On the cohomology of almost complex and symplectic manifolds and proper surjective maps

Let $(X,J)$ be an almost-complex manifold. In \cite{li-zhang} Li and Zhang introduce H^{(p,q),(q,p)}_J(X)_{\rr} as the cohomology subgroups of the $(p+q)$-th de Rham cohomology group formed by classes represented by real pure-type forms. Given a proper, surjective, pseudo-holomorphic map between two almost-complex manifolds we study the relationship among such cohomology groups. Similar results are proven in the symplectic setting for the cohomology groups introduced in \cite{tsengyauI} by Tseng and Yau and a new characterization of the Hard Lefschetz condition in dimension $4$ is provided

Some remarks on Hermitian manifolds satisfying K\"ahler-like conditions

We study Hermitian metrics whose Bismut connection $\nabla^B$ satisfies the first Bianchi identity in relation to the SKT condition and the parallelism of the torsion of the Bimut connection. We obtain a characterization of complex surfaces admitting Hermitian metrics whose Bismut connection satisfy the first Bianchi identity and the condition $R^B(x,y,z,w)=R^B(Jx,Jy,z,w)$, for every tangent vectors $x,y,z,w$, in terms of Vaisman metrics. These conditions, also called Bismut K\"ahler-like, have been recently studied in [D. Angella, A. Otal, L. Ugarte, R. Villacampa, On Gauduchon connections with K\"ahler-like curvature, to appear in Commun. Anal. Geom., arXiv:1809.02632 [math.DG]], [Q. Zhao, F. Zheng, Strominger connection and pluriclosed metrics, arXiv:1904.06604 [math.DG]], [S. T. Yau, Q. Zhao, F. Zheng, On Strominger K\"ahler-like manifolds with degenerate torsion, arXiv:1908.05322 [math.DG]]. Using the characterization of SKT almost abelian Lie groups in [R. M. Arroyo, R. Lafuente, The long-time behavior of the homogeneous pluriclosed flow, Proc. London Math. Soc. (3), 119, (2019), 266-289], we construct new examples of Hermitian manifolds satisfying the Bismut K\"ahler-like condition. Moreover, we prove some results in relation to the pluriclosed flow on complex surfaces and on almost abelian Lie groups. In particular, we show that, if the initial metric has constant scalar curvature, then the pluriclosed flow preserves the Vaisman condition on complex surfaces.Comment: Theorem B and Lemma 5.1 modified. Added Remark 5.

Symplectic cohomologies and deformations

In this note we study the behavior of symplectic cohomology groups under symplectic deformations. Moreover, we show that for compact almost-K\"ahler manifolds $(X,J,g,\omega)$ with $J$ $\mathcal{C}^\infty$-pure and full the space of de Rham harmonic forms is contained in the space of symplectic-Bott-Chern harmonic forms. Furthermore, we prove that the second non-HLC degree measures the gap between the de Rham and the symplectic-Bott-Chern harmonic forms

Cohomologies of locally conformally symplectic manifolds and solvmanifolds

We study the Morse-Novikov cohomology and its almost-symplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz Condition. We consider solvmanifolds and Oeljeklaus-Toma manifolds. In particular, we prove that Oeljeklaus-Toma manifolds with precisely one complex place, and under an additional arithmetic condition, satisfy the Mostow property. This holds in particular for the Inoue surface of type $S^0$

Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms

We construct a simply-connected compact complex non-K\"ahler manifold satisfying the $\partial\bar\partial$-Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the $\partial\bar\partial$-Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in \cite{rao-yang-yang, yang-yang, stelzig-blowup, stelzig-doublecomplex} with different techniques. Here, we provide a different approach using \v{C}ech cohomology theory to study the Dolbeault cohomology of the blow-up $\tilde X_Z$ of a compact complex manifold $X$ along a submanifold $Z$ admitting a holomorphically contractible neighbourhood

Relative Čech–Dolbeault homology and applications

We define the relative Dolbeault homology of a complex manifold with currents via a Čech approach, and we prove its equivalence with the relative Čech–Dolbeault cohomology as defined by Suwa (in: Singularities–Niigata–Toyama 2007. Advanced studies in pure mathematics, vol 56. Mathematical Society of Japan, Tokyo, pp 321–340, 2009). This definition is then used to compare the relative Dolbeault cohomology groups of two complex manifolds of the same dimension related by a suitable proper surjective holomorphic map. Finally, an application to blow-ups is considered and a blow-up formula for the Dolbeault cohomology in terms of relative cohomology is presented

HKT manifolds: Hodge theory, formality and balanced metrics

Let $(M,I,J,K,\Omega)$ be a compact HKT manifold and denote with $\partial$ the conjugate Dolbeault operator with respect to $I$, $\partial_J:=J^{-1}\overline\partial J$, $\partial^\Lambda:=[\partial,\Lambda]$ where $\Lambda$ is the adjoint of $L:=\Omega\wedge-$. Under suitable assumptions, we study Hodge theory for the complexes $(A^{\bullet,0},\partial,\partial_J)$ and $(A^{\bullet,0},\partial,\partial^\Lambda)$ showing a similar behavior to K\"ahler manifolds. In particular, several relations among the Laplacians, the spaces of harmonic forms and the associated cohomology groups, together with Hard Lefschetz properties, are proved. Moreover, we show that for a compact HKT $\mathrm{SL}(n,\mathbb{H})$-manifold the differential graded algebra $(A^{\bullet,0},\partial)$ is formal and this will lead to an obstruction for the existence of an HKT $\mathrm{SL}(n,\mathbb{H})$-structure $(I,J,K,\Omega)$ on a compact complex manifold $(M,I)$. Finally, balanced HKT structures on solvmanifolds are studied.Comment: 17 pages. Comments are welcom

Adjoint Monte Carlo Simulation of Fusion Product Activation Probe Experiment in ASDEX Upgrade tokamak

The activation probe is a robust tool to measure flux of fusion products from a magnetically confined plasma. A carefully chosen solid sample is exposed to the flux, and the impinging ions transmute the material making it radioactive. Ultra-low level gamma-ray spectroscopy is used post mortem to measure the activity and, thus, the number of fusion products. This contribution presents the numerical analysis of the first measurement in the ASDEX Upgrade tokamak, which was also the first experiment to measure a single discharge. The ASCOT suite of codes was used to perform adjoint/reverse Monte Carlo calculations of the fusion products. The analysis facilitates, for the first time, a comparison of numerical and experimental values for absolutely calibrated flux. The results agree to within a factor of about two, which can be considered a quite good result considering the fact that all features of the plasma cannot be accounted in the simulations. Also an alternative to the present probe orientation was studied. The results suggest that a better optimized orientation could measure the flux from a significantly larger part of the plasma.Comment: Contribution in 1st EPS Conference on Plasma Diagnostics. First two versions are for PoS(ECPD 2015)055. This 3rd version was accepted for publishing in Journal of Instrumentatio