Related papers: The Calabi conjecture and K-stability

We prove some general statements on stability conditions of Calabi-Yau surfaces and discuss the stability manifold of the cotangent bundle of P^1. Our primary interest is in spherical objects.

We show that a compact weighted extremal Kahler manifold (as defined by the third named author) has coercive weighted Mabuchi energy with respect to a maximal complex torus in the reduced group of complex automorphisms. This provides a vast…

In this paper, we directly prove that if the limit of microscopic stability thresholds introduced by Berman for a polarized manifold satisfies some condition, then there exists a unique constant scalar curvature K\"{a}hler metric. This is…

For a given polarized toric variety, we define the notion of $\lambda$-stability which is a natural generalization of uniform K-stability. At the neighbourhoods of the vertices of the corresponding moment polytope $\Delta$, we consider…

We introduce a notion of stable spherical variety which includes the spherical varieties under a reductive group $G$ and their flat equivariant degenerations. Given any projective space $\bP$ where $G$ acts linearly, we construct a moduli…

Let $X$ be a normal projective variety admitting a polarized endomorphism $f$, i.e., $f^*H\sim qH$ for some ample divisor $H$ and integer $q>1$. Then Broustet and Gongyo proposed the conjecture that $X$ is of Calabi-Yau type (CY for short),…

Let $B$ be a K\"ahler-Einstein Fano manifold, and $L \to B$ be a suitable root of the canonical bundle. We give a construction of complete Calabi-Yau metrics and gradient shrinking, steady, and expanding K\"ahler-Ricci solitons on the total…

We prove a new rigidity criterion for families of polarized Calabi-Yau manifolds. Motivated by known non-rigid examples, we conjecture that a family over a quasi-projective curve is rigid if it admits a smooth compactification whose…

For $g,n\geq 0$ a 3-dimensional Calabi-Yau $A_\infty$-category $\mathcal C_{g,n}$ is constructed such that a component of the space of Bridgeland stability conditions, $\mathrm{Stab}(\mathcal C_{g,n})$, is a moduli space of quadratic…

Let $X$ be a toric variety and $u$ be a normalized symplectic potential of the corresponding polytope $P$. Suppose that the Riemannian curvature is bounded by 1 and $ \int_{\partial P} u ~ d \sigma < C_1, $ then there exists a constant…

In this paper, we study the local properties of the moduli space of a polarized Calabi-Yau manifold. Let $U$ be a neighborhood of the moduli space. Then we know the universal covering space $V$ of $U$ is a smooth manifold. Suppose $D$ is…

In this short note, we investigate the existence of orbifold K\"ahler-Einstein metrics on toric varieties. In particular, we show that every $\mathbb{Q}$-factorial normal projective toric variety allows an orbifold K\"ahler-Einstein metric.…

K\"ahler-Einstein metrics for polarized families of Calabi-Yau manifolds define a natural hermitian metric on the relative canonical bundle. The fact that the curvature form is equal to the pull-back of the Weil-Petersson form up to a…

We extend an argument of Stoppa to make some prgress towards a proof that K\"ahler-Einstein manifolds are "b-stable". We point out some algebro-geometric questions, involving finite generation, that arise.

The main goal of this paper is to prove the polystability of the logarithmic tangent sheaf $\mathscr T_X(-D)$ of a log canonical pair $(X,D)$ whose canonical bundle $K_X+D$ is ample, generalizing in a significant way a theorem of Enoki. We…

We prove that families of Calabi-Yau threefolds (CY3's) admit Bridgeland stability conditions when they are obtained via orbifolding from a family of CY3's admitting Bridgeland stability conditions. In particular, we prove that the quintic…

Let X be a compact Kahler orbifold without \C-codimension-1 singularities. Let D be a suborbifold divisor in X such that D \supset Sing(X) and -pK_X = q[D] for some p, q \in \N with q > p. Assume that D is Fano. We prove the following two…

Let $(M,\omega)$ be a K\"ahler manifold and let $K$ be a compact group that acts on $M$ in a Hamiltonian fashion. We study the action of $K^\mathbb{C}$ on probability measures on $M$. First of all we identify an abstract setting for the…

We construct Bridgeland stability conditions on the the following hyper-K\"ahler or strict Calabi--Yau manifolds: - Generalized Kummer varieties associated to an abelian surface that is isogenous to a product of elliptic curves. - Universal…

Mirror symmetry for a semi-stable degeneration of a Calabi-Yau manifold was first investigated by Doran-Harder-Thompson when the degeneration fiber is a union of two (quasi)-Fano manifolds. They propose a topological construction of a…