Julius Ross

Julius Ross

Mathematician

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mscsuic/juliusross

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Minimalistic — Theme by vaibhavvikas

Research

[43] Generalized Bogomolov Inequalities

with M. Pavel and M. Toma
Preprint: arXiv:2510.04663

We introduce two notions, the first being a Hodge-Riemann pair and the second is that of a Bogomolov Pair. In both cases this can be either a pair of cohomology classes or a pair of forms; for simplicity in this summary we just consider the case of classes.

A Hodge-Riemann pair consists of a pair of cohomology classes on a compact Kahler manifold (or projective manifold) that generalizes the Hodge-Riemann property on $H^{1,1}(X)$. A Bogomolov pair is a pair of cohomology classes that generalizes the classical Bogomolov-Gieseker inequality for semistable sheaves.

We give a list of known Hodge-Riemann pairs (essentially all taken from my previous work with Toma). We are able to show that nearly all these known Hodge-Riemann pairs are also Bogomolov pairs, and conjecture that this is always the case.

[42] Realization of Cohomology Classes in Grassmannians

with I. Coskun
Preprint: arXiv:2509.03747

In this work we look at which cohomology classes in the Grassmannian $G(k,n)$ can be represented by an irreducible variety. This question can also be asked over $\mathbb{Q}$ by which we mean to find which cohomology classes have a positive multiple that can be represented by an irreducible variety. In this paper we solve this problem for dimension 2, dimension 3, codimension 2, and codimension 3 classes in all Grassmannians. We also do more for other specific Grassmannians.

This work was influenced by work of Huh and others. Huh in particular has looked at which cohomology classes in products of projective spaces can be represented by an irreducible variety, and with collaborators has solved this completely in dimension 2. All of this is related to Huh’s work on Lorentzian polynomials since the volume polynomial of an irreducible variety is Lorentzian by the Hodge-Riemann relations. The problem of classifying which cohomology classes on products of projective spaces and for Grassmannians are represented by an irreducible variety are related, and both appear to be extremely hard. My hope is that the problem for Grassmannians is easier (at least over $\mathbb{Q}$).

[41] Accessibility for the Working Mathematician

Preprint: arXiv:2505.22667

This document is an informal introduction to the topic of creating accessible electronic documents, in particular those that are created by mathematics teachers and researchers. When I came to this topic I was hoping to find a one-page “guide” that would tell me all that I needed to know. Upon exploration I now understand that asking for such a guide was a little naive since the topic is a little more complicated (but not that complicated), and so I set about writing this slightly longer guide.

I hope that this will be useful to others, at least it may save them time that I spent gathering this knowledge. I hope that future versions of this document will contain more detailed user documentation about how to write accessible documents using TeXML and/or LaTeXML and/or Word.

[40] Uniform boundedness of semistable pure sheaves on projective manifolds

with M. Pavel and M. Toma
Preprint: arXiv2403.12855

When considering moduli spaces of sheaves, a first property that one wants to prove is boundedness to ensure that these moduli spaces are of finite type (i.e. have finitely many irreducible components). For sheaves one usually imposes some kind of stability notion to ensure this boundedness. Once such moduli spaces have been constructed, it is interesting to ask how they change when this choice of stability changes and to do this in the best possible way one really wants some kind of uniform boundedness (that holds, say, as the stability condition varies within a compact set).

For torsion-free sheaves the first general uniform boundedness result I am aware of is that of Greb–Toma (expanded upon in [21]). In this paper we give the first general uniform boundedness result for pure sheaves (specifically it is the first result I am aware of that provides uniform boundedness for pure sheaves of any rank that holds for an infinite collection of slope-stable conditions). We do even better for pure sheaves of rank 2, answering in particular a question of Dominic Joyce coming from his work on enumerative invariants from moduli spaces of pure sheaves and their wall-crossing phenomena.

More should be true, and we have conjectures about even stronger uniform boundedness results that we hope to address in a future work.

