arXiv research

Proof of complex geometry theorem for specific singular spaces.

Proves Kähler-Ricci shrinkers are complex analytic varieties.

Study two types of singular Kähler-Einstein metrics on complex varieties.

Study on singularities of Chern-Ricci flow on complex manifolds.

Proves orbifold singularities for Ricci-flat metrics on certain Kähler varieties.

The study of projective varieties with nef anticanonical divisors and log terminal singularities.

Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.

We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log te…

Reverse Hölder inequalities on Fano metrics with applications to geodesics and singularities.

Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural ass…

Constructs Kahler-Einstein metrics near isolated log canonical singularities.

Researchers prove birational invariance of BCOV invariant using motivic integration.

Researchers find Kähler-Einstein metrics near isolated log terminal singularities.

Let X be a projective variety which is algebraic Lang hyperbolic. We show that Lang's conjecture holds (one direction only): X and all its subvarieties are of general type and the canonical divisor K_X is ample at smooth points and Kawamata log terminal points of X, provided that K_X is Q-Cartier, no Calabi-Yau variety…

We prove the existence and uniqueness of Kähler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler-Ricci flow, and of Keller, Rubinstein on…

Paper extends Hodge correspondence to singular Kähler spaces.

Defines and studies solutions to complex equations on Hermitian manifolds.

We prove the existence and uniqueness of the weak Kahler-Ricci flow on projective varieties with log terminal singularities. It is also shown that the weak Kahler-Ricci flow can be uniquely continued through divisorial contractions and flips if they exist. We then propose an analytic version of the Minimal Model Progra…

Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces proved.

The study proves a key inequality for specific types of three-dimensional spaces.

We prove that the sum of the $α$-invariants of two different Kollár components of a Kawamata log terminal singularity is less than $1$.

We survey some recent topics on singularities, with a focus on their connection to the minimal model program. This includes the construction and properties of dual complexes, the proof of the ACC conjecture for log canonical thresholds and the recent progress on the `local stability theory' of an arbitrary Kawamata log…

We consider the space KR(n,F) of Kahler-Ricci solitons on n-dimensional Fano manifolds with Futaki invariant bounded by F. We prove a partial C^0 estimate for KR(n,F) as a generalization of the recent work of Donaldson-Sun for Fano Kahler-Einstein manifolds. In particular, any sequence in KR(n,F) has a convergent sub…

In this paper, we study transcendental aspects of the cohomology groups of adjoint bundles of log canonical pairs, aiming to establish an analytic theory for log canonical singularities. As a result, in the case of purely log terminal pairs, we give an analytic proof of the injectivity theorem originally proved by the …

In this short paper, we improve the result of Phong-Song-Sturm on degeneration of Fano Kähler-Ricci solitons by removing the assumption on the uniform bound of the Futaki invariant. Let $\mathcal{KR}(n)$ be the space of Kähler-Ricci solitons on $n$-dimensional Fano manifolds. We show that after passing to a subsequence…

Let $π: \mathcal{X}^* \rightarrow B^*$ be an algebraic family of compact Kähler manifolds of complex dimension $n$ with negative first Chern class over a punctured disc $B^*\in \mathbb{C}$. Let $g_t$ be the unique Kähler-Einstein metric on $\mathcal{X}_t= π^{-1}(t)$. We show that as $t\rightarrow 0$, $(\mathcal{X}_t, g…

In this paper, we give a lower bound of Bergman kernels for a sequence of almost Kähler-Einstein Fano manifolds, or more general, a sequence of Fano manifolds with almost Kähler-Ricci solitons. This generalizes a result by Donaldson-Sun, Tian for Kähler-Einstein manifolds sequence with positive scalar curvature. As an …

Study optimal portfolio choice with risk control for log-returns.

Complex projective manifolds without rational curves are quotients of Abelian varieties.

Solves non-Archimedean Calabi-Yau equation on complex log pairs.

Uniformizes varieties with log-canonical singularities using ball quotients.

Proves finitely generated graded rings for klt singularities.

In this paper, we prove that the zero-locus of any global holomorphic log-one-form on a projective log-smooth pair $\left(X,D\right)$ of log-general type must be non-empty. Applying this result, we give an answer to the algebraic hyperbolicity part of Shafarevich's conjecture, with the generic fiber being Kawamata-log-…

New methods for convex optimization with locally Lipschitz gradient, achieving faster convergence.

We study the behaviour of Chern numbers of three dimensional terminal varieties under divisorial contractions.

Holomorphic torsion invariant for log-Enriques surfaces derived from Borcherds products.

We study the convergence of volume forms on a degenerating holomorphic family of log-Calabi-Yau varieties to a non-Archimedean measure, extending a result of Boucksom and Jonsson. More precisely, let $(X,B)$ be a holomorphic family of sub log canonical, log-Calabi-Yau complex varieties parameterized by the punctured un…

Formula for sections on complex manifolds with non-isolated components.

The paper studies fundamental groups of compact Kahler varieties with nef anti-canonical bundles.

Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.

A graph (digraph) $G=(V,E)$ with a set $T\subseteq V$ of terminals is called inner Eulerian if each nonterminal node $v$ has even degree (resp. the numbers of edges entering and leaving $v$ are equal). Cherkassky and Lovász showed that the maximum number of pairwise edge-disjoint $T$-paths in an inner Eulerian graph $G…

Let $M$ be a compact Kähler manifold and $N$ be a subvariety with codimension greater than or equal to 2. We show that there are no complete Kähler--Einstein metrics on $M-N$. As an application, let $E$ be an exceptional divisor of $M$. Then $M-E$ cannot admit any complete Kähler--Einstein metric if blow-down of $E$ is…

Extends results on smoothability of singular Fano and Calabi-Yau varieties.

Sharp bounds on Fano varieties' heights proven for specific cases.

Let $X$ be a canonically polarized variety, i.e. a complex projective variety such that its canonical class $K_{X}$ defines an ample $\Q-$line bundle, and satisfying the conditions $G_1$ and $S_2$. Our main result says that $X$ admits a Kähler-Einstein metric iff $X$ has semi-log canonical singularities i.e. iff $X$ is…

Motivated by the study of Fano type varieties we define a new class of log pairs that we call asymptotically log Fano varieties and strongly asymptotically log Fano varieties. We study their properties in dimension two under an additional assumption of log smoothness, and give a complete classification of two dimension…

The notion of asymptotically log Fano varieties was given by Cheltsov and Rubinstein. We show that, if an asymptotically log Fano variety $(X, D)$ satisfies that $D$ is irreducible and $-K_X-D$ is big, then $X$ does not admit Kähler-Einstein edge metrics with angle $2πβ$ along $D$ for any sufficiently small positive ra…