arXiv subjects

Carlo Scarpa

Publications and source records attributed to Carlo Scarpa.

13 recordsLinked to original sources

arXiv2026

The toric CR Yamabe problem

We study the equivariant CR Yamabe problem for a fixed-point-free torus action on a co-oriented compact contact manifold. Such examples arise naturally in the study of Sasakian manifolds with constant scalar curvature. In the toric case, we show that this problem is equivalent to a kind of boundary value problem for an elliptic PDE on a pair of functions defined on a convex, polyhedral domain. We provide examples of solutions and prove that many contact toric manifolds admit compatible CR structures with distinct signs of CR Yamabe invariants.

math.DG↗

arXiv2026

Yamabe-type problems on compact Hermitian manifolds

We introduce and study a one-parameter Hermitian deformation of the Yamabe problem on compact complex manifolds. The deformation is defined by adding natural torsion terms to the Riemannian scalar curvature, and includes both the classical Yamabe equation and a scalar-curvature equation arising in locally conformally K\"ahler geometry as a momentum map. We analyze criteria for the existence of solutions, and discuss several examples.

math.DG↗

arXiv2025

The CR Yamabe invariant and constant scalar curvature Sasaki metrics

We propose a new approach to the existence of constant transversal scalar curvature Sasaki structures drawing on ideas and tools from the CR Yamabe problem, establishing a link between the CR Yamabe invariant, the existence of Sasaki structures of constant transversal scalar curvature, and the K-stability of Sasaki manifolds. Assuming that the Sasaki-Reeb cone contains a regular vector field, we show that if the CR Yamabe invariant of a compact Sasaki manifold attains a specific value determined by the geometry of the Reeb cone, then the Sasaki manifold is K-semistable. Under the additional assumption of non-positive average scalar curvature, the CR Yamabe invariant attains this topological value if the manifold admits approximately constant scalar curvature Sasaki structures, and we also show a partial converse. As an application, we provide a new numerical criterion for the K-semistability of polarised compact complex manifolds.

math.DG↗

arXiv2025

Polynomial stability conditions for vector bundles: Positivity, equivariance and blow-ups

We introduce the notion of P-critical connections for hermitian holomorphic vector bundles over compact balanced manifolds: integrable hermitian connections whose curvature solves a polynomial equation. Such connections include HYM and dHYM connections, as well as solutions to higher rank Monge-Amp\`ere or J-equations, and are a slight generalisation of Dervan-McCarthy-Sektnan's Z-critical connections motivated by Bayer's polynomial Bridgeland stability conditions. The associated equations come with a moment map interpretation, and we provide numerical conditions that are expected to characterise existence of solutions in suitable cases: P-positivity and P-stability. We then provide some devices to check those numerical conditions in practice. First, we observe that P-positivity is equivalent to its equivariant version over T-varieties. In the toric case, we thus obtain an explicit finite set of subvarieties to test P-positivity on, independently on the choice of the polynomial equation. We also introduce equivariant P-stability and discuss its relation to P-stability. Secondly, we show that a uniform version of P-positivity is preserved by pulling back along a blow-up of points. We apply those results to some examples, such as blow-ups of Hirzebruch surfaces, or a Fano 3-fold.

math.AG↗

arXiv2024

$Z$-critical equations for holomorphic vector bundles on K\"ahler surfaces

We prove that the existence of a $Z$-positive and $Z$-critical Hermitian metric on a rank 2 holomorphic vector bundle over a compact K\"ahler surface implies that the bundle is $Z$-stable. As particular cases, we obtain stability results for the deformed Hermitian Yang-Mills equation and the almost Hermite-Einstein equation for rank 2 bundles over surfaces. We show examples of $Z$-unstable bundles and $Z$-critical metrics away from the large volume limit.

math.DG↗

arXiv2023

The Einstein-Hilbert functional and the Donaldson-Futaki invariant

Given a Kaehler manifold polarised by a holomorphic ample line bundle, we consider the circle bundle associated to the polarisation with the induced transversal holomorphic structure. The space of contact structures compatible with this transversal structure is naturally identified with a bundle, of infinite rank, over the space of Kaehler metrics in the first Chern class of the polarisation. We show that the Einstein-Hilbert functional of the associated Tanaka-Webster connections is a functional on this bundle, whose critical points are constant scalar curvature Sasaki structures. In particular, when the group of automorphisms is discrete, these critical points correspond to constant scalar curvature Kaehler metrics in the first Chern class of the polarisation. We show that the Einstein--Hilbert functional satisfies some monotonicity properties along some one-parameter families of CR-contact structures that are naturally associated to test configurations, and that its limit on the central fibre of a test configuration is related to the Donaldson-Futaki invariant through an expansion in terms of an extra real parameter. As a by-product, we obtain an original proof that the existence of constant scalar curvature Kaehler metrics on a polarized manifold implies K-semistability. We also show that the limit of the Einstein-Hilbert functional on the central fibre coincides with the ratio of the equivariant index characters pole coefficients of the central fibre.

math.DG↗

arXiv2023

A K-energy functional for complexified K\"ahler classes

The K-energy functional is extended to complexified K\"ahler classes, providing a variational approach to study the scalar curvature equation with B-field introduced by Schlitzer and Stoppa. The extended K-energy is convex along geodesics in the space of almost calibrated representatives of the complexified K\"ahler class. This fact is used to show that, in some situations, solutions of the scalar curvature equation with B-field are unique in their class, up to pullbacks by reduced automorphisms of the manifold.

math.DG↗

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↗

arXiv2022

Scalar curvature and deformations of complex structures

We study a system of equations on a compact complex manifold, that couples the scalar curvature of a Kaehler metric with a spectral function of a first-order deformation of the complex structure. The system comes from an infinite-dimensional Kaehler reduction, which is a hyperkaehler reduction for a particular choice of the spectral function. The system can be formally complexified using a flat connection on the space of first-order deformations that are compatible with a Kaehler metric. We describe a variational characterization of the equations, a Futaki invariant for the system, and a generalization of K-stability that is conjectured to characterize the existence of solutions to the system. We verify a particular case of this conjecture in the context of toric manifolds.

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

The Hitchin-cscK system

We present an infinite-dimensional hyperkähler reduction that extends the classical moment map picture of Fujiki and Donaldson for the scalar curvature of Kähler metrics. We base our approach on an explicit construction of hyperkähler metrics due to Biquard and Gauduchon. The construction is motivated by how one can derive Hitchin's equations for harmonic bundles from the Hermitian Yang-Mills equation, and yields a system of moment map equations which modifies the constant scalar curvature Kähler (cscK) condition by adding a "Higgs field" to the cscK equation. In the special case of complex curves, we recover previous results of Donaldson, while for higher-dimensional manifolds the system of equations has not yet been studied. We study the existence of solutions to the system in some special cases. On a Riemann surface, we extend an existence result for Donaldson's equation to our system. We then study the existence of solutions to the moment map equations on a class of ruled surfaces which do not admit cscK metrics, showing that adding a suitable Higgs term to the cscK equation can stabilize the manifold. Lastly, we study the system of equations on abelian and toric surfaces, taking advantage of a description of the system in symplectic coordinates analogous to Abreu's formula for the scalar curvature.

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↗