[39] Schur positivity of difference of products of derived Schur polynomials

with K. Wu
Preprint: arXiv:2403.04101

This follows up on work in [35] in which we consider again our so-called derived Schur polynomials $s_{\lambda}^{(i)}$ defined by the rule

\[s_\lambda(x_1+t,\ldots,x_n+t) = s_\lambda(x_1,\ldots,x_n) + t\, s_\lambda^{(1)}(x_1,\ldots,x_n) + t^2 s_\lambda^{(2)}(x_1,\ldots,x_n) + \cdots\]

In [35] we show that for any Schur polynomial $s_{\lambda}$ the quantity

\[p := (s_{\lambda}^{(i)})^2 - s_{\lambda}^{(i-1)} s_{\lambda}^{(i+1)}\]

is positive, in the sense that $p(x_1,\ldots,x_n) \ge 0$ whenever $x_1,\ldots,x_n \ge 0$. This should be thought of as a kind of log-concavity for these derived Schur polynomials.

In this paper we conjecture that actually a much stronger log-concavity condition holds, namely that $p$ is always Schur positive, and we prove this conjecture for certain partitions $\lambda$ and verify it numerically for other partitions.

Update: This conjecture has been proved by Lu-Zheng arXiv:2506.16992.

[38] Harmonic Interpolation and a Brunn-Minkowski Theorem for Random Determinants

with D. Witt Nyström
In Convex and Complex: Perspectives on Positivity in Geometry, Publication: Contemporary Mathematics Publication Year: 2025; Volume 810 Preprint: arXiv:2310.09697

This builds on [36]. We give a self-contained description of our harmonic interpolation of convex bodies, and prove a strong form of the Brunn–Minkowski inequality and characterize its equality case. As an application we improve a theorem of Berndtsson on the volume of slices of a pseudoconvex domain. We furthermore apply this to prove subharmonicity of the expected absolute value of the determinant of a matrix of random vectors through the connection with zonoids.

[37] Dually Lorentzian Polynomials

with H. Seuss and T. Wannerer
Monatsh Math 208, 495–524 (2025) Preprint: arXiv:2304.08399

One of the questions that arose from my work relating Schur polynomials and the Hodge-Riemann property (say from [34,35]) was whether this gave genuinely new inequalities between characteristic classes of ample line bundles (or Kahler classes) or whether these were combinatorial consequences of known inequalities. In this paper we prove that it is the latter, and the inequalities that arise are combinatorial consequences of the mixed Alexandrov-Fenchel inequality.

In fact we go further, by introducing a notion of dually Lorentzian polynomial, and prove that if $s\in\mathbb{R}[x_1,\ldots,x_n]$ is dually Lorentzian then the operator

\[s(\partial_{x_1},\ldots,\partial_{x_n}) : \mathbb{R}[x_1,\ldots,x_n] \to \mathbb{R}[x_1,\ldots,x_n]\]

takes (strictly) Lorentzian polynomials to (strictly) Lorentzian polynomials. Since Schur polynomials are dually Lorentzian we can use this along with the classical mixed Alexandrov-Fenchel inequality to reprove all of the characteristic class inequalities deduced in [34].

In fact we get much more since not only do we see that these inequalities extend beyond Schur polynomials, but they also hold whenever one has a “Kahler-package”. As such we get generalized Alexandrov–Fenchel inequalities also in the context of mixed volumes and valuations.

The paper does not negate earlier work entirely: in [34] the main focus was on ample vector bundles, a case where it is unclear how Lorentzian polynomial theory applies; and the current paper has restrictions on the number of Kahler classes considered, restrictions absent in [35] (we expect these can be removed in future work).

[36] Interpolation, Prekopa and Brunn–Minkowski for F-subharmonicity

with D. Witt Nyström
Advances in Mathematics,Volume 436,2024 Preprint: arXiv:2206.00576

The Brunn–Minkowski inequality is one of the central results in convex geometry. One version states that if $A_0$ and $A_1$ are convex bodies and

\[A_t := t A_1 + (1 - t) A_0, \qquad t \in [0,1],\]

then $\mathrm{vol}(A_t)$ is log-concave in $t$.

We generalize this to infinite families of convex sets $A_t$ parameterized by the boundary of a suitable domain $\Omega \subset \mathbb{R}^n$. Using the Harvey–Lawson theory of $F$‑subharmonicity, we construct an interpolation of these convex bodies for each subequation $F$.

