Related papers: On flops and canonical metrics
We study a one parameter degeneration of Calabi Yau threefolds whose central fiber contains a single ordinary double point. Using the nearby and vanishing cycle formalism, we construct a canonical perverse object on the singular fiber from…
We prove that the Gromov-Hausdorff compactification of the moduli space of Kahler-Einstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tian's theorem on the…
In this paper we show how Einstein metrics are naturally described using the quantization of the algebra of functions on a Kahler manifold M. In this setup one interprets M as the phase space itself, equipped with the Poisson brackets…
We prove a technical result which allows us to establish the non-degeneracy of potentials on quivers in some previously unknown or non-obvious cases. Our result applies to certain McKay quivers and also to potentials derived from geometric…
We give a simple sufficient condition for K-stability of polarized del Pezzo surfaces and for the existence of a constant scalar curvature Kahler metric in the Kahler class corresponding to the polarization.
A classical approach to the Calder\'on problem is to estimate the unknown conductivity by solving a nonlinear least-squares problem. It leads to a nonconvex optimization problem which is generally believed to be riddled with bad local…
In this note, we give a proof of the famous theorem of M. Morse dealing with the cancellation of a pair of non-degenerate critical points of a smooth function. Our proof consists of a reduction to the one-dimensional case where the question…
Infinitely many new examples of compact Lorentzian surfaces without conjugate points are given. Further, we study the existence and the stability of this property among Lorentzian metrics with a Killing field. We obtain a new obstruction…
The existence of incompatible measurements is often believed to be a feature of quantum theory which signals its inconsistency with any classical worldview. To prove the failure of classicality in the sense of Kochen-Specker…
A classical result by Casten-Holland and Matano asserts that constants are the only positive and stable solutions to semilinear elliptic PDEs subject to homogeneous Neumann boundary condition in bounded convex domains. In other terms, this…
We prove a compactness theorem for K\"ahler metrics with various bounds on Ricci curvature and the $\mathcal I$ functional. We explore applications of our result to the continuity method and the Calabi flow.
In this paper, we prove several formulas related to Hodge theory, and using them to prove the deformations of a compact $H$-twisted generalized Calabi-Yau manifold are unobstructed and $L^2$ convergence in a neighborhood in another power…
Let $X$ be a compact normal K\"ahler space whose canonical sheaf is a rank-one free $\mathcal O_X$ module and whose singularities are isolated, rational and quasi-homogeneous. We prove then that under a topological hypothesis the…
We study the structure of the stable norm of Finsler metrics on the 2-torus with a focus to points of irrational slope. By our results, the stable norm detects KAM-tori and hyperbolicity in the geodesic flow. Moreover, we study the stable…
We study the space of Ricci-flat Kahler metrics on a given Calabi-Yau manifold, pose a number of questions about their possible degenerations, and survey some recent results on these questions.
In this paper we prove that generic small partial smoothings of Kahler-Einstein (KE) Del Pezzo orbifolds with only nodal singularities, and with no non-zero holomorphic vector fields, admit orbifold KE metrics which are close in the…
In this paper we develop a systematic deformation theory for conic constant curvature metrics on a closed surface when all cone angles are less than $2\pi$; in particular, we define and study the Teichm\"uller space…
In this paper, we consider a classical Hamiltonian normal form with degeneracy in normal direction. In previous results, one needs to assume that the perturbation satisfies certain non-degenerate conditions in order to remove the degeneracy…
Cone spherical metrics are conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces. A cone spherical metric is called irreducible if each developing map of the metric does not have…
Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat K\"ahler metrics on a minimal K\"ahler surface whose K\"ahler classes stay in a compact…