Applications of pseudoholomorphic subvarieties to J-anti-invariant forms and spectral geometry

The results contained in this thesis can be split into two categories, namely those involving the analysis of J-anti-invariant forms and those in the realm of spectral geometry. We primarily study the relation of J-anti-invariant 2-forms with pseudoholomorphic curves in the first half of the thesis. We show that the zero set of a closed J-antiinvariant 2-form on an almost complex 4-manifold supports a J-holomorphic subvariety in the canonical class. This confirms a conjecture of Draghici-Li-Zhang. A higher dimensional analogue is also established. Furthermore, the local model built for the bundle of J-anti-invariant forms can be used to prove that, on an almost complex 4- manifold, the dimension of the cohomology group associated to closed J-anti-invariant 2-forms is a birational invariant in the sense that it is invariant under degree one pseudoholomorphic maps. In the latter half we study the eigenvalues of the Laplacian of an almost Kähler metric. In particular we find that the bounds established by Kokarev [33] in the case of a Kähler metric with respect to an integrable almost complex structure also hold in the almost Kähler setting. That is, we show that if a compact almost Kähler manifold admits a pseudoholomorphic map into a projective space then the k-th eigenvalue of the Laplacian, with respect to a given Kähler metric, can be bounded above by a constant depending only on dimension, the map into projective space and the Kähler class. We provide examples of strictly almost Kähler manifolds which admit a nontrivial pseudoholomorphic map into a projective space. Similarly to Kokarev [33] we establish a version of the estimate for pseudoholomorphic subvariety. Finally we prove that the estimate holds for almost Kähler manifolds admitting a pseudoholomorphic map into projective space in a class of non-smooth maps. In particular we obtain that the estimate holds for Kähler manifolds which admit a rational map into projective space