Under mild assumptions on $F$, this family satisfies:

In the special case where $F$ is the usual subharmonicity condition, we obtain the harmonic interpolation, which admits a set-valued integral representation.

This framework likely contains many previously known Brunn–Minkowski generalizations and suggests deeper connections between convexity, PDEs, and interpolation theory.

[35] Hodge–Riemann Relations for Schur Classes in the Linear and Kähler Cases

with M. Toma International Mathematics Research Notices, 2023, Issue 16, pp. 13780–13816 Preprint: arXiv:2202.13816

This is the third paper in a series on the Hodge–Riemann property for Schur classes. The main advance here is a linear algebra machine abstracted from the proof in [32]. Together with [34], this gives a unified approach to some of the main results in [32].

The original geometric motivation was to prove that Schur polynomials of Kähler forms satisfy the Hodge–Riemann relations. Rational classes can be treated with algebraic geometry; irrational ones required this new linear‑algebraic approach.

On tori with maximal Picard rank, every Kähler class is approximable by rational classes, giving the weak Hodge–Riemann property. The linear algebra machine upgrades this to the full property. A globalization argument (related to work of Gromov) extends this to all manifolds.

A purely linear algebra proof — avoiding geometry entirely — remains open.

[34] On Hodge–Riemann Cohomology Classes

with M. Toma
In Birational Geometry, Kähler–Einstein Metrics and Degenerations, Springer Proc. Math. Stat., vol. 409, pp. 763–793
Preprint: arXiv:2106.11285

Continuing the theme of [32], this paper studies Hodge–Riemann properties of Schur classes of nef and ample vector bundles.
The main theorem here is slightly weaker than in [32], but the proofs are much simpler.

Rather than showing that Schur classes of nef bundles have the Hodge–Riemann property, we show that they arise as limits of classes with this property — sufficient for most applications.

Additional results include:

[33] Twisted Kähler–Einstein metrics

with G. Székelyhidi
Pure and Applied Mathematics Quarterly 17(3): 1025-1044 (June 2021) Preprint: arXiv:1911.03442

We prove a twisted version of the Yau–Tian–Donaldson conjecture.

Let $M$ be a Fano manifold and $T \to M$ a line bundle with a smooth semi-positive representative $\beta \in c_1(T)$ expressible as

\[\beta = \int_{|T|} [D] \, d\mu(D).\]

A twisted Kähler–Einstein metric is a Kähler metric $\omega$ satisfying

\[\mathrm{Ric}(\omega) = \omega + \beta.\]

Main theorem: If $(M,\beta)$ is K-stable, then $M$ admits a twisted Kähler–Einstein metric.

This builds on the continuity‑method framework of Datar–Székelyhidi and recent results of Liu–Székelyhidi.

[32] Hodge–Riemann bilinear relations for Schur Classes of Ample Bundles

with M. Toma
Annales Scientifiques de l École Normale Supérieure 56(1):197-241**
Preprint: arXiv:1905.13636

We extend the Hard Lefschetz theorem and Hodge–Riemann bilinear relations to new classes.

If $X$ is a projective manifold of dimension $d$ and $E$ is a holomorphic vector bundle, then for any partition $\lambda$ of $d-2$, the Schur class

\[s_\lambda(E) \in H^{d-2,d-2}(X)\]

satisfies:

Consequences include new Hodge‑index‑type inequalities such as log‑concavity of

\[i \mapsto \int_X c_i(E)\, h^{d-i},\]

a higher‑rank analogue of Khovanskii–Tessier inequalities.

[31] Differentiability of the argmin function and a minimum principle for semiconcave subsolutions

with D. Witt Nyström
Journal of Convex Analysis 27(3): 811–832 (2020) Preprint: arXiv:1808.04402

This paper has two main components:

1. Differentiability of the argmin function
For $f : \mathbb{R}^n \to \mathbb{R}$, define the marginal function

\[g(x) = \inf_y f(x,y),\]

and the set‑valued function

\[\mathrm{argmin}_f(x) = \{y : f(x,y) \le f(x,z) \ \forall z\}.\]

Assume:

  1. $f(x,y) + \frac{\kappa}{2}|x|^2 - \frac{\sigma}{2}|y|^2$ is convex,
  2. for each $x$, there is a unique minimizer $\gamma(x)$.

