Related papers: Extremal metrics and K-stability

We present a simple geometric construction linking geometric to deformation quantization. Both theories depend on some apparently arbitrary parameters, most importantly a polarization and a symplectic connection, and for real polarizations…

We consider coupled K\"ahler-Einstein metrics and weighted solitons on Fano manifolds. These are natural generalizations of K\"ahler-Einstein metrics. As in the case of K\"ahler-Einstein metrics, the existence is known to be equivalent to…

Any oriented $4$-dimensional Einstein metric with semi-definite sectional curvature satisfies the pointwise inequality \[ \frac{|s|}{\sqrt{6}}\geq|W^+|+|W^-|, \] where $s$, $W^+$ and $W^-$ are respectively the scalar curvature, the…

We consider four (real or complex) dimensional hyper-K\"ahler metrics with a conformal symmetry K. The three-dimensional space of orbits of K is shown to have an Einstein-Weyl structure which admits a shear-free geodesics congruence for…

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…

In this paper, we propose a coupled system of complex Hessian equations which generalizes the equation for constant scalar curvature K\"ahler (cscK) metrics. We show this system can be realized variationally as the Euler-Lagrange equation…

Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin we consider the corresponding K{\"a}hler-Einstein equation $e^{-\Phi} = \det D^2 \Phi$. If $K$ is a simplex, then the Ricci tensor of the Hessian metric $D^2…

In this paper we study a special case of the completion of cusp K\"{a}hler-Einstein metric on the regular part of varieties by taking the continuity method proposed by La Nave and Tian. The differential geometric and algebro-geometric…

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…

We annnounce a proof of the fact that a K-stable Fano manifold admits a Kahler-Einstein metric and give a brief outline of the proof.

In their seminal work (\cite{CC}, \cite{CC2}), Chen and Cheng proved apriori estimates for the constant scalar curvature metrics on compact K\"ahler manifolds. They also proved $C^{3,\alpha}$ estimate for the potential of the \ka metrics…

Assume that a projective variety together with a polarization is uniformly K-stable. If the polarization is canonical or anti-canonical, then the projective variety is uniformly K-stable with respects to any polarization sufficiently close…

A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kaehler metric of constant scalar curvature on the blow-up according to…

We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kaehler manifold can be solved using a variational method independent of Yau's theorem. Our formulation yields in particular a natural…

The Wu--Yau theorem asserts that a compact K\"ahler manifold with negative holomorphic sectional curvature admits a cohomologous metric with negative Ricci curvature. We introduce a conjectural positive analog of the Wu--Yau theorem and…

In complex Finsler geometry, an open problem is: does there exist a weakly K\"ahler Finsler metric which is not K\"ahler? In this paper, we give an affirmative answer to this open problem. More precisely, we construct a family of the weakly…

We study Laplace eigenvalues $\lambda_k$ on K\"ahler manifolds as functionals on the space of K\"ahler metrics with cohomologous K\"ahler forms. We introduce a natural notion of a $\lambda_k$-extremal K\"ahler metric and obtain necessary…

We give a survey of our recent work describing a method which combines the Sasaki join construction with the admissible K\"ahler construction of to obtain new extremal and new constant scalar curvature Sasaki metrics, including…

We calculate Chow quotients of some families of symmetric \(T\)-varieties. In complexity two we obtain new examples of K\"ahler-Einstein metrics by bounding the symmetric alpha invariant of their orbifold quotients. As an additional…

These lecture notes are written for a PhD mini-course I gave at the CIRM in Luminy in 2019. Their intended purpose was to present, in the context of smooth toric varieties, a relatively self-contained and elementary introduction to the…