Ruadhaí Dervan

I'm a Royal Society University Research Fellow at the University of Cambridge, and a Research Fellow at Gonville & Caius College. I've previously spent time in École Polytechnique, Université libre de Bruxelles and Trinity College Dublin.

My first name is pronounced "Rui" (the "dh" is silent, it's Irish).

I research complex geometry. I'm interested in both the algebro-geometric and differential-geometric sides of complex geometry, and am particularly interested in the interplay between algebraic geometry and geometric analysis. Most of my research focuses on the existence of canonical metrics in complex geometry (e.g. Kähler-Einstein, extremal), notions of algebro-geometric stability (e.g. K-stability), and more general aspects of moduli theory and geometric analysis. My work is mostly motivated by the "Yau-Tian-Donaldson" conjecture, which links stability notions to the existence of canonical metrics; see Donaldson's survey for an introduction.

I taught Part III Complex Manifolds in Lent Term 2020 and Lent Term 2019.

My CV. My email address is rd430(at)cam.ac.uk.

Papers and preprints:

Valuative stability of polarised varieties (with Eveline Legendre) arXiv:2010.04023 (abstract)

Fujita and Li have given a characterisation of K-stability of a Fano variety in terms of quantities associated to valuations, which has been essential to all recent progress in the area. We introduce a notion of valuative stability for arbitrary polarised varieties, and show that it is equivalent to K-stability with respect to test configurations with integral central fibre. The numerical invariant governing valuative stability is modelled on Fujita's beta invariant, but includes a term involving the derivative of the volume. We give several examples of valuatively stable and unstable varieties, including the toric case. We also discuss the role that the delta invariant plays in the study of valuative stability and K-stability of polarised varieties.
Uniqueness of optimal symplectic connections (with Lars Sektnan) arXiv:2003.13626 (video) (abstract)

Consider a holomorphic submersion between compact Kähler manifolds, such that each fibre admits a constant scalar curvature Kähler metric. When the fibres admit continuous automorphisms, a choice of fibrewise constant scalar curvature Kähler metric is not unique. An optimal symplectic connection is choice of fibrewise constant scalar curvature Kähler metric satisfying a geometric partial differential equation. The condition generalises the Hermite-Einstein condition for a holomorphic vector bundle, through the induced fibrewise Fubini-Study metric on the associated projectivisation.
We prove various foundational analytic results concerning optimal symplectic connections. Our main result proves that optimal symplectic connections are unique, up to the action of the automorphism group of the submersion, when they exist. Thus optimal symplectic connections are canonical relatively Kähler metrics when they exist. In addition, we show that the existence of an optimal symplectic connection forces the automorphism group of the submersion to be reductive, and that an optimal symplectic connection is automatically invariant under a maximal compact subgroup of this automorphism group.
Moduli theory, stability of fibrations and optimal symplectic connections (with Lars Sektnan) arXiv:1911.12701 (to appear in Geom. Topol.) (video) (abstract)

K-polystability is, on the one hand, conjecturally equivalent to the existence of certain canonical Kähler metrics on polarised varieties, and, on the other hand, conjecturally gives the correct notion to form moduli. We introduce a notion of stability for families of K-polystable varieties, extending the classical notion of slope stability of a bundle, viewed as a family of K-polystable varieties via the associated projectivisation. We conjecture that this is the correct condition for forming moduli of fibrations.
Our main result relates this stabiility condition to Kähler geometry: we prove that the existence of an optimal symplectic connection implies semistability of the fibration. An optimal symplectic connection is a choice of fibrewise constant scalar curvature Kähler metric, satisfying a certain geometric partial differential equation. We conjecture that the existence of such a connection is equivalent to polystability of the fibration. We prove a finite dimensional analogue of this conjecture, by describing a GIT problem for fibrations embedded in a fixed projective space, and showing that GIT polystability is equivalent to the existence of a zero of a certain moment map.
Optimal symplectic connections on holomorphic submersions (with Lars Sektnan) arXiv:1907.11014 (to appear in Comm. Pure Appl. Math.) (abstract)

