arXiv research

The flow contracts cone divisors on Kähler surfaces to points.

We develop some foundations for the study of Kahler-Einstein metrics with cone singularities transverse to a divisor. The main goal is a treatment of the deformation of the cone angle.

The paper proves conditions for existence of constant scalar curvature Kähler metrics with cone singularities.

Study locates divisors in Hodge bundle with specific properties.

Study shows flow convergence to smooth K-Ricci outside a divisor with cusp singularity.

We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor on a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-Einstein metrics with cone singularities. As…

Study finds existence and non-uniqueness of cone spherical metrics on compact Riemann surfaces.

Optimal family of Calabi-Yau cone metrics found for toric Kähler cones.

New Kähler metrics found with cone singularities on complex manifolds.

Study of zero-divisors in sedenions via determinant factorization.

In the present paper we prove that, on a hyperkähler manifold, walls of the kähler cone and extremal rays of the Mori cone are determined by all divisors satisfying certain numerical conditions.

The study finds infinitely many divisors in a specific space of geometric objects.

The continuity method is used to deform the cone angle of a weak conical Kähler-Einstein metric with cone singularities along a smooth anti-canonical divisor on a smooth Fano manifold. This leads to an alternative proof of Donaldson's Openness Theorem on deforming cone angle \cite{Don} by combining it with the regulari…

Constructs a map from stable extensions to irreducible metrics on Riemann surfaces.

Let X be a Kähler manifold and D be a R-divisor with simple normal crossing support and coefficients between 1/2 and 1. Assuming that K_X+D is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on X\D having mixed Poincaré and cone singularities according to the coefficients of D…

Constructs scalar-flat Kähler metrics with varying conical singularities.

Unique conical Kähler-Einstein metric found on Fano manifold.

Study tangent cones of reflexive sheaves, proving existence and uniqueness.

Symplectic vortex equations link Sasakian manifolds to Kahler cones.

Proves Matsushima's theorem for Kähler-Einstein metrics on Fano manifolds with cone singularities.

In this paper we consider a canonical compactification of Hitchin's moduli space of stable Higgs bundles with fixed determinant of odd degree over a Riemann surface, producing a projective variety by gluing in a divisor at infinity. We give a detailed study of the compactified space, the divisor at infinity and the mod…

This is the second of a series of three papers which provide proofs of results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle is less than 2π. We show that these are in a natrual way projective algebraic var…

We study the scalar curvature of Kähler metrics that have cone singularities along a divisor, with a particular focus on certain specific classes of such metrics that enjoy some curvature estimates. Our main result is that, on the projective completion of a pluricanonical bundle over a product of Kähler--Einstein Fano …

The paper studies Kähler-Einstein metrics with singularities and their limits.

Generalizes Schwarz lemma for conical Kähler metrics with arbitrary cone angles.

Study Schauder estimates for conical singularities using Kähler metrics.

Study shows non-polyhedral structure in moduli spaces for n≥8.

The paper proves stability of Kähler-Ricci flows on Fano manifolds.

In this note, we prove that on a compact Kähler manifold $X$ carrying a smooth divisor $D$ such that $K_X+D$ is ample, the Kähler-Einstein cusp metric is the limit (in a strong sense) of the Kähler-Einstein conic metrics when the cone angle goes to $0$. We further investigate the boundary behavior of those and prove th…

Constructs ALE Calabi-Yau metrics with cone singularities.

Establishes convexity and coercivity of K-energy functional for complex tori.

New Einstein RCD spaces found with cone singularities.

Let $X$ be a non-singular compact Kähler manifold, endowed with an effective divisor $D= \sum (1-β_k) Y_k$ having simple normal crossing support, and satisfying $β_k \in (0,1)$. The natural objects one has to consider in order to explore the differential-geometric properties of the pair $(X, D)$ are the so-called metri…

Tian initiated the study of incomplete Kähler-Einstein metrics on quasi-projective varieties with cone-edge type singularities along a divisor, described by the cone-angle $2π(1-α)$ for $α\in (0, 1)$. In this paper we study how the existence of such Kähler-Einstein metrics depends on $α$. We show that in the negative s…

This article considers the existence and regularity of Kahler-Einstein metrics on a compact Kahler manifold $M$ with edge singularities with cone angle $2πβ$ along a smooth divisor $D$. We prove existence of such metrics with negative, zero and some positive cases for all cone angles $2πβ\leq 2π$. The results in the po…

We discuss a technique to construct Ricci-flat hermitian metrics on complements of (some) anticanonical divisors of almost homogeneous manifolds and discuss when this metric is complete and Kähler. This construction has a strong interplay with invariance groups of the same dimension as the manifold acting with an open …

We give some non-existence results for Kähler-Einstein metrics with conical singularities along a divisor on Fano manifolds. In particular we show that the maximal possible cone angle is in general smaller than the invariant R(M). We study this discrepancy from the point of view of log K-stability.

Study finds limits for conical Kähler-Einstein metrics on unstable surfaces.

In this paper we introduce the "interpolation-degneration" strategy to study Kahler-Einstein metrics on a smooth Fano manifold with cone singularities along a smooth divisor that is proportional to the anti-canonical divisor. By "interpolation" we show the angles in $(0, 2π]$ that admit a conical Kahler-Einstein metric…

Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. …

The study finds metrics with constant scalar curvature on complex manifolds.

Study limits of Kähler-Einstein metrics with cone singularities on complex projective manifolds.

Uniformizes branched surfaces into Higgs bundles.

Using results by Donaldson and Auroux on pseudo-holomorphic curves as well as Duval's rational convexity construction, the paper investigates the existence of smooth Lagrangian surfaces representing 2-dimensional homology classes in complex projective surfaces. We prove that if the projective surface X is minimal, of g…

Let $X$ be a complex manifold and $S\hookrightarrow X$ be an embedding of complex submanifold. Assuming that the embedding is $(k-1)$-linearizable or $(k-1)$-comfortably embedded, we construct via the deformation to the normal cone a diffeomorphism $F$ from a small neighborhood of the zero section in the normal bundle …

Compactifies Riemann surface metrics with conical singularities.

Formulae count square-tiled surfaces in genus two.

In this paper, we consider the twisted Kähler-Ricci soliton, and show that the existence of twisted Kähler-Ricci soliton with semi-positive twisting form is closely related to the properness of some energy functionals. We also consider the conical Kähler-Ricci soliton, and obtain some existence results. In particular, …