Then $\gamma$ is differentiable almost everywhere.

2. A minimum principle for subequations
This generalizes earlier results (including [29]) to very general subequations using viscosity techniques.

[30] The Minimum Principle for Convex Subequations

with D. Witt Nyström
Journal of Geometric Analysis 32:28 (2022)
Preprint: arXiv:1806.06033

For convex functions $f$ and $g$, $\max{f,g}$ is convex but $\min{f,g}$ generally is not.

However, if $f(x,y)$ is convex in $(x,y)$, then the marginal function

\[g(y) = \min_x f(x,y)\]

is convex in $y$. This is the classical “minimum principle,” with a complex-analytic analogue due to Kiselman.

This paper generalizes the minimum principle by introducing product subequations.
If $F$ is a convex subequation and $\phi$ is $F$‑subharmonic, then the associated marginal function satisfies a generalized minimum principle.

This extends convexity theory to a wide new class of degenerate elliptic PDEs.

[29] The Dirichlet Problem for the Complex Homogeneous Monge–Ampère Equation

with D. Witt Nyström
In Modern Geometry: A Celebration of the Work of Simon Donaldson, Proc. Sympos. Pure Math. 99 (2018), pp. 289–330
Preprint: arXiv:1712.00405

This survey introduces the complex Homogeneous Monge–Ampère Equation (HMAE) both locally on $\mathbb{C}^n$ and globally on families of compact Kähler manifolds over Riemann surfaces with boundary.

We summarize earlier work relating the Hele–Shaw flow to the HMAE (including [19], [22], [26]) and extend this to general compact Riemann surfaces, beyond earlier cases such as $\mathbb{P}^1$ or the unit disc.

New results include:

Subsequent improvements by McCleary show strict monotonicity of weak Hele–Shaw flows on any compact Riemann surface.

[28] Stable Maps in Higher Dimension

with R. Dervan
Mathematische Annalen 374(3–4): 1033–1073 (2019)
Preprint: arXiv:1708.09750

We introduce a notion of K‑stability for morphisms, not only varieties, inspired by Grothendieck’s viewpoint that properties of spaces should be viewed through the lens of morphisms.

K‑stable maps appear naturally, such as:

Main result: Construction of a projective moduli space of stable general type maps, characterized by semi-log-canonical singularities.
This generalizes both:

The theory also suggests that moduli of Fano stable maps should exist, analogous to the relative simplicity of Fano moduli compared to general-type moduli.

[27] On the Maximal Rank Problem for the Complex Homogeneous Monge–Ampère Equation

with D. Witt Nyström
Analysis & PDE 12(2): 493–503 (2019)
Preprint: arXiv:1610.02280

Classically, convex solutions of certain elliptic PDEs satisfy constant rank properties for their Hessians.
We investigate an analogue for the complex HMAE.

Main result: The constant rank property fails.

We construct boundary data on the unit disc whose HMAE solution becomes completely degenerate on an open set, hence the Hessian rank is not constant.

We conjecture that the phenomenon we observe — regularity on a nonempty open set and complete degeneracy on its complement — should be generic.

[26] A Master Space for Moduli Spaces of Gieseker‑Stable Sheaves

with D. Greb and M. Toma
Transformation Groups 24(2): 379–401 (2019)
Preprint: arXiv:1605.06642

Building on [20], we show that any two moduli spaces of multi‑Gieseker‑stable sheaves of fixed topological type arise as GIT quotients of a single master space. Thus all such moduli spaces are connected by a finite sequence of Thaddeus flips.

Unlike earlier results, we allow degenerate stability parameters.
A corollary: moduli spaces of Gieseker‑stable sheaves with respect to arbitrary polarizations are related by Thaddeus flips.

This construction also refines [21], where intermediate spaces can themselves be identified as moduli spaces.