The main result of this paper gives a new construction of extremal Kähler metrics on the total space of certain holomorphic submersions, giving a vast generalisation and unification of results of Hong, Fine and others. The principal new ingredient is a novel geometric partial differential equation on such fibrations, which we call the optimal symplectic connection equation.
We begin with a smooth fibration for which all fibres admit a constant scalar curvature Kähler metric. When the fibres admit automorphisms, such metrics are not unique in general, but rather are unique up to the action of the automorphism group of each fibre. We define an equation which, at least conjecturally, determines a canonical choice of constant scalar curvature Kähler metric on each fibre. When the fibration is a projective bundle, this equation specialises to asking that the hermitian metric determining the fibrewise Fubini-Study metric is Hermite-Einstein.
Assuming the existence of an optimal symplectic connection, and the existence of an appropriate twisted extremal metric on the base of the fibration, we show that the total space of the fibration itself admits an extremal metric for certain polarisations making the fibres small.
K-semistability of optimal degenerations arXiv:1905.11334 (Q. J. Math. Vol. 71 (2020) Issue 3, pp. 989-995) (abstract)

K-polystability of a polarised variety is an algebro-geometric notion conjecturally equivalent to the existence of a constant scalar curvature Kähler metric. When a variety is K-unstable, it is expected to admit a "most destabilising" degeneration. In this note we show that if such a degeneration exists, then the limiting scheme is itself relatively K-semistable.
Moduli of polarised manifolds via canonical Kähler metrics (with Philipp Naumann) arXiv:1810.02576 (video) (abstract)

We construct a moduli space of polarised manifolds which admit a constant scalar curvature Kähler metric. We show that this space admits a natural Kähler metric.
Extremal metrics on fibrations (with Lars Sektnan) arXiv:1712.05374 (Proc. Lond. Math. Soc. (3) 120 (2020) pp. 587-616) (video) (abstract)

Consider a fibred compact Kähler manifold X endowed with a relatively ample line bundle, such that each fibre admits a constant scalar curvature Kähler metric and has discrete automorphism group. Assuming the base of the fibration admits a twisted extremal metric where the twisting form is a certain Weil-Petersson type metric, we prove that X admits an extremal metric for polarisations making the fibres small. Thus X admits a constant scalar curvature Kähler metric if and only if the Futaki invariant vanishes. This extends a result of Fine, who proved this result when the base admits no continuous automorphisms.
As consequences of our techniques, we obtain the following analogues for maps of various fundamental results for varieties: if a map admits a twisted constant scalar curvature Kähler metric metric, then its automorphism group is reductive; a twisted extremal metric is invariant under a maximal compact subgroup of the automorphism group of the map; there is a geometric interpretation for uniqueness of twisted extremal metrics on maps.
Stable maps in higher dimensions (with Julius Ross) arXiv:1708.09750 (Math. Ann. Vol. 374 (2019) no. 3-4, pp. 1033-1073) (abstract)

We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. We give some examples, such as Kodaira embeddings and fibrations. We prove the existence of a projective moduli space of canonically polarised stable maps, generalising the Kontsevich-Alexeev moduli space of stable maps in dimensions one and two. We also state an analogue of the Yau-Tian-Donaldson conjecture in this setting, relating stability of maps to the existence of certain canonical Kähler metrics.
Hermitian Yang-Mills connections on blowups (with Lars Sektnan) arXiv:1707.07638 (to appear in J. Geom. Anal.) (abstract)

