arXiv subjects

Jacopo Stoppa

Publications and source records attributed to Jacopo Stoppa.

At least 19 recordsLinked to original sources

arXiv2026

Special Lagrangian smoothings, Calabi ansatz and stability conditions

As part of his work on special Lagrangian (sLag) submanifolds with isolated conical singularities, Joyce proved a criterion for the existence of sLag smoothings, along a small variation of complex structure, for the union of two connected, compact, embedded sLags, with the same phase, intersecting transversely. Here we construct infinitely many examples of pairs of non-compact, embedded sLags, of the same phase and with arbitrary dimension, intersecting only at infinity in a non-transverse way, which satisfy Joyce's criterion: along a small variation of complex structure, a sLag smoothing of their union exists on the stable locus where a slope inequality for periods of the holomorphic volume form holds. At least under a natural symmetry assumption, this slope inequality is also necessary for the existence of such smoothing. Our approach uses the Leung-Yau-Zaslow transform and the analysis of deformed Hermitian Yang-Mills connections with Calabi ansatz, due to Jacob and Sheu. In the unstable case, we prove that if a family of Lagrangian smoothings evolving under the natural Calabi-symmetric version of the mean curvature flow (due to Chan and Jacob) admits a limit, then this must be the union of the original sLags. As an application we show that in our examples, in dimension two, the condition for the existence of the sLag smoothing is in fact equivalent to the stability of the corresponding object in the Fukaya-Seidel category, with respect to a known Bridgeland stability condition imported from algebraic geometry, and in the unstable case the limit of the Calabi-symmetric mean curvature flow in our result coincides with the Harder-Narasimhan decomposition, consistently with a general conjecture of Joyce. A similar (although weaker) result also holds in dimension three.

math.DG↗

arXiv2025

Special Lagrangian sections and stability conditions on threefolds

We study a class of Lagrangian submanifolds, given by sections of a special Lagrangian fibration, contained in certain almost Calabi-Yau threefolds (mirrors of polarised toric threefolds satisfying suitable assumptions). We show that, for a Lagrangian section $L$ in this class, the shift $L[2]$ defines an object in the heart of a natural Bridgeland stability condition on the relevant Fukaya-Seidel category, and that if $L[2]$ is semistable with respect to this stability condition, then it is isomorphic to a special Lagrangian. For mirrors of weak Fanos, the central charge of the stability condition is very close to periods of the holomorphic volume form. These results are consistent with Joyce's interpretation of the Thomas-Yau conjecture. As part of the proof we describe a set of line bundles and polarisations on suitable toric threefolds for which semistability with respect to the Bridgeland stability conditions constructed by Bernardara-Macr\`i-Schmidt-Zhao implies the existence of a deformed Hermitian Yang-Mills connection.

math.AG↗

arXiv2025

Nakai-Moishezon criteria and the toric Thomas-Yau conjecture

We consider a class of Lagrangian sections $L$ contained in certain Calabi-Yau Lagrangian fibrations (mirrors of toric weak Fano manifolds). We prove that a form of the Thomas-Yau conjecture holds in this case: $L$ is Hamiltonian isotopic to a special Lagrangian section in this class if and only if a stability condition holds, in the sense of a slope inequality on objects in a set of exact triangles in the Fukaya-Seidel category. This agrees with general proposals by Li. We use the SYZ transform, the toric gamma theorem, and toric homological mirror symmetry in order to reduce the statement to one about supercritical deformed Hermitian Yang-Mills connections, known as the Nakai-Moishezon criterion. As an application, we prove that, on the mirror of a toric weak del Pezzo surface, if $L$ defines a Bridgeland stable object in the Fukaya-Seidel category in a natural sense, then it is Hamiltonian isotopic to a special Lagrangian section in the class. The converse also holds for the mirror of the projective plane blown-up at one or two points, and always holds assuming a conjecture of Arcara and Miles. When $L$ is Bridgeland unstable, we obtain a morphism from $L$ to a weak solution of the special Lagrangian equation with phase angle satisfying a minimality condition. These results are consistent with general conjectures due to Joyce. We discuss some generalisations, including a weaker analogue of our main result for general projective toric manifolds, and a similar obstruction, related to Lagrangian multi-sections, in a special case.

math.DG↗

arXiv2024

Toric mirrors and test configurations

We obtain results that relate Donaldson-Futaki type invariants (that is, the numerical invariants used to define K-stability for general polarised manifolds) for a toric polarised manifold and for a compactification of its mirror Landau-Ginzburg model, nearby the large volume limit. In general, these have the form of expansions containing terms which involve the base loci of certain linear systems determined by the Landau-Ginzburg potential (as expected from known constructions of compactified mirrors), and we give a condition under which these terms are subleading. As an application we show that recently proposed notions of K-stability involving elements of the extended K\"ahler moduli space, i.e. Z-stability for polarised varieties, appear naturally from considerations of mirror symmetry (as a mirror to classical K-stability).

math.AG↗

arXiv2024

Some applications of canonical metrics to Landau-Ginzburg models

It is known that a given smooth del Pezzo surface or Fano threefold $X$ admits a choice of log Calabi-Yau compactified mirror toric Landau-Ginzburg model (with respect to certain fixed K\"ahler classes and Gorenstein toric degenerations). Here we consider the problem of constructing a corresponding map $\Theta$ from a domain in the complexified K\"ahler cone of $X$ to a well-defined, separated moduli space $\mathfrak{M}$ of polarised manifolds endowed with a canonical metric. We prove a complete result for del Pezzos and a partial result for some special Fano threefolds. The construction uses some fundamental results in the theory of constant scalar curvature K\"ahler metrics. As a consequence $\mathfrak{M}$ parametrises $K$-stable manifolds and the domain of $\Theta$ is endowed with the pullback of a Weil-Petersson form.

math.AG↗

arXiv2023

Scattering diagrams and Jeffrey-Kirwan residues

We show that the consistent completion of an initial scattering diagram in $M_{\mathbb{R}}$ (for a finite rank lattice $M$) can be expressed quite generally in terms of the Jeffrey-Kirwan residues of certain explicit meromorphic forms, by using the Maurer-Cartan asymptotic analysis developed by Chan-Leung-Ma and Leung-Ma-Young. A similar result holds for the associated theta functions.

math.AG↗

arXiv2023

K-stability and large complex structure limits

We discuss how, under suitable assumptions, a K\"ahler test configuration admits a mirror Landau-Ginzburg model, giving a corresponding expression for the Donaldson-Futaki invariant as a residue pairing. We study the general behaviour of such mirror formulae under large scaling of the K\"ahler form. We exploit the observation that this scaling trivially preserves $K$-stability, but takes the mirror Landau-Ginzburg model to a large complex structure limit. In certain cases the mirror formulae for the Donaldson-Futaki invariant simplify in this limit. We focus on a special type of limiting behaviour, when the Donaldson-Futaki invariant concentrates at a single critical point of the Landau-Ginzburg potential, and show that this leads to new formulae for the Donaldson-Futaki invariant in terms of theta functions on the mirror. We provide a main application, which shows that such limiting behaviour actually occurs for test configurations in several nontrivial examples, both toric and non-toric, in the case of slope (in)stability for polarised surfaces.

math.AG↗

arXiv2022

Special representatives of complexified K\"ahler classes

Motivated by constructions appearing in mirror symmetry, we study special representatives of complexified K\"ahler classes, which extend the notions of constant scalar curvature and extremal representatives for usual K\"ahler classes. In particular, we provide a moment map interpretation, discuss a possible correspondence with compactified Landau-Ginzburg models, and prove existence results for such special complexified K\"ahler forms and their large volume limits in certain toric cases.

math.DG↗

arXiv2021

The HcscK equations in symplectic coordinates

The Donaldson-Fujiki Kähler reduction of the space of compatible almost complex structures, leading to the interpretation of the scalar curvature of Kähler metrics as a moment map, can be lifted canonically to a hyperkähler reduction. Donaldson proposed to consider the corresponding vanishing moment map conditions as (fully nonlinear) analogues of Hitchin's equations, for which the underlying bundle is replaced by a polarised manifold. However this construction is well understood only in the case of complex curves. In this paper we study Donaldson's hyperkähler reduction on abelian varieties and toric manifolds. We obtain a decoupling result, a variational characterisation, a relation to $K$-stability in the toric case, and prove existence and uniqueness under suitable assumptions on the ``Higgs tensor''. We also discuss some aspects of the analogy with Higgs bundles.

math.DG↗

arXiv2021

Log Calabi-Yau surfaces and Jeffrey-Kirwan residues

We prove an equality, predicted in the physical literature, between the Jeffrey-Kirwan residues of certain explicit meromorphic forms attached to a quiver without loops or oriented cycles and its Donaldson-Thomas type invariants. In the special case of complete bipartite quivers we also show independently, using scattering diagrams and theta functions, that the same Jeffrey-Kirwan residues are determined by the the Gross-Hacking-Keel mirror family to a log Calabi-Yau surface.

math.AG↗

arXiv2020

Deformed Hermitian Yang-Mills connections, extended gauge group and scalar curvature

The deformed Hermitian Yang-Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background Kähler metric, to be constant. In this paper we introduce and study dHYM equations with variable Kähler metric. These are coupled equations involving both the Lagrangian phase and the radius function, at the same time. They are obtained by using the extended gauge group to couple the moment map interpretation of dHYM connections, due to Collins-Yau and mirror to Thomas' moment map for special Lagrangians, to the Donaldson-Fujiki picture of scalar curvature as a moment map. As a consequence one expects that solutions should satisfy a mixture of K-stability and Bridgeland-type stability. In special limits, or in special cases, we recover the Kähler-Yang-Mills system of Álvarez-Cónsul, Garcia-Fernandez and García-Prada, and the coupled Kähler-Einstein equations of Hultgren-Witt Nyström. After establishing several general results we focus on the equations and their large/small radius limits on abelian varieties, with a source term, following ideas of Feng and Székelyhidi.

math.DG↗

arXiv2020

Examples of dHYM connections in a variable background

We study deformed Hermitian Yang-Mills (dHYM) connections on ruled surfaces explicitly, using the momentum construction. As a main application we provide many new examples of dHYM connections coupled to a variable background K\"ahler metric. These are solutions of the moment map partial differential equations given by the Hamiltonian action of the extended gauge group, coupling the dHYM equation to the scalar curvature of the background. The large radius limit of these coupled equations is the K\"ahler-Yang-Mills system of \'Alvarez-C\'onsul, Garcia-Fernandez and Garc\'ia-Prada, and in this limit our solutions converge smoothly to those constructed by Keller and T{\o}nnesen-Friedman. We also discuss other aspects of our examples including conical singularities, realisation as B-branes, the small radius limit and canonical representatives of complexified K\"ahler classes.

math.DG↗

arXiv2020

Scalar curvature and an infinite-dimensional hyperkähler reduction

We discuss a natural extension of the Kähler reduction of Fujiki and Donaldson, which realises the scalar curvature of Kähler metrics as a moment map, to a hyperkähler reduction. Our approach is based on an explicit construction of hyperkähler metrics due to Biquard and Gauduchon. This extension is reminiscent of how one derives Hitchin's equations for harmonic bundles, and yields real and complex moment map equations which deform the constant scalar curvature Kähler (cscK) condition. In the special case of complex curves we recover previous results of Donaldson. We focus on the case of complex surfaces. In particular we show the existence of solutions to the moment map equations on a class of ruled surfaces which do not admit cscK metrics.

math.DG↗

arXiv2019

Solutions to Donaldson's hyperkähler reduction on a curve

We study an infinite-dimensional hyperkähler reduction introduced by Donaldson and associated with the constant scalar curvature equation on a Riemann surface. It is known that the corresponding moment map equations admit special solutions constructed from holomorphic quadratic differentials. Here we obtain a more general existence result and so a larger hyperkähler moduli space.

math.DG↗

arXiv2019

A note on BPS structures and Gopakumar-Vafa invariants

We regard the work of Maulik and Toda, proposing a sheaf-theoretic approach to Gopakumar-Vafa invariants, as defining a BPS structure, that is, a collection of BPS invariants together with a central charge. Assuming their conjectures, we show that a canonical flat section of the flat connection corresponding to this BPS structure, at the level of formal power series, reproduces the Gromov-Witten partition function for all genera, up to some error terms in genus 0 and 1. This generalises a result of Bridgeland and Iwaki for the contribution from genus 0 Gopakumar-Vafa invariants.

math.AG↗

arXiv2019

A construction of Frobenius manifolds from stability conditions

A finite quiver $Q$ without loops or 2-cycles defines a 3CY triangulated category $D(Q)$ and a finite heart $A(Q)$. We show that if $Q$ satisfies some (strong) conditions then the space of stability conditions $Stab(A(Q))$ supported on this heart admits a natural family of semisimple Frobenius manifold structures, constructed using the invariants counting semistable objects in $D(Q)$. In the case of $A_n$ evaluating the family at a special point we recover a branch of the Saito Frobenius structure of the $A_n$ singularity $y^2 = x^{n+1}$. We give examples where applying the construction to each mutation of $Q$ and evaluating the families at a special point yields a different branch of the maximal analytic continuation of the same semisimple Frobenius manifold. In particular we check that this holds in the case of $A_n$, $n \leq 5$.

math.AG↗

arXiv2019

A quantized Riemann-Hilbert problem in Donaldson-Thomas theory

We introduce Riemann-Hilbert problems determined by refined Donaldson-Thomas theory. They involve piecewise holomorphic maps from the complex plane to the group of automorphisms of a quantum torus algebra. We study the simplest case in detail and use the Barnes double gamma function to construct a solution.

math.AG↗

arXiv2018

Torus equivariant K-stability

It is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group. We prove partial results towards this conjecture. We also show that it would give a new proof of the K-polystability of constant scalar curvature polarised manifolds.

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