[25] K‑stability for Kähler manifolds

with R. Dervan
Mathematics Research Letters 24(3): 689–739 (2017)
Preprint: arXiv:1602.08983

We extend K‑stability to general (not necessarily projective) Kähler manifolds.

Following Odaka–Wang, we view the Donaldson–Futaki invariant as a topological invariant of the total space of a test configuration.
This perspective allows the definition of K‑stability in the Kähler setting.

Main theorem: We prove boundedness (resp. coercivity) of the Mabuchi functional implies K‑semistability (resp. uniform K‑stability).

In particular:

Dyrefelt independently obtained similar results using different methods.

[24] Moduli of vector bundles on higher-dimensional base manifolds — construction and variation

with D. Greb and M. Toma
International Journal of Mathematics 27(7): 1650054 (2016)
Preprint: arXiv:1503.00319

A survey of recent progress on moduli of vector bundles over higher-dimensional manifolds, summarizing results from [20], [21], and [26].

[23] Quantization of Hitchin’s equations for Higgs Bundles I

with M. Garcia‑Fernandez and J. Keller
Preprint: arXiv:1601.04960

Motivated by earlier work ([18]), we develop a new parameter space adapted to quantization of Higgs bundles, covering decorated bundles beyond the reach of [18].

We construct an algebraic framework for quantizing Hermitian metrics solving the Hitchin equations over projective manifolds.

We had planned to do the following, but it never materialized: - extend known quantization results for bundles to the Higgs setting,
- compare the resulting structures with natural complex-symplectic geometry on the cotangent bundle of the moduli space.

[22] Applications of the Duality between the Complex Monge–Ampère Equation and the Hele–Shaw flow

with D. Witt Nyström
Annales de l’Institut Fourier 69(1): 1–30 (2019)
Preprint: arXiv:1509.02665

Building on [19], we use the fact (due to Berndtsson) that almost any increasing planar flow arises as the Hele–Shaw flow associated to some permeability. This yields a method of constructing Kähler metrics on $\mathbb{P}^1$ or the unit disc with prescribed properties.

A key consequence: starting from a planar flow, we obtain geodesic rays in the space of Kähler metrics that are smooth as long as the Hele–Shaw flow remains simply connected; if it becomes non-simply-connected, the geodesic acquires controlled singularities.

We produce examples of:

[21] Semi-continuity of Stability for Sheaves and Variation of Gieseker Moduli Spaces

with D. Greb and M. Toma
Journal für die reine und angewandte Mathematik (Crelle) vol. 2019, no. 749, 2019, pp. 227-265) Preprint: arXiv:1501.04440

We continue the study of variation of moduli spaces as stability conditions change.
In [20], certain moduli were shown to be related by Thaddeus flips; here we investigate whether the intermediate spaces also have modular interpretations.

A major obstacle: many stability notions (e.g., Gieseker stability) fail to satisfy a needed semi-continuity property.

Key contribution: identification of uniform paths of stability conditions that do satisfy semi-continuity.
There are enough such paths to deduce that, on threefolds, most Gieseker moduli spaces are related by Thaddeus flips via moduli of multi‑Gieseker‑stable sheaves.

The ideas should extend to manifolds of any dimension.

[20] Variation of Gieseker Moduli Spaces of Sheaves via Quiver GIT

with D. Greb and M. Toma
Geometry & Topology 20(3): 1539–1610 (2016)
Preprint: https://arxiv.org/abs/1409.7564

We resolve difficulties in describing how moduli spaces of sheaves change as the polarization varies on manifolds of arbitrary dimension.

The key innovation is multi‑Gieseker stability, depending on several polarizations.
Using quiver constructions of Álvarez‑Cónsul–King, we show:

Consequences include:

[19] Harmonic Discs of Solutions to the Complex Homogeneous Monge–Ampère Equation

with D. Witt Nyström
Publications Mathématiques de l’IHÉS 122(1): 315–335 (2015)
Preprint: arXiv:1408.6663

Geodesics in the space of Kähler potentials correspond to solutions of the complex Homogeneous Monge–Ampère Equation (HMAE). Understanding their regularity is a central problem.

