The dHYM equation on crepant resolutions of Calabi-Yau cones
Abstract
In this paper, we study the deformed Hermitian-Yang-Mills (dHYM) equation on the non-compact Calabi-Yau manifold Z = Tot(bfKX), where X is a rational homogeneous variety. Since X is a Fano variety, Z is a resolution of the singularity at the vertex of the affine cone Aff(X), provided the cone is built using the anticanonical polarization bfL = bfKX−1. Using the cohomogeneity-one symmetry of the Ricci-flat Kähler metric obtained via the Calabi ansatz on Z, we reduce the fully nonlinear PDE underlying the dHYM equation to a scalar, asymptotically autonomous ordinary differential equation (ODE). From this, we determine the exact condition on the topological phase that guarantees global existence of solutions. As an application, we show that every holomorphic line bundle over Z admits a smooth, globally defined Hermitian connection solving the dHYM equation, provided the total phase lies in an explicit open interval determined by the Lie-theoretic data. Also, we prove a rigidity result classifying the exact geometric conditions under which the dHYM solution collapses into a classical Hermitian-Yang-Mills (HYM) connection. The results established generalize previous constructions and provide a substantial new class of examples. Furthermore, the approach presented allows one to study the behavior of the dHYM solutions through ODE methods. Using this approach, we construct the first explicit non-trivial example of a Hermitian-Einstein connection on a line bundle over a non-toric Calabi-Yau manifold which is not dHYM.
Related
- Crepant partial resolutions of the nilpotent cone via Hamiltonian reduction
- Crepant resolutions of four-dimensional finite quotients
- Equivariant noncommutative crepant resolutions and G-maximal modifying modules
- Crepant resolutions of fiber products and Kummer fibrations of rational elliptic surfaces with non-reduced fibers
- One-Dimensional Symmetry Reductions of the Monge-Ampère Equation on Some Semi-Homogeneous Convex Cones in ℝ4
BibTeX
@misc{correa2026dhym,
title = {{The dHYM equation on crepant resolutions of Calabi-Yau cones}},
author = {Eder M. Correa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25309},
eprint = {2609.25309},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25309},
}
RIS
TY - UNPB
TI - The dHYM equation on crepant resolutions of Calabi-Yau cones
AU - Eder M. Correa
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25309
UR - https://arxiv.org/abs/2609.25309
AB - In this paper, we study the deformed Hermitian-Yang-Mills (dHYM) equation on the non-compact Calabi-Yau manifold Z = Tot(bfK_X), where X is a rational homogeneous variety. Since X is a Fano variety, Z is a resolution of the singularity at the vertex of the affine cone Aff(X), provided the cone is built using the anticanonical polarization bfL = bfK_X^−1. Using the cohomogeneity-one symmetry of the Ricci-flat Kähler metric obtained via the Calabi ansatz on Z, we reduce the fully nonlinear PDE underlying the dHYM equation to a scalar, asymptotically autonomous ordinary differential equation (ODE). From this, we determine the exact condition on the topological phase that guarantees global existence of solutions. As an application, we show that every holomorphic line bundle over Z admits a smooth, globally defined Hermitian connection solving the dHYM equation, provided the total phase lies in an explicit open interval determined by the Lie-theoretic data. Also, we prove a rigidity result classifying the exact geometric conditions under which the dHYM solution collapses into a classical Hermitian-Yang-Mills (HYM) connection. The results established generalize previous constructions and provide a substantial new class of examples. Furthermore, the approach presented allows one to study the behavior of the dHYM solutions through ODE methods. Using this approach, we construct the first explicit non-trivial example of a Hermitian-Einstein connection on a line bundle over a non-toric Calabi-Yau manifold which is not dHYM.
ER -