Yau Metrics Research Articles

Metric flows with neural networks

Abstract We develop a general theory of flows in the space of Riemannian metrics induced by neural network (NN) gradient descent. This is motivated in part by recent advances in approximating Calabi–Yau metrics with NNs and is enabled by recent advances in understanding flows in the space of NNs. We derive the corresponding metric flow equations, which are governed by a metric neural tangent kernel (NTK), a complicated, non-local object that evolves in time. However, many architectures admit an infinite-width limit in which the kernel becomes fixed and the dynamics simplify. Additional assumptions can induce locality in the flow, which allows for the realization of Perelman’s formulation of Ricci flow that was used to resolve the 3d Poincaré conjecture. We demonstrate that such fixed kernel regimes lead to poor learning of numerical Calabi–Yau metrics, as is expected since the associated NNs do not learn features. Conversely, we demonstrate that well-learned numerical metrics at finite-width exhibit an evolving metric-NTK, associated with feature learning. Our theory of NN metric flows therefore explains why NNs are better at learning Calabi–Yau metrics than fixed kernel methods, such as the Ricci flow.

Smooth asymptotics for collapsing Calabi–Yau metrics

AbstractWe prove that Calabi–Yau metrics on compact Calabi–Yau manifolds whose Kähler classes shrink the fibers of a holomorphic fibration have a priori estimates of all orders away from the singular fibers. To this end, we prove an asymptotic expansion of these metrics in terms of powers of the fiber diameter, with ‐order remainders that satisfy uniform ‐estimates with respect to a collapsing family of background metrics. The constants in these estimates are uniform not only in the sense that they are independent of the fiber diameter, but also in the sense that they only depend on the constant in the estimate for known from previous work of the second‐named author. For , the new estimates are proved by blowup and contradiction, and each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.

Complete Calabi–Yau metrics from smoothing Calabi–Yau complete intersections

We construct complete Calabi–Yau metrics on non-compact manifolds that are smoothings of an initial complete intersection V0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_0$$\\end{document} that is a Calabi–Yau cone, extending the work of Székelyhidi (Duke Math J 168(14):2651–2700, 2019). The constructed Calabi–Yau manifold has tangent cone at infinity given by C×V0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}\ imes V_0$$\\end{document}. This construction produces Calabi–Yau metrics with fibers having varying complex structures and possibly isolated singularities.

On the collapsing of Calabi–Yau manifolds and Kähler–Ricci flows

Abstract We study the collapsing of Calabi–Yau metrics and of Kähler–Ricci flows on fiber spaces where the base is smooth. We identify the collapsed Gromov–Hausdorff limit of the Kähler–Ricci flow when the divisorial part of the discriminant locus has simple normal crossings. In either setting, we also obtain an explicit bound for the real codimension-2 Hausdorff measure of the Cheeger–Colding singular set and identify a sufficient condition from birational geometry to understand the metric behavior of the limiting metric on the base.

CYJAX: A package for Calabi-Yau metrics with JAX

We present the first version of CYJAX, a package for machine learning Calabi–Yau metrics using JAX. It is meant to be accessible both as a top-level tool and as a library of modular functions. CYJAX is currently centered around the algebraic ansatz for the Kähler potential which automatically satisfies Kählerity and compatibility on patch overlaps. As of now, this implementation is limited to varieties defined by a single defining equation on one complex projective space. We comment on some planned generalizations.More documentation can be found at: https://cyjax.readthedocs.io.The code is available at: https://github.com/ml4physics/cyjax.

Geodesics on a K3 surface near the orbifold limit

This article studies Kummer K3 surfaces close to the orbifold limit. We improve upon estimates for the Calabi–Yau metrics due to Kobayashi. As an application, we study stable closed geodesics. We use the metric estimates to show how there are generally restrictions on the existence of such geodesics. We also show how there can exist stable, closed geodesics in some highly symmetric circumstances due to hyperkähler identities.

SU(2)^2-Invariant Gauge Theory on Asymptotically Conical Calabi–Yau 3-Folds

We give a complete description of the behaviour of Calabi–Yau instantons and monopoles with an SU(2)^2-symmetry, on Calabi–Yau 3-folds with asymptotically conical geometry and SU(2)^2 acting with co-homogeneity one. We consider gauge theory on the smoothing and small resolution of the conifold, and on the canonical bundle of mathbb{C}mathbb{P}^1 times mathbb{C}mathbb{P}^1, with their known asymptotically conical co-homogeneity one Calabi–Yau metrics, and find new one-parameter families of invariant instantons. We also entirely classify the relevant moduli-spaces of instantons and monopoles satisfying a natural curvature decay condition, and show that the expected bubbling phenomena occur in these families of instantons.

Calabi–Yau metrics with conical singularities along line arrangements

Given a finite collection of lines $L_j \subset \mathbb{CP}^2$ together with real numbers $0 \lt \beta_j \lt 1$ satisfying natural constraint conditions, we show the existence of a Ricci–flat Kähler metric $g_{RF}$ with cone angle $2\pi\beta_j$ along each line $L_j$ asymptotic to a polyhedral Kähler cone at each multiple point. Moreover, we discuss a Chern–Weil formula that expresses the energy of $g_{RF}$ as a logarithmic Euler characteristic with points weighted according to the volume density of the metric.

Machine-learned Calabi–Yau metrics and curvature

Machine-learned Calabi–Yau metrics and curvature

Numerical metrics for complete intersection and Kreuzer–Skarke Calabi–Yau manifolds

We introduce neural networks (NNs) to compute numerical Ricci-flat Calabi–Yau (CY) metrics for complete intersection and Kreuzer–Skarke (KS) CY manifolds at any point in Kähler and complex structure moduli space, and introduce the package cymetric which provides computation realizations of these techniques. In particular, we develop and computationally realize methods for point-sampling on these manifolds. The training for the NNs is carried out subject to a custom loss function. The Kähler class is fixed by adding to the loss a component which enforces the slopes of certain line bundles to match with topological computations. Our methods are applied to various manifolds, including the quintic manifold, the bi-cubic manifold and a KS manifold with Picard number two. We show that volumes and line bundle slopes can be reliably computed from the resulting Ricci-flat metrics. We also apply our results to compute an approximate Hermitian–Yang–Mills connection on a specific line bundle on the bi-cubic.

Degenerating Kähler–Einstein cones, locally symmetric cusps, and the Tian–Yau metric

Let $X$ be a complex projective manifold and let $D\subset X$ be a smooth divisor. In this article, we are interested in studying limits when $\beta\to 0$ of K\"ahler-Einstein metrics $\omega_\beta$ with a cone singularity of angle $2\pi \beta$ along $D$. In our first result, we assume that $X\setminus D$ is a locally symmetric space and we show that $\omega_\beta$ converges to the locally symmetric metric and further give asymptotics of $\omega_\beta$ when $X\setminus D$ is a ball quotient. Our second result deals with the case when $X$ is Fano and $D$ is anticanonical. We prove a folklore conjecture asserting that a rescaled limit of $\omega_\beta$ is the complete, Ricci flat Tian-Yau metric on $X\setminus D$. Furthermore, we prove that $(X,\omega_\beta)$ converges to an interval in the Gromov-Hausdorff sense.

Calabi–Yau metrics, energy functionals and machine learning

In this paper, we apply machine learning to the problem of finding numerical Calabi–Yau metrics. We extend previous work on learning approximate Ricci-flat metrics calculated using Donaldson’s algorithm to the much more accurate “optimal” metrics of Headrick and Nassar. We show that machine learning is able to predict the Kähler potential of a Calabi–Yau metric having seen only a small sample of training data.

On the Hölder Estimate of Kähler–Ricci Flow

AbstractIt is conjectured by Song–Tian that the normalized Kähler–Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle is uniformly smooth away from the singular fibers. In this work, we adapt the method in the work of Hein–Tosatti on collapsing Calabi–Yau metrics and obtain a uniform spatial Hölder estimate along the normalized Kähler–Ricci flow for all time away from the singular fiber.

Complete noncompact G2-manifolds from asymptotically conical Calabi–Yau 3-folds

We develop a powerful new analytic method to construct complete noncompact Ricci-flat 7-manifolds, more specifically G2-manifolds, that is, Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction gives the first general analytic construction of complete noncompact Ricci-flat metrics in any odd dimension and establishes a link with the Cheeger–Fukaya–Gromov theory of collapse with bounded curvature. The construction starts with a complete noncompact asymptotically conical Calabi–Yau 3-fold B and a circle bundle M→B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics gϵ on M that collapses with bounded curvature as ϵ→0 to the original Calabi–Yau metric on the base B. The G2-metrics we construct have controlled asymptotic geometry at infinity, so-called asymptotically locally conical (ALC) metrics; these are the natural higher-dimensional analogues of the asymptotically locally flat (ALF) metrics that are well known in 4-dimensional hyper-Kähler geometry. We give two illustrations of the strength of our method. First, we use it to construct infinitely many diffeomorphism types of complete noncompact simply connected G2-manifolds; previously only a handful of such diffeomorphism types was known. Second, we use it to prove the existence of continuous families of complete noncompact G2-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete noncompact G2-metrics were known.

On the degeneration of asymptotically conical Calabi–Yau metrics

We study the degenerations of asymptotically conical Ricci-flat Kähler metrics as the Kähler class degenerates to a semi-positive class. We show that under appropriate assumptions, the Ricci-flat Kähler metrics converge to a incomplete smooth Ricci-flat Kähler 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.

On collapsing Calabi–Yau fibrations

We develop some techniques to study the adiabatic limiting behaviour of Calabi–Yau metrics on the total space of a fibration, and obtain strong control near the singular fibres by imposing restrictions on the singularity types. We prove a uniform lower bound on the metric up to the singular fibre, under fairly general hypotheses. Assuming a result in pluripotential theory, we prove a uniform fibre diameter bound for a Lefschetz K3 fibred Calabi–Yau $3$‑fold, which reduces the study of the collapsing metric to a locally non-collapsed situation, and we identify the Gromov–Hausdorff limit of the rescaled neighbourhood of the singular fibre.

A “boundedness implies convergence” principle and its applications to collapsing estimates in Kähler geometry

We establish a general “boundedness implies convergence” principle for a family of evolving Riemannian metrics. We then apply this principle to collapsing Calabi–Yau metrics and normalized Kähler–Ricci flows on minimal models to obtain convergence results.

Renormalized characteristic forms of the Cheng–Yau metric and global CR invariants

For each invariant polynomial Φ, we construct a global CR invariant via the renormalized characteristic form of the Cheng–Yau metric on a strictly pseudoconvex domain. When the degree of Φ is 0, the invariant agrees with the total Q′-curvature. When the degree is equal to the CR dimension, we construct a primed pseudo-hermitian invariant IΦ′ which integrates to the corresponding CR invariant. These are generalizations of the I′-curvature on CR five-manifolds, introduced by Case–Gover.

Machine Learning Calabi–Yau Metrics

AbstractWe apply machine learning to the problem of finding numerical Calabi–Yau metrics. Building on Donaldson's algorithm for calculating balanced metrics on Kähler manifolds, we combine conventional curve fitting and machine‐learning techniques to numerically approximate Ricci‐flat metrics. We show that machine learning is able to predict the Calabi–Yau metric and quantities associated with it, such as its determinant, having seen only a small sample of training data. Using this in conjunction with a straightforward curve fitting routine, we demonstrate that it is possible to find highly accurate numerical metrics much more quickly than by using Donaldson's algorithm alone, with our new machine‐learning algorithm decreasing the time required by between one and two orders of magnitude.

Uniqueness of some Calabi–Yau metrics on $${\mathbf {C}}^{{n}}$$

We consider the Calabi–Yau metrics on $$\mathbf {C}^n$$ constructed recently by Yang Li, Conlon–Rochon, and the author, that have tangent cone $$\mathbf {C}\times A_1$$ at infinity for the $$(n-1)$$ -dimensional Stenzel cone $$A_1$$ . We show that up to scaling and isometry this Calabi–Yau metric on $$\mathbf {C}^n$$ is unique. We also discuss possible generalizations to other manifolds and tangent cones.