We perform a systematic study in the simplest nontrivial case: the base $\mathbb{P}^1$.

Main result: A duality between HMAE solutions and the Hele–Shaw flow and regularity of the HMAE solution corresponds exactly to simple connectivity of the Hele–Shaw domains.

Using this:

[18] Balanced Metrics on Twisted Higgs Bundles

with M. Garcia‑Fernandez
Mathematische Annalen 367(3–4): 1429–1471 (2017)
Preprint: arXiv:1401.7108

We develop a framework for quantization of metrics on twisted Higgs bundles.

Balanced metrics in this setting are shown to approximate solutions to the Hitchin equations. We also prove a finite‑dimensional Hitchin–Kobayashi correspondence: existence of a balanced metric is equivalent to a suitable algebro‑geometric stability condition.

Motivation came partly from physics (Donagi–Wijnholt). In the Higgs‑free case, these results reduce to Wang’s classical theory. A subtle point is identifying the correct finite‑dimensional parameter spaces for quantization — several viable choices exist, but only some interact well with balanced metrics.

[17] Homogeneous Monge–Ampère Equations and Canonical Tubular Neighbourhoods in Kähler Geometry

with D. Witt Nyström
International Mathematics Research Notices 2017(23): 7069–7108
Preprint: arXiv:1403.3282

We study tubular neighborhoods in Kähler geometry.
Unlike in differential geometry, complex submanifolds rarely admit holomorphic tubular neighborhoods.

Using regular solutions of the HMAE on the deformation to the normal cone, we construct a canonical tubular neighborhood intertwining holomorphic and symplectic structures.

This construction yields:

The relationship between these formulas is of independent interest.

[16] Asymptotics of Partial Density Functions for Divisors

with M. Singer
Journal of Geometric Analysis 27(3): 1803–1854 (2017)
Preprint: arXiv:1312.1145

Zooming in on a Kähler manifold locally resembles $\mathbb{C}^n$, a fact reflected in the Kähler identities and in Hörmander-type global analytic techniques.

Here we study a phenomenon that is not fully local: the asymptotics of partial density functions associated to a divisor.

For $\epsilon > 0$ and $k \in \mathbb{N}$, consider monomials $z^{\epsilon k}, z^{\epsilon k + 1}, \ldots$ normalized with respect to the standard $L^2$ inner product using the weight $e^{-k|z|^2}$. The sum of their squared pointwise norms defines a partial density function.
As $k \to \infty$, this function:

We show an analogous picture holds near divisors in general Kähler manifolds under an $S^1$‑symmetry hypothesis, and give a full asymptotic expansion whose leading term is the error function. The work suggests a connection with the tubular neighborhoods studied in [17], especially the possibility of similar asymptotics without $S^1$ symmetry.

[15] The Hele–Shaw Flow and Moduli of Holomorphic Discs

with D. Witt Nyström
Compositio Mathematica 151(12): 2301–2328 (2015)
Preprint: arXiv:1212.2337

The Hele–Shaw flow models fluid injected between two closely spaced plates. In this paper, we study a version where the medium has spatially varying permeability.

We show a close connection between such flows and holomorphic discs whose boundaries lie on a specific Lagrangian submanifold.
Applications include:

We also apply these ideas to quadrature domains, showing the space of quadrature domains is a smooth manifold of computable dimension.

This paper is the first in a series linking Hele–Shaw flow with complex geometry; it is the only one that uses holomorphic discs directly to prove results about the flow.

[14] Envelopes of plurisubharmonic metrics with prescribed singularities

with D. Witt Nyström
Annales de la Faculté des Sciences de Toulouse, Ser. 6, Vol. 26(3): 687–727 (2017)
Preprint: arXiv:1210.2220

We study extremal plurisubharmonic envelopes associated with a background potential and a prescribed singularity type.

Main result: such envelopes are $C^{1,1}$ on the locus where they are locally bounded.

The proof follows Berman’s approach, itself inspired by Bedford–Taylor theory.

Connections:

We also compute a product formula for envelopes, analogous to classical formulas for Siciak extremal functions, using partial density functions and Mustaţă’s summation formula.

