A Free Boundary Monge–Ampère Equation and Applications to Complete Calabi–Yau Metrics

After providing an explicit K-stability condition for a ℚ-Gorenstein log spherical cone, we prove the existence and uniqueness of an equivariant K-stable degeneration of the cone, and deduce uniqueness of the asymptotic cone of a given complete 𝐾-invariant Calabi–Yau metric in the trivial class of an affine 𝐺-spherical manifold, 𝐾 being the maximal compact subgroup of 𝐺. Next, we prove that the valuation induced by 𝐾-invariant Calabi–Yau metrics on affine 𝐺-spherical manifolds is in fact 𝐺-invariant. As an application, we point out an affine smoothing of a Calabi–Yau cone that does not admit any 𝐾-invariant Calabi–Yau metrics asymptotic to the cone. Another corollary is that, on C 3 \mathbb{C}^{3} , there are no complete Calabi–Yau metrics with maximal volume growth and spherical symmetry other than the standard flat metric and the Li–Conlon–Rochon–Székelyhidi metrics with horospherical asymptotic cone. This answers the question whether there is a nontrivial asymptotic cone with smooth cross section on C 3 \mathbb{C}^{3} raised by Conlon–Rochon when the symmetry is spherical.

A method for constructing explicit Calabi–Yau metrics in six dimensions in terms of an initial hyperkahler structure is presented. The equations to solve are nonlinear in general, but become linear when the objects describing the metric depend only on one complex coordinate of the hyperkahler four-dimensional space and its complex conjugate. This situation in particular gives a dual description of D6-branes wrapping a complex 1-cycle inside the hyperkahler space, Fayyazuddin (2007 Class. Quantum Grav. 24 3151). The present work generalizes the construction given in that reference. But the explicit solutions we present correspond to the nonlinear problem. This is a nonlinear equation with respect to two variables which, with the help of some specific anzatz, is reduced to a nonlinear equation with a single variable solvable in terms of elliptic functions. In these terms we construct an infinite family of non-compact Calabi–Yau metrics.

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.

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.

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.

We 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.

We 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.

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.

We construct infinitely many complete Calabi–Yau metrics on $\\mathbf{C}^{n}$ for $n\\geq 3$ , with maximal volume growth and singular tangent cones at infinity. In addition, we construct Calabi–Yau metrics in neighborhoods of certain isolated singularities whose tangent cones have singular cross section, generalizing work of Hein and Naber.

Explicit Calabi–Yau metrics in [formula omitted] possessing an isometry group with orbits of codimension one

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.

Invariant forms, associated bundles and Calabi–Yau metrics

We first determine the asymptotic cone of the steady gradient Kähler–Ricci soliton of the Taub-NUT type constructed by Apostolov and Cifarelli. Then we study a special case and prove that it is an ALF Calabi–Yau metric in a certain sense. Finally, we construct new ALF Calabi–Yau metrics on crepant resolution of its quotients modeled on it using the method of Tian–Yau–Hein.

We prove that the deformation theory of compactifiable asymptotically cylindrical Calabi–Yau manifolds is unobstructed. This relies on a detailed study of the Dolbeault–Hodge theory and its description in terms of the cohomology of the compactification. We also show that these Calabi–Yau metrics admit a polyhomogeneous expansion at infinity, a result that we extend to asymptotically conical Calabi–Yau metrics as well. We then study the moduli space of Calabi–Yau deformations that fix the complex structure at infinity. There is a Weil–Petersson metric on this space, which we show is Kahler. By proving a local families L 2-index theorem, we exhibit its Kahler form as a multiple of the curvature of a certain determinant line bundle.

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.