Related papers: On K\"ahler-Einstein surfaces with edge singularit…
This is an announcement of conjectures and results concerning the generating series of Euler characteristics of Hilbert schemes of points on surfaces with simple (Kleinian) singularities. For a quotient surface C^2/G with G a finite…
The Wu--Yau theorem asserts that a compact K\"ahler manifold with negative holomorphic sectional curvature admits a cohomologous metric with negative Ricci curvature. We introduce a conjectural positive analog of the Wu--Yau theorem and…
We study the 2-systole on compact K\"ahler surfaces of positive scalar curvature. For any such surface $(X,\omega)$, we prove the sharp estimate $\min_X S(\omega)\cdot\operatorname{sys}_2(\omega)\le 12\pi$, with equality if and only if…
In this paper we prove that, under natural assumptions, the scalar curvature of a Kaehler-Einstein metric on a compactification of C^n is strictly positive.
Among log canonical surface singularities, the ones which have a rational homology disk smoothing are the cyclic quotient singularities $\frac{1}{n^2}(1,na-1)$ with gcd$(a,n)=1$, and three distinguished elliptic quotient singularities. We…
We give some non-existence results for K\"ahler-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…
Mechanical fields over thin elastic surfaces can develop singularities at isolated points and curves in response to constrained deformations (e.g., crumpling and folding of paper), singular body forces and couples, distributions of isolated…
Indefinite Kaehler solutions of the Einstein equations are studied, and it is almost completely determined which compact complex surfaces admit such metrics.
The problem of prescribing Gaussian curvature on Riemann surface with conical singularity is considered. Let $(\Sigma,\beta)$ be a closed Riemann surface with a divisor $\beta$, and $K_\lambda=K+\lambda$, where…
We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive…
For hypersurfaces moving by standard mean curvature flow with boundary, we show that if a tangent flow at a boundary singularity is given by a smoothly embedded shrinker, then the shrinker must be non-orientable. We also show that there is…
In this paper we define and analyze singularities of discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms. In particular, we discuss singularities of discrete…
We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the…
Smooth Kahler-Einstein metrics have been studied for the past 80 years. More recently, singular Kahler-Einstein metrics have emerged as objects of intrinsic interest, both in differential and algebraic geometry, as well as a powerful tool…
We use a Simons type equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.
In this short note, we prove the existence of constant scalar curvature K\"ahler metrics on compact K\"ahler manifolds with semi-ample canonical bundles.
In this paper, we investigate automorphisms of compact K\"ahler manifolds with different levels of topological triviality. In particular, we provide several examples of smooth complex projective surfaces X whose groups of…
In this work, singular surfaces are obtained from smooth orientable closed surfaces by applying three basic simple loop operations, collapsing operation, zipping operation and double loop identification, each of which produces different…
In this paper, we study the collapsing behaviour of negative K\"{a}hler-Einstein metrics along degenerations of canonical polarized manifolds. We prove that for a toroidal degeneration of canonical polarized manifolds with the total space…
We show that there exists a metric with positive scalar curvature on S2xS1 and a sequence of embedded minimal cylinders that converges to a minimal lamination that, in a neighborhood of a strictly stable 2-sphere, is smooth except at two…