[13] Analytic test configurations and geodesic rays

with D. Witt Nyström
Journal of Symplectic Geometry 12(1): 125–169 (2014)
Preprint: arXiv:1101.1612

A central conjecture (Yau–Tian–Donaldson) connects cscK metrics with algebraic stability via test configurations.

We introduce analytic test configurations, defined as convex curves of singularity types with mild regularity.
They are far easier to manipulate than algebraic test configurations: one may interpolate linearly between them.

Main theorem: Each analytic test configuration produces a weak geodesic ray via the Legendre transform of the associated family of envelopes (as in [14]). Moreover, every weak geodesic ray arises this way.

This generalizes Phong–Sturm’s construction for algebraic test configurations, and we show how to recover their results within our framework.

[12] Limits of balanced metrics on vector bundles and polarised manifolds

with M. Garcia‑Fernandez
Proceedings of the London Mathematical Society 106(5): 1143–1156 (2013)
Preprint: arXiv:1111.2819

We combine the Hitchin–Kobayashi correspondence for vector bundles, and the Yau–Tian–Donaldson framework for constant scalar curvature Kähler metrics, to study stability of pairs $(X,E)$ where both the manifold and the bundle vary. The stability notion includes a parameter; for special values, the analytic objects are coupled equations intertwining:

The motivation was to find a framework encompassing coupled equations previously introduced by Álvarez‑Cónsul, Garcia‑Prada, and Garcia‑Fernandez from physical considerations.

The relationship with those equations remains an interesting direction for future development.

[11] A note on Chow stability of the projectivisation of Gieseker stable bundles

with J. Keller
Journal of Geometric Analysis 24(3): 1526–1546 (2014)
Preprint: arXiv:1110.4489

A common principle is that stability of the projectivization $\mathbb{P}(E)$ (as a polarized manifold with small fibers) should correspond to stability of the underlying vector bundle $E$ and stability of the base.

For Mumford‑stable bundles, the natural stability notion for $\mathbb{P}(E)$ is K‑stability. Here we instead consider the case where $E$ is only Gieseker‑stable.

We show that in this situation, the corresponding notion for $\mathbb{P}(E)$ should be Chow stability. We prove results supporting this viewpoint and give examples of manifolds that are Chow stable but not asymptotically Chow stable.

[10] Weighted Bergman Kernels on Orbifolds

with R. P. Thomas
Journal of Differential Geometry 88: 87–108 (2011)
Preprint: arXiv:0907.5215

Classical links between cscK metrics and stability rely on asymptotics of the Bergman (density of states) function.
However, for orbifolds with cyclic quotient singularities, the usual Bergman function develops distributional terms at the singular points.

We introduce a weighted sum of Bergman kernels whose distributional terms cancel, yielding a smooth asymptotic expansion exactly analogous to the manifold case.

This leads to a meta‑theorem:
Any manifold result relying on Bergman kernel asymptotics extends to orbifolds if one uses this weighted sum.

Companion work [9] develops further applications.

[9] Weighted Projective Embeddings, Stability of Orbifolds, and Constant Scalar Curvature Kähler Metrics

with R. P. Thomas
Journal of Differential Geometry 88: 109–160 (2011)
Preprint: arXiv:0907.5214

We formulate a Yau–Tian–Donaldson type correspondence for orbifolds.

Key ingredients:

Main theorem:
If an orbifold admits a constant scalar curvature Kähler metric, then it is K‑semistable.

We also define an orbifold version of slope stability, yielding obstructions to existence of cscK orbifold metrics.
This connects classical results of Troyanov, Ghigi–Kollár, and Rollin–Singer to algebro‑geometric stability.

Extensions include possible orbifold analogues of Stoppa’s results on K‑stability.

[8] An inequality between Multipoint Seshadri constants

with J. Roé
Geometriae Dedicata 140: 175–181 (2009)
Preprint: arXiv:0804.1662

We generalize an inequality of Roé for multipoint Seshadri constants.
The method: divide the points into subcollections, perform a degeneration to the normal cone, and apply semicontinuity of Seshadri constants.

This technique parallels methods used by Biran in related work.