Consider a vector bundle over a Kähler manifold which admits a Hermitian Yang-Mills connection. We show that the pullback bundle on the blowup of the Kähler manifold at a collection of points also admits a Hermitian Yang-Mills connection, for Kähler classes on the blowup which make the exceptional divisors small. Our proof uses gluing techniques, and is hence asymptotically explict. This recovers, through the Hitchin-Kobayashi correspondence, algebro-geometric results due to Buchdahl and Sibley.
The Kähler-Ricci flow and optimal degenerations (with Gábor Székelyhidi) arXiv:1612.07299 (J. Differential Geom. Vol. 116 (2020), no. 1 pp. 187-203) (abstract)

We prove that on Fano manifolds, the Kähler-Ricci flow produces a "most destabilising" degeneration, with respect to a new stability notion related to the H-functional. This answers questions of Chen-Sun-Wang and He.
We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the supremum of Perelman's μ-functional on Fano manifolds, resolving a conjecture of Tian-Zhang-Zhang-Zhu as a special case. Secondly, we use this to prove that if a Fano manifold admits a Kähler-Ricci soliton, then the Kähler-Ricci flow converges to it modulo the action of automorphisms, with any initial metric. This extends work of Tian-Zhu and Tian-Zhang-Zhang-Zhu, where either the manifold was assumed to admit a Kähler-Einstein metric, or the initial metric of the flow was assumed to be invariant under a maximal compact group of automorphism.
Relative K-stability for Kähler manifolds arXiv:1611.00569 (Math. Ann. Vol. 372 (2018), no. 3-4, pp. 859-889) (addendum) (abstract)

We study the existence of extremal Kähler metrics on Kähler manifolds. After introducing a notion of relative K-stability for Kähler manifolds, we prove that Kähler manifolds admitting extremal Kähler metrics are relatively K-stable. Along the way, we prove a general L^p lower bound on the Calabi functional involving test configurations and their associated numerical invariants, answering a question of Donaldson.
When the Kähler manifold is projective, our definition of relative K-stability is stronger than the usual definition given by Székelyhidi. In particular our result strengthens the known results in the projective case (even for constant scalar curvature Kähler metrics), and rules out a well known counterexample to the "naïve" version of the Yau-Tian-Donaldson conjecture in this setting.

K-stability for Kähler manifolds (with Julius Ross) arXiv:1602.08983 (Math. Res. Lett. Vol. 24, No. 3 (2017), pp. 689-739) (abstract)

We formulate a notion of K-stability for Kähler manifolds, and prove one direction of the Yau-Tian-Donaldson conjecture in this setting. More precisely, we prove that the Mabuchi functional being bounded below (resp. coercive) implies K-semistability (resp. uniformly K-stable). In particular this shows that the existence of a constant scalar curvature Kähler metric implies K-semistability, and K-stability if one assumes the automorphism group is discrete. We also show how Stoppa's argument holds in the Kähler case, giving a simpler proof of this K-stability statement.
A finite dimensional approach to Donaldson's J-flow (with Julien Keller) arXiv:1507.03461 (Comm. Anal. Geom. (2019) Vol. 27, no. 5. pp. 1025-1085) (abstract)

Consider a projective manifold with two distinct polarisations L₁ and L₂. From this data, Donaldson has defined a natural flow on the space of Kähler metrics in c₁(L₁), called the J-flow. The existence of a critical point of this flow is closely related to the existence of a constant scalar curvature Kähler metric in c₁(L₁) for certain polarisations L₂.
Associated to a quantum parameter k>>0, we define a flow over Bergman type metrics, which we call the J-balancing flow. We show that in the quantum limit of k→∞, the rescaled J-balancing flow converges towards the J-flow. As corollaries, we obtain new proofs of uniqueness of critical points of the J-flow and also that these critical points achieve the absolute minimum of an associated energy functional.
We show that the existence of a critical point of the J-flow implies the existence of J-balanced metrics for k >> 0. Defining a notion of Chow stability for linear systems, we show that this in turn implies the linear system |L₂| is asymptotically Chow stable. Asymptotic Chow stability of |L₂| implies an analogue of K-semistability for the J-flow introduced by Lejmi-Székelyhidi, which we call J-semistability. We prove also that J-stability holds automatically in a certain numerical cone around L₂, and that if L₂ is the canonical class of the manifold that J-semistability implies K-stability. Eventually, this leads to new K-stable polarisations of surfaces of general type.
On K-stability of finite covers arXiv:1505.07754 (Bull. London Math. Soc. (2016) 48 (4) pp. 717-728) (abstract)

