In this paper, we construct complete constant scalar curvature Khler (cscK) metrics on the complement of the zero section in the total space of O ( - 1 ) 2 over O ( - 1 ) 2 , which is biholomorphic to the smooth part of the cone <i>C</i> ₀ in O ( - 1 ) 2 defined by equation O ( - 1 ) 2 . On its small resolution and its deformation, we also consider complete cscK metrics and find that if the cscK metrics are homogeneous, then they must be Ricci-flat.

Let X be a compact Khler manifold and X a subvariety of X with higher co-dimension. The aim is to study complete constant scalar curvature Khler metrics on non-compact Khler manifold X with Poincar--Mok--Yau asymptotic property (see Definition\ref {def}). In this paper, the methods of Calabi's ansatz and the moment construction are used to provide some special examples of such metrics.

In this paper, we study the zero loci of local systems of the form \delta\Pi , where \delta\Pi is the period sheaf of the universal family of CY hypersurfaces in a suitable ambient space \delta\Pi , and \delta\Pi is a given differential operator on the space of sections \delta\Pi . Using earlier results of three of the authors and their collaborators, we give several different descriptions of the zero locus of \delta\Pi . As applications, we prove that the locus is algebraic and in some cases, non-empty. We also give an explicit way to compute the polynomial defining equations of the locus in some cases. This description gives rise to a natural stratification to the zero locus.

Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further simplified by explicitly constructing the Forman-Ricci tensor associated to each edge, thereby diagonalizing the first-order differential operator and avoiding the need to invert large sparse matrices at each time step. We recently showed analytically and numerically that these equations converge for axisymmetric 3-geometries to the corresponding continuum RF equations. We demonstrate here that these DRF equations yield an explicit numerical realization of Thurston's geometrization procedure for a discrete 3D axially-symmetric neckpinch geometry by using surgery to explicitly integrate through its Type-1 neck pinch singularity. A cubic-spline-based adaptive mesh was required to complete the evolution. Our numerically efficient simulations yield the expected Thurston decomposition of the sufficiently pinched axially symmetric geometry into its unique geometric structure--a direct product of two lobes, each collapsing toward a 3-sphere geometry. The structure of our curvature may be used to better inform one of the vertex and edge weighting factors that appear in the Forman's expression of Ricci curvature on graphs.

This erratum will correct the classification of Theorem 1 in Lin-Lu-Yau, Comm. Anal. Geom., 2014, that misses the Triplex graph.