Carl Tipler

• PhD
• Professor (Assistant) at Université de Bretagne Occidentale

Kaneyama and Klyachko have shown that any torus equivariant vector bundle of rank r over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}\mathbb {P}^n$$\end{...

We study the zero locus of the Futaki invariant on K-polystable Fano threefolds, seen as a map from the Kähler cone to the dual of the Lie algebra of the reduced automorphism group. We show that, apart from families 3.9, 3.13, 3.19, 3.20, 4.2, 4.4, 4.7 and 5.3 of the Iskovskikh–Mori–Mukai classification of Fano threefolds, the Futaki invariant of s...

We investigate hermitian Yang–Mills connections on pullback bundles with respect to adiabatic classes on the total space of holomorphic submersions with connected fibres. Under some technical assumptions on the graded object of a Jordan–Hölder filtration, we obtain a necessary and sufficient criterion for when the pullback of a strictly semistable...

Kaneyama and Klyachko have shown that any torus equivariant vector bundle of rank $r$ over $\mathbb{CP}^n$ splits if $r < n$. In particular, any such bundle is not slope stable. In contrast, we provide explicit examples of stable equivariant reflexive sheaves of rank $r$ on any polarised toric variety $(X, L)$, for $2 \leq r < \mathrm{dim}(X) + \ma...

We study the zero locus of the Futaki invariant on K-polystable Fano threefolds, seen as a map from the K\"ahler cone to the dual of the Lie algebra of the reduced automorphism group. We show that, apart from families 3.9, 3.13, 3.19, 3.20, 4.2, 4.4, 4.7 and 5.3 of the Iskovskikh-Mori-Mukai classification of Fano threefolds, the Futaki invariant of...

We consider a sufficiently smooth semi-stable holomorphic vector bundle over a compact K\"ahler manifold. Assuming the automorphism group of its graded object to be abelian, we provide a semialgebraic decomposition of a neighbourhood of the polarisation in the K\"ahler cone into chambers characterising (in)stability. For a path in a stable chamber...

For a polarized Kähler manifold ( X , L ) (X,L) , we show the equivalence between relative balanced embeddings introduced by Mabuchi and σ \sigma -balanced embeddings introduced by Sano, answering a question of Hashimoto. We give a GIT characterization of the existence of a σ \sigma -balanced embedding, and relate the optimal weight σ \sigma to the...

We investigate hermitian Yang--Mills connections for pullback vector bundles on blow-ups of K\"ahler manifolds along submanifolds. Under some mild asumptions on the graded object of a simple and semi-stable vector bundle, we provide a necessary and sufficent numerical criterion for the pullback bundle to admit a sequence of hermitian Yang--Mills co...

For a small deformation of a constant scalar curvature K\"ahler manifold with abelian reduced automorphism group, we prove that K-polystability along nearby polarisations implies the existence of a constant scalar curvature K\"ahler metric. In this setting, we reduce K-polystability to the computation of the classical Futaki invariant on the cscK d...

For an equivariant reflexive sheaf over a normal polarised toric variety, we study slope stability of its reflexive pullback along a toric fibration. Examples of such fibrations include equivariant blow-ups and toric locally trivial fibrations. We show that stability (resp. unstability) is preserved under such pullbacks for so-called adiabatic pola...

We give a short proof of the Zariski–Lipman conjecture for toric varieties: any complex toric variety with locally free tangent sheaf is smooth.

We give a short proof of the Zariski-Lipman conjecture for toric varieties: any complex toric variety with locally free tangent sheaf is smooth.

For $(X,\,L)$ a polarized toric variety and $G\subset \mathrm {Aut}(X,\,L)$ a torus, denote by $Y$ the GIT quotient $X/\!\!/G$ . We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on $Y$ to the category of torus equivariant reflexive sheaves on $X$ . We show, under a genericity assumption on $G$ ,...

We investigate stability of pullback bundles with respect to adiabatic classes on the total space of holomorphic submersions. We show that the pullback of a stable (resp. unstable) bundle remains stable (resp. unstable) for these classes. Assuming the graded object of a Jordan-H\"older filtration to be locally free, we obtain a necessary and suffic...

We consider $G_2$-structures with torsion coupled with $G_2$-instantons, on a compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms which appears in supergravity and generalized geometry, known as the Bianchi identity. First studied by Friedrich and Ivanov, the resulting system of partial differential equations describes c...

We introduce a moment map picture for holomorphic string algebroids where the Hamiltonian gauge action is described by means of Morita equivalences, as suggested by higher gauge theory. The zero locus of our moment map is given by the solutions of the Calabi system, a coupled system of equations which provides a unifying framework for the classical...

We introduce the category of holomorphic string algebroids , whose objects are Courant extensions of Atiyah Lie algebroids of holomorphic principal bundles and whose morphisms correspond to inner morphisms of the underlying holomorphic Courant algebroids. This category provides natural candidates for Atiyah Lie algebroids of holomorphic principal b...

For (X,L) a polarized toric variety and G a torus of automorphisms of (X,L), denote by Y the GIT quotient X/G. We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on Y to the category of torus equivariant reflexive sheaves on X. We show, under a genericity assumption on G, that slope stability is p...

We introduce the category of holomorphic string algebroids, whose objects are Courant extensions of Atiyah Lie algebroids of holomorphic principal bundles, as considered by Bressler, and whose morphisms correspond to inner morphisms of the underlying holomorphic Courant algebroids in the sense of Severa. This category provides natural candidates fo...