[7] Slope Stability and Exceptional Divisors of High Genus

with D. Panov
Math. Ann. 343(1): 79–101 (2009)
Preprint: arXiv:0710.4078

This paper studies slope stability for polarized surfaces ($\dim = 2$).

Main result:
To test slope stability, one need only consider effective divisors, not arbitrary subschemes.
Moreover, these divisors must be exceptional in a precise sense (e.g., negative self‑intersection, genus $\ge 2$).

The resulting criterion is concrete and computable, and yields explicit examples.

It is shown that slope stability is strictly weaker than K‑stability.
The search for a strengthened slope‑type notion equivalent to K‑stability remains active (work of Odaka, Fujita, etc.).

[6] Deligne pairings and the Knudsen–Mumford expansion

with D. Phong and J. Sturm
Journal of Differential Geometry 78(3): 475–496 (2008)
Preprint: arXiv:0612555

We revisit the CM line bundle, which appears in the algebraic formulation of K‑stability via the Hilbert–Mumford criterion.
The main result is an expression for the CM line bundle of a family of polarized varieties using the Deligne pairing construction.

This yields equivalences between multiple formulations of the CM line bundle (e.g., Paul–Tian and Zhang).
We connect the algebraic Donaldson–Futaki invariant with asymptotic behavior of the K‑energy along test configurations.

This circle of ideas appears prominently in later foundational work of Berman on the Yau–Tian–Donaldson conjecture.

[5] Seshadri constants on symmetric products of curves

Math. Res. Lett. 14(1): 63–75 (2007)
Preprint: arXiv:0608224

Seshadri constants measure local positivity of line bundles.
Although all known examples are rational, it is widely suspected that irrational examples exist.

This paper studies Seshadri constants on symmetric products of curves and formulates several conjectures about their (ir)rationality.

Using a degeneration to the Franchetta curve, following Ciliberto–Kouvidakis, we show that these conjectures follow from the classical Nagata conjecture.

The results raise the possibility that Seshadri constants on symmetric products may be as difficult to compute as the Nagata problem itself.

[4] A note on Positivity of the CM line bundle

with J. Fine
International Mathematics Research Notices (2006), Art. ID 95875
Preprint: arXiv:0605302

The CM line bundle arises when interpreting K‑stability via Geometric Invariant Theory.
However, its positivity properties are subtle.

In this paper, we study the CM line bundle as the leading term in an expansion of determinant line bundles.

Main results:

These examples illustrate complexities in the relationship between GIT and K‑stability.

[3] Unstable products of smooth curves

Inventiones Mathematicae 165(1): 153–162 (2006)
Preprint: arXiv:0506447

A folklore conjecture claimed that any Kähler class on a manifold with negative first Chern class admits a constant scalar curvature Kähler (cscK) metric.

Using slope stability from [1], we disprove this:
certain products of smooth curves do not admit cscK metrics.

Although the counterexamples initially involve special curves, later results show similar behavior for self‑products of any smooth curve.
This reflects the nongeneric nature of being a product in moduli.

[2] A study of the Hilbert–Mumford criterion for the stability of projective varieties

with R. P. Thomas
Journal of Algebraic Geometry 16(2): 201–255 (2007)
Preprint: arXiv:0412519

We analyze K‑stability by restricting the class of test configurations that need to be considered.

Under technical hypotheses, the entire problem reduces to slope stability, as in [1].
This reduction holds unconditionally for curves — yielding the first purely algebro‑geometric proof that smooth curves are K‑stable.

The work anticipates later developments by Donaldson, Odaka, Székelyhidi, Li–Xu, and others.

[1] An obstruction to the existence of constant scalar curvature Kähler metrics

with R. P. Thomas
Journal of Differential Geometry 72(3): 429–466 (2006)
Preprint: arXiv:0412518

This work and [2] grew out of my PhD thesis.

We introduce slope stability for manifolds, modeled on Mumford stability for sheaves.
Slope stability is implied by K‑stability, and thus provides computable obstructions to the existence of cscK metrics.

Results include:

Odaka later proved analogous results for K‑stability, paralleling Yau’s resolution of the Calabi conjecture.