Marlon de Oliveira Gomes

Research

My Dissertation concerns a branch of Differential Geometry called Twistor Theory, which lies at the crossroads of Conformal Geometry (where angles are defined, but lengths are not), and Complex Geometry.

The relation between these two geometries dates back to the 18th century. Claims of the equivalence between the two in the two-dimensional setting were laid out by Riemann, Klein, Poincaré, and others, and simultaneuously led to momentous growth in Algebraic Geometry, Function Theory, Topology and Differential Geometry.

In higher dimensions, the relation is not so simple. At any point on an even-dimensional manifold, there is a plethora of linear complex structures compatible with a given conformal structure. On the other hand, globally defined complex structures are scarce (and may even be obstructed by topology), whereas global conformal structures are abundant.

Twistor Theory, pioneered by Penrose in the 1960s, makes use of the many pointwise complex structures on a 4-dimensional manifold in an unexpected way: instead of trying to find a most suitable complex structure compatible with the conformal structure (as one can do in dimension 2), we study them all at once!

The gadget we use to parametrize linear complex structures is called twistor space. It has a tautological almost-complex structure of its own, which is a faithful complex structure when a suitable curvature condition, called anti-self-duality, is satisfied by the underlying oriented conformal manifold. By restricting our attention to anti-self-dual manifolds, we can once again bring the full power of Complex Analysis into Conformal Geometry.

Twistor theory bears its fruits in the fact that the twistor construction is reversible: there are certain conditions on a complex manifold which ensure it is a Twistor Space. Sufficient knowlegde about the complex structure of Twistor Space should allow one to reverse-engineer anti-self-dual 4-manifolds. The challege lies on the fact that complex structures arising from the Twistor construction are rather mysterious. For instance, in the compact case, they are almost never Kähler, so one expects them not to be amenable to the techniques of Classical Algebraic Geometry.

While we have many results asserting the existence of anti-self-dual structures, the ones we understand explicitly are few and far between. My work exploits examples of Twistor Spaces with just enough of an algebraic flavor, namely, those which possess canonical rational maps to the projective plane, to shed light on the anti-self-duality equations.