Yau's solution of the Calabi Conjecture implies that every K\"ahler Calabi-Yau manifold $X$ admits a metric with holonomy contained in $\mathrm{SU}(n)$, and that these metrics are parametrized by the positive cone in $H^2(X,\mathbb{R})$. In this work we give evidence of an extension of Yau's theorem to non-K\"ahler manifolds, where $X$ is replaced...

The solution of the Calabi Conjecture by Yau implies that every K\"ahler Calabi-Yau manifold $X$ admits a metric with holonomy contained in $\textrm{SU}(n)$, and that these metrics are parametrized by the positive cone in $H^{1,1}(X,\mathbb{R})$. In this work we give evidence of an extension of Yau's theorem to non-K\"ahler manifolds, where $X$ is...

For a polarized K\"ahler manifold $(X, L)$, we show the equivalence between relative balanced embeddings introduced by Mabuchi and $\sigma$-balanced embeddings introduced by Sano, answering a question of Hashimoto. We give a GIT characterization of the existence of a $\sigma$-balanced embedding, and relate the optimal weight $\sigma$ to the action...

We construct the space of infinitesimal variations for the Strominger system and an obstruction space to integrability, using elliptic operator theory. Motivated by physics, we provide refinements of these finite-dimensional vector spaces using generalized geometry and establish a comparison with previous work by de la Ossa--Svanes and Anderson--Gr...

We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature K\"ahler metrics to the case of extremal metrics. Namely, we show that a given extremal metric is the limit of some specific relatively balanced metrics. As a corollary, we recover uniqueness...

We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature K\"ahler metrics to the case of extremal metrics. Namely, we show that a given extremal metric is the limit of some specific relatively balanced metrics. As a corollary, we recover uniqueness...

We endow the group of automorphisms of an exact Courant algebroid over a compact manifold with an infinite dimensional Lie group structure modeled on the inverse limit of Hilbert spaces (ILH). We prove a slice theorem for the action of this Lie group on the space of generalized metrics. As an application, we show that the moduli space of generalize...

We endow the group of automorphisms of an exact Courant algebroid over a compact manifold with an infinite dimensional Lie group structure modelled on the inverse limit of Hilbert spaces (ILH). We prove a slice theorem for the action of this Lie group on the space of generalized metrics. As an application, we show that the moduli space of generaliz...

We consider $G_2$ structures with torsion coupled with $G_2$-instantons, on a compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms which appears in supergravity and generalized geometry, known as the Bianchi identity. The resulting system of partial differential equations can be regarded as an analogue of the Strominger s...

Let X be a compact toric surface. Then there exists a sequence of torus equivariant blow-ups of X such that the blown-up toric surface admits a cscK metric.

Extending the work of G. Sz\'ekelyhidi and T. Br\"onnle to Sasakian manifolds we prove that a small deformation of the complex structure of the cone of a constant scalar curvature Sasakian manifold admits a constant scalar curvature structure if it is K-polystable. This also implies that a small deformation of the complex structure of the cone of a...

Let (X,L) be a polarized K\"ahler manifold that admits an extremal K\"ahler metric in c1(L). We show that on a nearby polarized deformation that preserves the symmetry induced by the extremal vector field of (X,L), the modified K-energy is bounded from below. This generalizes a result of Chen, Sz\'ekelyhidi and Tosatti to extremal metrics. Our proo...

In this work we define a deformation theory for the Coupled K\"ahler-Yang-Mills equations in arXiv:1102.0991, generalizing work of Sz\'ekelyhidi on constant scalar curvature K\"ahler metrics. We use the theory to find new solutions of the equations via deformation of the complex structure of a polarised manifold endowed with a holomorphic vector bu...

We provide a new proof of a result of X.X.Chen and G.Tian : for a polarized extremal K\"ahler manifold, an extremal metric attains the minimum of the modified K-energy. The proof uses an idea of C.Li adapted to the extremal metrics using some weighted balanced metrics.

Let X be a compact toric surface. There exists a sequence of torus equivariant blow-ups of X such that the blown-up toric surface obtained admits a cscK metric.

Let $X$ be a compact toric extremal K\"ahler manifold. Using the work of Sz\'ekelyhidi, we provide a combinatorial criterion on the fan describing $X$ to ensure the existence of complex deformations of $X$ that carry extremal metrics. As an example, we find new CSC metrics on 4-points blow-ups of $\C\P^1\times\C\P^1$.

The main subject of interest in this thesis is the existence of extremal metrics. Let (M, J, g) be a compact Kähler manifold. An extremal metric on M is a Kähler metricwhose L2 norm of the scalar curvature is minimal amongst the metrics representing the same Kähler class. New constructions of extremal metrics are explained using perturbative method...

Let (X,\Omega) be a closed polarized complex manifold, g be an extremal metric on X that represents the K\"ahler class \Omega, and G be a compact connected subgroup of the isometry group Isom(X,g). Assume that the Futaki invariant relative to G is nondegenerate at g. Consider a smooth family $(M \to B)$ of polarized complex deformations of (X,\Omeg...

New examples of extremal K\"ahler metrics on blow-ups of parabolic ruled surfaces are constructed. The method is based on the gluing construction of Arezzo, Pacard and Singer. This enables to endow ruled surfaces of the form $\mathbb{P}(\mathcal{O}\oplus L)$ with special parabolic structures such that the associated iterated blow-up admits an extre...

Let $(\mathcal {X},\Omega)$ be a closed polarized complex manifold, $g$ be an extremal metric on $\mathcal X$ that represents the K\"ahler class $\Omega$, and $G$ be a compact connected subgroup of the isometry group $Isom(\mathcal{X},g)$. Assume that the Futaki invariant relative to $G$ is nondegenerate at $g$. Consider a smooth family $(\mathcal{...