Journal of the London. Math. So