Julien Keller

About canonical Kähler metrics on Mumford semistable projective bundles over a curve

We study the existence of canonical Kahler metrics on the projec- tivisation of strictly Mumford semistable vector bundles over a curve. Consider the projectivisation P(E) of a holomorphic vector bundle E over a smooth base manifold B polarized by the ample line bundle LB. Various results have established a relationship between the stability of the underlying bundle E and the existence of Kahler metrics with special curva- ture on P(E), at least when c1(LB) can be endowed with an extremal metric. The case of a base manifold of complex dimension 1 has been intensively studied. Building on the work of D. Burns- P. de Bartolomeis, E. Calabi, A. Fujiki, C. Lebrun and many others, V. Apostolov, D. Calderbank, P. Gaudu- chon and C. Tonnesen-Friedman have provided a complete understanding of the situation for stable or polystable bundles over a smooth Riemann surface. They showed that there is a Kahler metric with constant scalar curvature (cscK metric in short) in any Kahler class on P(E) if and only if the bundle E is Mumford polystable (1, 2, 3). Another approach was also carried out in a series of paper of Y.-J. Hong, see (17) who investigated the case of higher dimensional base. Other results for ruled manifolds appeared recently in relationship with extremal Kahler metrics in (7, 21). Up to our knowledge, the case of strictly semistable bundle is still open in complete generality. We expect that for a base manifold of dimension ≥ 2 all the phe- nomena of stability for P(E) could happen when E is Mumford semistable (see for instance (19)). In this note, we essentially study the particular case of a ruled surface given by the projectivisation of a Mumford semistable vector bundle (which is not stable) over a Riemann surface of genus g ≥ 2. Some partial generalizations are given for higher dimensional base (note that the results of Section 3 will be extended in a forthcoming paper). Conventions: If π: E → B is a vector bundle then π: P(E) → B shall de- note the space of complex hyperplanes in the fibres of E. Thus π∗OP(E)(r) = S r E for r ≥ 0.We study the existence of canonical Kähler metrics on the projectivisation of strictly Mumford semistable vector bundles over a curve. Consider the projectivisation P(E) of a holomorphic vector bundle E over a smooth base manifold B polarized by the ample line bundle LB. Various results have established a relationship between the stability of the underlying bundle E and the existence of Kähler metrics with special curvature on P(E), at least when c1(LB) can be endowed with an extremal metric. The case of a base manifold of complex dimension 1 has been intensively studied. Building on the work of D. BurnsP. de Bartolomeis, E. Calabi, A. Fujiki, C. Lebrun and many others, V. Apostolov, D. Calderbank, P. Gauduchon and C. Tønnesen-Friedman have provided a complete understanding of the situation for stable or polystable bundles over a smooth Riemann surface. They showed that there is a Kähler metric with constant scalar curvature (cscK metric in short) in any Kähler class on P(E) if and only if the bundle E is Mumford polystable [1, 2, 3]. Another approach was also carried out in a series of paper of Y.-J. Hong, see [17] who investigated the case of higher dimensional base. Other results for ruled manifolds appeared recently in relationship with extremal Kähler metrics in [7, 21]. Up to our knowledge, the case of strictly semistable bundle is still open in complete generality. We expect that for a base manifold of dimension ≥ 2 all the phenomena of stability for P(E) could happen when E is Mumford semistable (see for instance [19]). In this note, we essentially study the particular case of a ruled surface given by the projectivisation of a Mumford semistable vector bundle (which is not stable) over a Riemann surface of genus g ≥ 2. Some partial generalizations are given for higher dimensional base (note that the results of Section 3 will be extended in a forthcoming paper). Conventions: If π : E → B is a vector bundle then π : P(E) → B shall denote the space of complex hyperplanes in the fibres of E. Thus π∗OP(E)(r) = SE for r ≥ 0.

Construction of constant scalar curvature Kähler cone metrics

Over a compact Kahler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a Kahler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature Kahler metrics with conic singularities: existence result under small deformations of Kahler classes, existence result over a Fano manifold, existence result over certain ruled manifolds. In this last case, we consider the projectivization of a parabolic stable holomorphic bundle. This leads us to prove that the existing Hermitian–Einstein metric on this bundle enjoys a regularity property along the divisor on the base.

Canonical metrics and Härder-Narasimhan filtration

Modulei Space of Killing Helices of Low Orders on a Complex Space Form (T Adachi) KCC and Linear Stability for the Parkinson Tremor Model (V Balan) The Camassa-Holm Equation as a Geodesic Flow For H1 Right-Invariant Metric (A Constantin & R I Ivanov) Fermi-Walker Parallel Transport, Time Evolution of a Space Curve and the Schrodinger Equation as a Moving Curve (R Dandoloff) Complex Submanifolds and Lagrangian Submanifolds Associate with Minimal Surfaces in Tori (N Ejiri) Soliton Equations with Deep Reductions. Generalized Fourier Transforms (V Gerdjikov et al.) Existence and Uniqueness Results for the Schrodinger -- Poisson System Below the Energy Norm (A M Ivanov & G P Venkov) Exact Solutions of the Manakov System (N A Kostov) Cluster Sets and Periodicity in Some Structure Fractals (J awrynowicz et al.) Dispersion and Asymptotic Profiles for Kichhoff Equations (T Matsuyama & M Ruzhansky) On the Hypoellipticity of Some Classes of Overdetermined Systems of Differential and Pseudodifferential Operators (P R Popivanov) Baklund Transformations and Riemann-Hilbert Problem for N Wave Equations with Additional Symmetries (T Valchev) and other papers.The algebraic notion of Gieseker stability is related to the existence of balanced metrics which are zeros of a certain moment map. We investigate some properties of balanced metrics relative to the Harder-Narasimhan filtration of a vector bundle and to blowups in the case of projective surfaces.

Quantization of Donaldson’s heat flow over projective manifolds

Consider E a holomorphic vector bundle over a projective manifold X polarized by an ample line bundle L. Fix k large enough, the holomorphic sections

Geometric Quantization of Complex Monge-Ampère Operator for Certain Diffusion Flows

H^0(E\otimes L^k)

Numerical Weil–Petersson metrics on moduli spaces of Calabi–Yau manifolds

H0(E⊗Lk) provide embeddings of X in a Grassmanian space. We define the balancing flow for bundles as a flow on the space of projectively equivalent embeddings of X. This flow can be seen as a flow of algebraic type hermitian metrics on E. At the quantum limit

Twisted balanced metrics

k\rightarrow \infty