We show that certain Galois covers of K-semistable Fano varieties are K-stable. We use this to give some new examples of Fano manifolds admitting Kähler-Einstein metrics, including hypersurfaces, double solids and threefolds.
Non-reductive automorphism groups, the Loewy filtration and K-stability (with Giulio Codogni) arXiv:1501.03372 (Annales de l'institut Fourier 66 no. 5 (2016) pp. 1895-1921) (corrigendum) (abstract)

We study the K-stability of a polarised variety with non-reductive automorphism group. We associate a canonical filtration of the co-ordinate ring to each variety of this kind, which destabilises the variety in several examples which we compute. We conjecture this holds in general. This is an algebro-geometric analogue of Matsushima's theorem regarding the existence of constant scalar curvature Kähler metrics. As an application, we give an example of an orbifold del Pezzo surface without a Kähler-Einstein metric.
Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds arXiv:1412.1426 (Ann. Fac. Sci. Toulouse Sér. 6, 25 no. 4 (2016), p. 919-934) (abstract)

We give a criterion for the coercivity of the Mabuchi functional for general Kähler classes on Fano manifolds in terms of Tian's alpha invariant. This generalises a result of Tian in the anti-canonical case implying the existence of a Kähler-Einstein metric. We also prove the alpha invariant is a continuous function on the Kähler cone. As an application, we provide new Kähler classes on a general degree one del Pezzo surface for which the Mabuchi functional is coercive.
Uniform stability of twisted constant scalar curvature Kähler metrics arXiv:1412.0648 (Int. Math. Res. Notices (2016) Vol 15 pp. 4728-4783) (abstract)

We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature Kähler metric. This notion of uniform K-stability is analogous to coercivity of the Mabuchi functional. We characterise the triviality of test configurations, by showing that a test configuration has zero minimum norm if and only if it has zero L²-norm, if and only if it is almost trivial.
We prove that the existence of a twisted constant scalar curvature Kähler metric implies uniform twisted K-stability with respect to the minimum norm, when the twisting is ample.
We give algebro-geometric proofs of uniform K-stability in the general type and Calabi-Yau cases, as well as in the Fano case under an alpha invariant condition. Our results hold for line bundles sufficiently close to the (anti)-canonical line bundle, and also in the twisted setting. We show that log K-stability implies twisted K-stability, and also that twisted K-semistability of a variety implies that the variety has mild singularities.
Alpha invariants and K-stability for general polarisations of Fano varieties arXiv:1307.6527 (Int. Math. Res. Notices (2015) Vol 16 pp. 7162-7189) (abstract)

We provide a sufficient condition for polarisations of Fano varieties to be K-stable in terms of Tian's alpha invariant, which uses the log canonical threshold to measure singularities of divisors in the linear system associated to the polarisation. This generalises a result of Odaka-Sano in the anti-canonically polarised case, which is the algebraic counterpart of Tian's analytic criterion implying the existence of a Kähler-Einstein metric. As an application, we give new K-stable polarisations of a general degree one del Pezzo surface. We also prove a corresponding result for log K-stability.

Students:

Michael Hallam, DPhil student at the University of Oxford, co-supervised by Frances Kirwan (2019 - present)
John McCarthy, PhD student at Imperial College, co-supervised by Simon Donaldson (2019 - present)
Qiangru Kuang, undergraduate summer research student at the University of Cambridge (Summer 2019)

I organised conferences in Cambridge (Postgraduate Conference in Complex Geometry, 2015), Rome (Moduli of K-stable varieties, 2017) and Newcastle (Newcastle Complex Geometry Workshop, 2018). The Rome conference has an associated conference proceedings, also edited by Codogni and Viviani.

