This chapter and the next are adapted from a lecture published in the Proceedings of the Third Physics Workshop, held at the Institute of Theoretical and Experimental Physics, Moscow, in 1975. They focus on the uses of homotopy theory in physics, and especially on the calculation of homotopy groups. The next chapter summarizes briefly the ways in which homotopy theory can be applied to quantum field theory.

Elements of Homotopy Theory

Let Z be a compact complex (2n+1)-manifold which carries a complex contact structure, meaning a codimension-1 holomorphic sub-bundle D⊂TZ which is maximally non-integrable. If Z admits a Kahler-Einstein metric of positive scalar curvature, we show that it is the Salamon twistor space of a quaternion-Kahler manifold (M4n, g). If Z also admits a second complex contact structure , then Z=CP2n+1. As an application, we give several new characterizations of the Riemannian manifold HPn= Sp(n+1)/(Sp(n)×Sp(1)).

Fano Manifolds, Contact Structures, and Quaternionic Geometry

Lectures on torus embeddings and applications

On Surfaces of Class VII 0

Locally conformally symplectic and Kähler geometry

We review the properties of the Morse-Novikov cohomology and compute it for all known compact complex surfaces with locally conformally Kahler metrics. We present explicit computations for the Inoue surfaces $$\mathcal {S}^0$$
 , $$\mathcal {S}^+$$
 , $$\mathcal {S}^-$$
 and classify the locally conformally Kahler (and the tamed locally conformally symplectic) forms on $$\mathcal {S}^0$$
 . We prove the nonexistence of LCK metrics with potential and more generally, of $$d_\theta $$
 -exact LCK metrics on Inoue surfaces and Oeljeklaus-Toma manifolds.

Morse-Novikov cohomology of locally conformally Kähler surfaces

Oeljeklaus-Toma (OT) manifolds are complex non-Kahler manifolds whose construction arises from specific number fields. In this note, we compute their de Rham cohomology in terms of invariants associated to the background number field. This is done by two distinct approaches, one using invariant cohomology and the other one using the Leray-Serre spectral sequence. In addition, we compute also their Morse-Novikov cohomology. As an application, we show that the low degree Chern classes of any complex vector bundle on an OT manifold vanish in the real cohomology. Other applications concern the OT manifolds admitting locally conformally Kahler (LCK) metrics: we show that there is only one possible Lee class of an LCK metric, and we determine all the possible Morse-Novikov classes of an LCK metric, which implies the nondegeneracy of certain Lefschetz maps in cohomology.

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De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds

Oeljkelaus-Toma (OT) manifolds are higher dimensional analogues of Inoue-Bombieri surfaces and their construction is associated to a finite extension $K$ of $\mathbb{Q}$ and a subgroup of units $U$. We characterize the existence of pluriclosed metrics (also known as strongly Kahler with torsion (SKT) metrics) on any OT-manifolds $X(K, U)$ purely in terms of number-theoretical conditions, yielding restrictions on the third Betti number $b_3$ and the Dolbeault cohomology group $H^{2,1}_{\overline{\partial}}$. We prove that in complex dimension 4, the existence of a pluriclosed metric on $X(K, U)$ is entirely topological, namely, it is equivalent to $b_3=2$. Moreover, we provide an explicit example of an OT manifold of complex dimension 4 carrying a pluriclosed metric, thus enlarging the list of examples known so far in the literature. Finally, we show that no OT-manifold admits balanced metrics, but all of them carry instead locally conformally balanced metrics.

Special Hermitian metrics on Oeljeklaus-Toma manifolds

We review the properties of the Morse-Novikov cohomology and compute it for all known compact complex surfaces with locally conformally Kahler metrics. We present explicit computations for the Inoue surfaces $\mathcal{S}^0$, $\mathcal{S}^+$, $\mathcal{S}^-$ and classify the locally conformally Kahler (and the tamed locally conformally symplectic) forms on $\mathcal{S}^0$. We prove the nonexistence of LCK metrics with potential and more generally, of $d_\theta$-exact LCK metrics on Inoue surfaces and Oeljeklaus-Toma manifolds.

Morse-Novikov cohomology of locally conformally K\"ahler surfaces

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 $$S^0$$
 .

Alexandra Otiman

Papers

Morse-Novikov cohomology of locally conformally Kähler surfaces

De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds

Special Hermitian metrics on Oeljeklaus-Toma manifolds

Morse-Novikov cohomology of locally conformally K\"ahler surfaces

Cohomologies of locally conformally symplectic manifolds and solvmanifolds