Bihermitian metrics on Del Pezzo surfaces

Computing $α$-invariants of singular del Pezzo surfaces

We prove new local inequality for divisors on surfaces and utilize it to compute $α$-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type $\mathbb{A}_{1}$, $\mathbb{A}_{2}$, $\mathbb{A}_{3}$, $\mathbb{A}_{4}$, $\mathbb{A}_{5}$ or $\mathbb{A}_{6}$ are Kähler-Einstein.

Exceptional del Pezzo hypersurfaces

We compute global log canonical thresholds of a large class of quasismooth well-formed del Pezzo weighted hypersurfaces in $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$. As a corollary we obtain the existence of orbifold Kähler--Einstein metrics on many of them, and classify exceptional and weakly exceptional quasismooth well-formed del Pezzo weighted hypersurfaces in $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$.

Compact Moduli Spaces of Del Pezzo Surfaces and Kähler-Einstein metrics

We prove that the Gromov-Hausdorff compactification of the moduli space of Kahler-Einstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tian's theorem on the existence of Kahler-Einstein metrics on smooth Del Pezzo surfaces and classifies the degenerations of such metrics. The proof is based on a combination of both algebraic and differential geometric techniques.

Lagrangian spheres in Del Pezzo surfaces

Lagrangian spheres in the symplectic Del Pezzo surfaces arising as blow-ups of the complex projective plane in 4 or fewer points are classified up to Lagrangian isotopy. Unlike the case of the 5-point blow-up, there is no Lagrangian knotting.

K-不稳定 del Pezzo 曲面上的锥形 Kähler-Einstein 度量

Conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces

该研究确定了 K-不稳定 del Pezzo 曲面上锥形 Kähler-Einstein 度量锥角的最优上界，为理解此类度量的存在性提供关键约束。

Comments on the del Pezzo cone

We describe a framework for constructing the general Ricci-flat metric on the anticanonical cone over the del Pezzo surface of rank one.

Poisson Cohomology of Del Pezzo surfaces

In this note, we compute the Poisson cohomology groups for any Poisson Del Pezzo surface.

Existence of Kähler-Einstein metrics and multiplier ideal sheaves on del Pezzo surfaces

We apply Nadel's method of multiplier ideal sheaves to show that every complex del Pezzo surface of degree at most six whose automorphism group acts without fixed points has a Kähler-Einstein metric. In particular, all del Pezzo surfaces of degree $4,5$, or $6$ and certain special del Pezzo surfaces of lower degree are shown to have a Kähler-Einstein metric. This result is not new, but the proofs given in the present paper are less involved than earlier ones by Siu, Tian and Tian-Yau.

Convergence of the Kähler-Ricci flow and multiplier ideal sheaves on del Pezzo surfaces

On certain del Pezzo surfaces with large automorphism groups, it is shown that the solution to the Kähler-Ricci flow with a certain initial value converges in $C^\infty$-norm exponentially fast to a Kähler-Einstein metric. The proof is based on the method of multiplier ideal sheaves.

Numerical Kaehler-Ricci soliton on the second del Pezzo

The second del Pezzo surface is known by work of Tian-Zhu and Wang-Zhu to admit a unique Kaehler-Ricci soliton. Applying a method described in hep-th/0703057, we use Ricci flow to numerically compute that soliton metric. We numerically compute the value of its Perelman entropy (or Gaussian density).