I ran the Cambridge Junior Geometry Seminar (called "Geometry Tea") in 2013-2014. It continues in 2019-2020.

Slightly outdated list of conferences I've been to or am going to:

K-stability, Simons Center for Geometry and Physics, 28 September - 2nd October
Fano Varieties, Cone Singularities and their Links, University of Edinburgh, 4 - 8 November 2019
Stability and Canonical Metrics on Bundles and Varieties, Brest, 23 - 26 June 2019
Current developments in Complex and Analytic Geometry, Cortona, 2 - 8 June 2019
Kähler and Special Toric Geometry, King's College London, 20 - 22 May 2019
British Algebraic Geometry meeting, University of Liverpool, 24 - 26 April 2019
Birational Geometry, Kähler-Einstein metrics and Degenerations, HSE Moscow, 8 - 13 April 2019
Singular metrics in Kähler geometry, CIRM Marseille, 4 - 8 February 2019
Degenerations of Calabi-Yau Manifolds, Institut Henri Poincaré Paris, 14 - 16 May 2018
Constant Scalar Curvature Metrics in Kähler and Sasaki Geometry, CIRM Marseille, January 15 - 19 2018
British Algebraic Geometry meeting, Cambridge, September 11 - 13 2017
Symplectic geometry - celebrating the work of Simon Donaldson, Cambridge, August 14 - 18 2017
Moduli of K-stable varieties, Rome, July 10 - 14 2017
Symposium in Geometry and Differential Equations, AMSS Beijing, July 3 - 7 2017
Current Developments and New Directions in Kähler Geometry, University of Notre Dame, June 26 - 30 2017
Positivity in Algebraic and Complex Geometry, ICMS Edinburgh, April 24 - 27 2017
Stability and moduli spaces, American Institute of Mathematics San Jose, January 9 - 13 2017
Algebraic Geometry: Old and New, University of Oxford, September 26 - 30 2016
Hitchin 70, University of Oxford, September 9 - 11 2016
European Congress of Mathematics, Berlin, July 18 - 22 2016
Recent Advances in Complex Differential Geometry, Institut de Mathématiques de Toulouse, June 13 - 22 2016
British Algebraic Geometry meeting, ICMS Edinburgh, 13 - 15 April 2016
Moduli spaces and singularities in algebraic and Riemannian geometry, Simons Center for Geometry and Physics, September 14 - October 11 2015
Postgraduate Conference in Complex Geometry, University of Cambridge, September 9 - 11 2015
Cones and Positivity, Loughborough University, August 4 - 7 2015
Summer Research Institute on Algebraic Geometry, University of Utah, July 13 - 17 2015
British Algebraic Geometry meeting, University of Warwick, 22 - 24 September 2014
Edge Days, University of Edinburgh, 5 - 7 July 2014
Asymptotic Aspects of Complex and Algebraic Geometry, Universita di Milano Bicocca, 23 - 28 June 2014
Edge Days, University of Edinburgh, 6 - 8 June 2014
Irish Workshop on Kähler Geometry and Geometric Analysis, University College Cork, 12 - 14 May 2014
Minicourses on Stability, University of Coimbra, 10 - 12 April 2014
Traumatic Workshop, Imperial College, 3 - 4 April 2014
William Rowan Hamilton Geometry and Topology Workshop, Trinity College Dublin, 27 - 31 August 2013
Ricci curvature: limit spaces and Kahler geometry, ICMS Edinburgh 1 - 5 July 2013
Edge Days, University of Edinburgh, 7 - 9 June 2013
Compactifying Moduli Spaces, CRM Barcelona, 27 - 31 May 2013

Pickles