RESEARCH INTERESTS
My research lies at the interface of complex differential geometry, algebraic geometry, and geometric analysis. I am especially interested in canonical metrics (including Kähler-Einstein, cscK, special non-Kähler metrics) and, in particular, in the geometric structures that arise from their degenerations. Beside their intrinsic importance in pure mathematics, such topics have deep connections with aspects of theoretical physics (e.g., spacetime geometries, String/M-Theory, AdS-CFT).
A recurring theme in my work is that analytic phenomena in limits of canonical metrics often have algebraic counterparts. This can lead to two interrelated outcomes:
construction of new canonical algebraic structures associated with moduli spaces and singularities (such as compact K-moduli spaces of Fano varieties and refined invariants of singularities) obtained by incorporating input from geometric PDEs into the algebro-geometric framework;
the emergence of an “algebraic computability” of highly non-explicit differential-geometric objects (such as Einstein metrics) through explicit descriptions of their degenerations via concrete algebraic tools.
This viewpoint underlies several of my works, including:
first constructions of compactified Kähler-Einstein/K-moduli spaces of Fano varieties in different contexts;
constructions and regularity results for Calabi-Yau metrics singular along divisors and their relations with the Bogomolov-Miyaoka-Yau inequality;
algebraic descriptions of multiscale bubbling phenomena in degenerations of Kähler-Einstein metrics;
gluing and adiabatic constructions for canonical metrics in both Kähler and non-Kähler geometry.
A further direction of my work concerns special metrics in non-Kähler geometry, where the appropriate notions of canonical metrics are still less understood. In particular, this has led me to study Yamabe-type problems and infinite-dimensional moment-map-type equations in this setting.
For a general introduction to the scientific area, see Wikipedia page on Complex Geometry.
Past collaborators: D. Angella, C. Arezzo, M. de Borbon, S. Calamai, P. Gallardo, F. Giusti, J. Martinez-Garcia, Y. Odaka, F. Pediconi, L.M. Sektnan, C. Scarpa, S. Sun, J. Windare, C. Yao.
Mentors: C. Arezzo, O. Biquard, S. K. Donaldson, J. Ross.