Remarks on derived equivalences of Ricci-flat manifolds

Remarks on Kahler Ricci Flow

We study some estimates along the Kahler Ricci flow on Fano manifolds. Using these estimates, we show the convergence of Kahler Ricci flow directly if the $α$-invariant of the canonical class is greater than $\frac{n}{n+1}$. Applying these convergence theorems, we can give a flow proof of Calabi conjecture on such Fano manifolds. In particular, the existence of Kahler Einstein metrics on a lot of Fano surfaces can be proved by flow method. Note that this geometric conclusion (based on the same assumption) was established earlier via elliptic method by G. Tian. However, a new proof based on Kah…

Note on equivalences for degenerations of Calabi-Yau manifolds

This note studies the equivalencies among convergences of Ricci-flat Kähler-Einstein metrics on Calabi-Yau manifolds, cohomology classes and potential functions.

关于三次四维流形和超Kähler流形的L等价的一些注记

Some remarks on L-equivalence for cubic fourfolds and hyper-Kähler manifolds

本文证明了非常一般的三次四维流形在L等价时必为同构，同时指出存在特殊三次四维流形在L等价但非同构。当三次四维流形在特定Hassett除子中非常一般时，L等价的流形也是傅里叶-莫克伴侣。此外，作者还提供了进一步的例子支持L等价的超Kähler流形应为D等价的猜想。

Levi-Civita Ricci-flat metrics on non-Kähler Calabi-Yau manifolds

In this paper, we provide new examples of Levi-Civita Ricci-flat Hermitian metrics on certain compact non-Kähler Calabi-Yau manifolds, including every compact Hermitian Weyl-Einstein manifold, every compact locally conformal hyperKähler manifold, certain suspensions of Brieskorn manifolds, and every generalized Hopf manifold provided by suspensions of exotic spheres. These examples generalize previous constructions on Hopf manifolds. Additionally, we also construct new examples of compact Hermitian manifolds with nonnegative first Chern class that admit constant strictly negative Riemannian sc…

关于Chow对应的一些评论

Some remarks on Chow correspondences

本文在Voevodsky的三角动机范畴中研究了Chow环中几种适当的等价关系及其性质。

Some remarks on moduli spaces of lattice polarized holomorphic symplectic manifolds

We construct quasi-projective moduli spaces of $K$-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily--Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds.