skip to main content

Attention:

The NSF Public Access Repository (NSF-PAR) system and access will be unavailable from 11:00 PM ET on Thursday, October 10 until 2:00 AM ET on Friday, October 11 due to maintenance. We apologize for the inconvenience.

Explore Research Products in the NSF-PAR

Title: Warped quasi-asymptotically conical Calabi-Yau metrics
We construct many new examples of complete Calabi-Yau metrics of maximal volume growth on certain smoothings of Cartesian products of Calabi-Yau cones with smooth cross-sections. A detailed description of the geometry at infinity of these metrics is given in terms of a compactification by a manifold with corners obtained through the notion of weighted blow-up for manifolds with corners. A key analytical step in the construction of these Calabi-Yau metrics is to derive good mapping properties of the Laplacian on some suitable weighted Hölder spaces.  more » « less
Award ID(s):
2109577
NSF-PAR ID:
10470642
Author(s) / Creator(s):
;
Publisher / Repository:
arXiv:2308.02155
Date Published:
Journal Name:
arXivorg
ISSN:
2331-8422
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ( , Mathematische Annalen)
    null (Ed.)
    We study the degenerations of asymptotically conical Ricci-flat Kahler metrics as the Kahler class degenerates to a semi-positive class. We show that under appropriate assumptions, the Ricci-flat Kahler metrics converge to a incomplete smooth Ricci-flat Kahler metric away from a compact subvariety. As a consequence, we construct singular Calabi–Yau metrics with asymptotically conical behaviour at infinity on certain quasi-projective varieties and we show that the metric geometry of these singular metrics are homeomorphic to the topology of the singular variety. Finally, we will apply our results to study several classes of examples of geometric transitions between Calabi–Yau manifolds. 
    more » « less
  2. ; ; ( , Journal of High Energy Physics)
    A bstract The 2 d (0 , 2) supersymmetric gauge theories corresponding to the classes of Y p,k (ℂℙ 1 × ℂℙ 1 ) and Y p,k (ℂℙ 2 ) manifolds are identified. The complex cones over these Sasaki-Einstein 7-manifolds are non-compact toric Calabi-Yau 4-folds. These infinite families of geometries are the largest ones for Sasaki-Einstein 7-manifolds whose metrics, toric diagrams, and volume functions are known explicitly. This work therefore presents the largest list of 2 d (0 , 2) supersymmetric gauge theories corresponding to Calabi-Yau 4-folds with known metrics. 
    more » « less
  3. ; ; ; ; ; ( , Journal of High Energy Physics)
    null (Ed.)
    A bstract We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string spectrum which plays a crucial role in swampland conjectures, to mirror symmetry and the SYZ conjecture. In the case of SU(3) structure, our machine learning approach allows us to engineer metrics with certain torsion properties. Our methods are demonstrated for Calabi-Yau and SU(3)-structure manifolds based on a one-parameter family of quintic hypersurfaces in ℙ 4 . 
    more » « less
  4. ; ( , Forum of Mathematics, Sigma)
    Abstract

    The goal of this paper is to describe certain nonlinear topological obstructions for the existence of first-order smoothings of mildly singular Calabi–Yau varieties of dimension at least$4$. For nodal Calabi–Yau threefolds, a necessary and sufficient linear topological condition for the existence of a first-order smoothing was first given in [Fri86]. Subsequently, Rollenske–Thomas [RT09] generalized this picture to nodal Calabi–Yau varieties of odd dimension by finding a necessary nonlinear topological condition for the existence of a first-order smoothing. In a complementary direction, in [FL22a], the linear necessary and sufficient conditions of [Fri86] were extended to Calabi–Yau varieties in every dimension with$1$-liminal singularities (which are exactly the ordinary double points in dimension$3$but not in higher dimensions). In this paper, we give a common formulation of all of these previous results by establishing analogues of the nonlinear topological conditions of [RT09] for Calabi–Yau varieties with weighted homogeneousk-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions.

     
    more » « less
  5. ; ( , Journal of High Energy Physics)
    A bstract Non-simply connected Calabi-Yau threefolds play a central role in the study of string compactifications. Such manifolds are usually described by quotienting a simply connected Calabi-Yau variety by a freely acting discrete symmetry. For the Calabi-Yau threefolds described as complete intersections in products of projective spaces, a classification of such symmetries descending from linear actions on the ambient spaces of the varieties has been given in [16]. However, which symmetries can be described in this manner depends upon the description that is being used to represent the manifold. In [24] new, favorable, descriptions were given of this data set of Calabi-Yau threefolds. In this paper, we perform a classification of cyclic symmetries that descend from linear actions on the ambient spaces of these new favorable descriptions. We present a list of 129 symmetries/non-simply connected Calabi-Yau threefolds. Of these, at least 33, and potentially many more, are topologically new varieties. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email