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In this article, we show the existence of a nontrivial Riemann surface lamination embedded in $mathbb{CP}^2$ by using Donaldsons construction of asymptotically holomorphic submanifolds. Further, the lamination we obtain has the property that each leaf is a totally geodesic submanifold of $mathbb{CP}^2 $ with respect to the Fubini-Study metric. This may constitute a step in understanding the conjecture on the existence of minimal exceptional sets in $mathbb{CP}^2$.
In this short note we show that the existence of bilaterally symmetric extremal Kahler metrics on $mathbb{CP}^2sharp 2bar{mathbb{CP}^2}$.
We prove the existence of Kahler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties, and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings.
We show that on every compact Riemannian 2-orbifold there exist infinitely many closed geodesics of positive length.
We prove that generalised Monge-Ampere equations (a family of equations which includes the inverse Hessian equations like the J-equation, as well as the Monge-Ampere equation) on projective manifolds have smooth solutions if certain intersection numb
We discuss the complex geometry of two complex five-dimensional Kahler manifolds which are homogeneous under the exceptional Lie group $G_2$. For one of these manifolds rigidity of the complex structure among all Kahlerian complex structures was prov