SURF FLOW · Parabolic Approach to the Classification of Complex Surfaces through the Pluriclosed Flow

One of the landmark achievements of 20th-century mathematics is the Enriques-Kodaira classification of compact complex surfaces. Yet, for odd Betti number, there is a long-standing open conjecture, known as the Global Spherical Shell (GSS) conjecture, which concerns Class VII surfaces. This remains the main open problem in low-dimensional complex geometry, with the latest advances exploiting deep techniques from algebraic geometry. Inspired by Perelman’s breakthrough use of Ricci flow in 3-manifolds topology, Streets and Tian introduced a parabolic flow of Hermitian metrics – the pluriclosed flow – which provides a promising analytical approach to the GSS conjecture. Despite the remarkable properties of the flow, well-adapted to the geometry and topology of Class VII surfaces, and its interesting relation to theoretical physics, this approach remains largely unexplored due to important technical difficulties. Recent progress, achieved by García-Fernández with Streets and Jordan, shows how combining advanced analytic techniques with Hitchin’s generalized geometry can overcome these barriers, opening a new research frontier.SURF FLOW aims at pushing well beyond the analytical approach to the GSS conjecture along this new path, eventually giving new insight for a combined effort with algebraic geometry. I will investigate the evolution of the pluriclosed flow up to its maximal existence time and the formation of singular loci by describing it on a significant class of manifolds. This is the first milestone in the Streets-Tian program for the classification of complex surfaces. Complementing this, I willl study the static points of the flow, which have deep implications to conjectural extensions of Calabi's programme on canonical metrics to complex non-Kähler geometry.The combination of the supervision by Garcia-Fernandez at ICMAT with the secondment visit to Prof. Streets at UC Irvine, is strategic for the goals of the present proposal.