Papers:
2022: K-Polystability of Smooth Fano SL₂ Threefolds [Preprint]
My Research:
I aim to find concrete, combinatorial conditions implying K-stability for Fano varieties equipped with certain reductive group actions. Suppose a reductive group G acts on a variety X. The minimal codimension in X of the orbits of a Borel subgroup B of G is called the complexity of the group action. Hence, when G is an algebraic torus, the normal G-varieties of complexity zero are the toric varieties. More broadly, normal G-varieties of complexity zero (i.e. those containing an open dense B-orbit) when G is an arbitrary reductive group are called spherical. I study complexity one G-varieties for general reductive groups G, i.e. the varieties whose largest B-orbits are of codimension one.
These varieties with large symmetry groups can be described (in the case where the complexity is at most one) by combinatorial data, i.e. certain cones, polytopes etc., generalising the well-known description of toric varieties by fans. I use the combinatorial description due to Timashev of complexity one G-varieties to look for combinatorial criteria guaranteeing various geometric properties, in particular K-stability.
Due to the work of Datar-Székelyhidi on equivariant K-stability, it is possible to find concrete criteria for K-stability of varieties with large symmetry groups. Combinatorial conditions implying equivariant K-stability have been found by Wang-Zhu in the toric case, by Delcroix in the spherical case, and by Ilten-Süß for torus varieties of complexity one. My work concerns the case of K-stability for varieties with complexity one actions by general reductive groups.
I am currently writing a paper in which myself and my supervisor Hendrik Süß have proved the K-stability of all smooth Fano SL₂ threefolds not admitting the action of a 2- or 3-dimensional torus. These varieties each contain an open SL₂-orbit within which lies a one-parameter family of codimension one B-orbits, where B is the Borel subgroup of upper triangular matrices. My expectation is that the methods therein should be generalisable to more broad classes of complexity one varieties.
Otherwise, there are different opportunities to use this combinatorial data to study properties other than K-stability. For example, there is a known formula due to Luna and Brion for the anticanonical divisor of a spherical variety, and a corresponding, easily checkable Fano condition. No such formula or condition is known in complexity one (to my knowledge), so I intend to investigate this and other similar problems.
K-Stability:
The existence of canonical metrics on compact Kähler manifolds has been a longstanding and fruitful area of research in both complex and algebraic geometry. The class of Kähler-Einstein metrics are of particular interest, in part because these metrics have constant scalar curvature. The problem of whether such metrics exist can be split into cases depending on the sign of the first Chern class of the underlying manifold, and the cases where this class is negative or zero were solved by Aubin and Yau in the 1970s - in particular all such manifolds admit Kähler-Einstein metrics.
The case of positive first Chern class (the Fano case) proved to be more difficult, and obstructions to the existence of KE metrics have been known for a long time (a result of Matsushima from the 1950s implies that Fano manifolds with non-reductive automorphism groups, e.g. the complex projective plane blown up at one point, cannot admit KE metrics). The Yau-Tian-Donaldson conjecture, now solved by Chen-Donaldson-Sun, posited that the existence of a Kähler-Einstein metric on a compact Fano manifold should be equivalent to an algebro-geometric condition called K-stability, inspired by the Hilbert-Mumford stability of geometric invariant theory.
The solution of the YTD conjecture has led to a wealth of research into the topic of K-stability, culminating in the recent solution of the finite generation conjecture by Liu, Xu and Zhuang. The more practical problem of finding methods to check the K-stability of particular varieties still requires further exploration.