Blowing up generalized Kahler 4-manifolds

We show that the blow-up of a generalized Kahler 4-manifold in a nondegenerate complex point admits a generalized Kahler metric. As with the blow-up of complex surfaces, this metric may be chosen to coincide with the original outside a tubular neighbourhood of the exceptional divisor. To accomplish this, we develop a blow-up operation for bi-Hermitian manifolds.Comment: 16 page

Ambitoric geometry I: Einstein metrics and extremal ambikaehler structures

We present a local classification of conformally equivalent but oppositely oriented 4-dimensional Kaehler metrics which are toric with respect to a common 2-torus action. In the generic case, these "ambitoric" structures have an intriguing local geometry depending on a quadratic polynomial q and arbitrary functions A and B of one variable. We use this description to classify Einstein 4-metrics which are hermitian with respect to both orientations, as well a class of solutions to the Einstein-Maxwell equations including riemannian analogues of the Plebanski-Demianski metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors. These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kaehler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kaehler metrics. For both the Einstein-Maxwell and the extremal ambitoric structures, A and B are quartic polynomials, but with different conditions on the coefficients. In the sequel to this paper we consider global examples, and use them to resolve the existence problem for extremal Kaehler metrics on toric 4-orbifolds with second betti number b2=2.Comment: 31 pages, 1 figure, partially replaces arXiv:1010.099

Ambitoric geometry II: Extremal toric surfaces and Einstein 4-orbifolds

We provide an explicit resolution of the existence problem for extremal Kaehler metrics on toric 4-orbifolds M with second Betti number b2(M)=2. More precisely we show that M admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Szekelyhidi et al.). Furthermore, in this case, the extremal Kaehler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kaehler metric, which turns out to be extremal as well. These results provide a computational test for the K-stability of labelled quadrilaterals. Extremal ambitoric structures were classified locally in Part I of this work, but herein we only use the straightforward fact that explicit Kaehler metrics obtained there are extremal, and the identification of Bach-flat (conformally Einstein) examples among them. Using our global results, the latter yield countably infinite families of compact toric Bach-flat Kaehler orbifolds, including examples which are globally conformally Einstein, and examples which are conformal to complete smooth Einstein metrics on an open subset, thus extending the work of many authors.Comment: 31 pages, 3 figures, partially replaces and extends arXiv:1010.099

The 22-year cycle in the geomagnetic 27-day recurrences reflecting on the F2-layer ionization

Solar cycle variations of the amplitudes of the 27-day solar rotation period reflected in the geomagnetic activity index <i>A<sub>p</sub></i>, solar radio flux F10.7cm and critical frequency <i>fo</i>F2 for mid-latitude ionosonde station Moscow from the maximum of sunspot cycle 18 to the maximum of cycle 23 are examined. The analysis shows that there are distinct enhancements of the 27-day amplitudes for <i>fo</i>F2 and <i>A<sub>p</sub></i> in the late declining phase of each solar cycle while the amplitudes for F10.7cm decrease gradually, and the <i> fo</i>F2 and <i>A<sub>p</sub></i> amplitude peaks are much larger for even-numbered solar cycles than for the odd ones. Additionally, we found the same even-high and odd-low pattern of <i>fo</i>F2 for other mid-latitude ionosonde stations in Northern and Southern Hemispheres. This property suggests that there exists a 22-year cycle in the F2-layer variability coupled with the 22-year cycle in the 27-day recurrence of geomagnetic activity.<br><br> <b>Key words.</b> Ionosphere (mid-latitude ionosphere; ionosphere- magnetosphere interactions) – Magnetospheric physics (solar wind-magnetosphere interactions

Self-induced tunable transparency in layered superconductors

We predict a novel nonlinear electromagnetic phenomenon in layered superconducting slabs irradiated from one side by an electromagnetic plane wave. We show that the reflectance and transmittance of the slab can vary over a wide range, from nearly zero to one, when changing the incident wave amplitude. Thus changing the amplitude of the incident wave can induce either the total transmission or reflection of the incident wave. In addition, the dependence of the superconductor transmittance on the incident wave amplitude has an unusual hysteretic behavior with jumps. This remarkable nonlinear effect (self-induced transparency) can be observed even at small amplitudes, when the wave frequency $\omega$ is close to the Josephson plasma frequency $\omega_J$.Comment: 9 pages, 7 figure