Surveys in Differential Geometry

We survey some recent progress on the deformed Hermitian–Yang–Mills (dHYM) equation. We discuss the role of geometric invariant theory (GIT) in approaching the solvability of the dHYM equation, following work of the first author and S.‑T. Yau. We compare the GIT picture with the conjectural picture for dHYM involving Bridgeland stability. In particular, following Arcara–Miles [3], we show that on the blow-up of $\mathbb{P}^2$ any line bundle admitting a solution of the deformed Hermitian–Yang–Mills equation is Bridgeland stable, but not conversely. Finally, we survey some recent progress on heat flows associated to the dHYM equation.

We discuss a model for associative submanifolds in $G_2$ manifolds with $K3$ fibrations, in the adiabatic limit. The model involves graphs in a $3$‑manifold whose edges are locally gradient flow lines. We show that this model produces analogues of known singularity formation phenomena for associative submanifolds. We propose conjectures on the existence of associative and special Lagrangian submanifolds in certain product spaces, corresponding to the vertices of the graphs.

Moduli spaces of stable sheaves on smooth projective surfaces are in general singular. Nonetheless, they carry a virtual class, which—in analogy with the classical case of Hilbert schemes of points—can be used to define intersection numbers, such as virtual Euler characteristics, Verlinde numbers, and Segre numbers.

Our purpose is to survey the recent results concerning the study of germs of singular varieties in a symplectic space. We formulate the theory of symplectic bifurcations with the symplectic group action on the reduced space and provide the complete symplectic classification of simple and unimodal singularities of planar curves. Their differential and symplectic invariants, e.g. symplectic defect and $\delta$‑invariant were distinguished and the corresponding cyclic moduli spaces were calculated. The classification problem of singular differentiable mappings to the symplectic space was considered and basic invariants for classification, symplectic codimension, symplectic isotropic codimension, symplectic multiplicity were constructed. The methods of geometric and algebraic restrictions of differential forms to singular varieties were presented and applied in symplectic classification of space curves and singular surfaces.

In this note, we survey the developments that led to the invention of Mixed‑Spin‑P field theory (MSP theory) which interpolates Gromov–Witten theory of quintic Calabi–Yau threefolds and Landau–Ginzburg theory of the corresponding quintic polynomials.

The paper is concerned with the maximization of Laplace eigenvalues on surfaces of given volume with a Riemannian metric in a fixed conformal class. A significant progress on this problem has been recently achieved by Nadirashvili–Sire and Petrides using related, though different methods. In particular, it was shown that for a given $k$, the maximum of the $k$‑th Laplace eigenvalue in a conformal class on a surface is either attained on a metric which is smooth except possibly at a finite number of conical singularities, or it is attained in the limit while a “bubble tree” is formed on a surface. Geometrically, the bubble tree appearing in this setting can be viewed as a union of touching identical round spheres. We present another proof of this statement, developing the approach proposed by the second author and Y. Sire. As a side result, we provide explicit upper bounds on the topological spectrum of surfaces.

The Yang–Mills–Higgs equations, while they arose in theoretical physics, have become a central object of study in geometric analysis. The moduli spaces of solutions are used to study problems in algebraic geometry as well as topology. To understand these moduli spaces, it is natural to ask what sort of objects are limits in a weak sense of sequences of solutions. In many examples, it can be shown that subsequences of solutions converge to a solution off a set of Hausdorff codimension $4$. In four dimensions, these point singularities are removable. The question arises of when this singular set of dimension $n-4$ is removable in higher dimensions. In a classic paper, Tao and Tian [11] prove an important step in the removability theorem. What they show is that, in general, independent of the equation, if the case the curvature of a connection defined on a set of Hausdorff codimension $4$ is sufficiently small in a certain Morrey space, that there exists a smaller ball and a gauge in which the connection is $d+A$, where $A$ is co-closed and bounded by norms on the curvature. One can now treat the Yang–Mills–Higgs equations as a standard elliptic system and obtain further regularity. This choice of Morrey space is natural, because monotonicity of solutions provides estimates exactly in this space. However, since the solution is only defined off a set of Hausdorff codimension $4$, it is not known that the monotonicity theorem is true for limiting singular solutions. Only by adding the assumption of stationary with respect to diffeomorphisms in the entire domain can one apply the theorem to limiting singular connections. Unfortunately, we cannot fill in this gap. We provide a new proof of the Tao–Tian result. Our proof uses the equation and a maximum principle for a differential inequality derived from the equation via averaging. In this manner, we can avoid the geometric construction of a good gauge in a weak setting via averaging exponential gauges, as we first obtain further estimates on the curvature from the inequality. Our main result Corollary 4 is that, if the curvature is sufficiently small in a Morrey space, and the solution is defined and smooth off a set of Hausdorff codimension $4$ with the covariant derivative of the Higgs field in the same Morrey space, then the solution is smoothly gauge equivalent to one which extends smoothly over the singular set. In the case that the solution has small energy, and is a critical point of the integral, or even just critical with respect to diffeomorphisms in the entire domain, not just off the singular set, the smallness of the curvature in the Morrey space in a smaller domain can be verified and leads to Theorem 11 as a corollary of Corollary 4. There may be other ways to obtain the bounds on the curvature in the Morrey space; hence we state the main result in terms of small curvature in the Morrey space. The results apply to most of the coupled Yang–Mills–Higgs equations, and we do not go into the details in this paper. We do assume that the Higgs field is bounded, and explain why this follows from $L^2$ bounds on it. Many interesting equations, such as the Kapustin–Witten equations and the equations for a complex flat connection do not come with such a bound on sequences. Moreover, the singular set of a renormalized limit is Hausdorff codimension $2$, and we sadly have nothing to say about these singularities.

We re-examine quantization via branes with the goal of understanding its relation to geometric quantization. If a symplectic manifold $M$ can be quantized in geometric quantization using a polarization $\mathcal{P}$, and in brane quantization using a complexification $Y$, then the two quantizations agree if $\mathcal{P}$ can be analytically continued to a holomorphic polarization of $Y$. We also show, roughly, that the automorphism group of $M$ that is realized as a group of symmetries in brane quantization of $M$ is the group of symplectomorphisms of $M$ that can be analytically continued to holomorphic symplectomorphisms of $Y$. We describe from the point of view of brane quantization several examples in which geometric quantization with different polarizations gives equivalent results.

We review a proof of the well know result stating that moduli spaces of stable sheaves with fixed Chern character on a polarized $K3$ surface are deformations of a hyperkähler variety of Type $K3^{[n]}$ (if a suitable numerical hypothesis is satisfied). In a recent work we have adapted that proof in order to prove results on moduli of vector bundles on polarized hyperkähler varieties of Type $K3^{[2]}$ — this is the content of the second part of the paper.

In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also includes statements about the structure of compact manifolds of positive scalar curvature extending the work of [SY1] to all dimensions. The technical work in this paper is to construct minimal slicings and associated weight functions in the presence of small singular sets and to show that the singular sets do not become too large in the lower dimensional slices. It is shown that the singular set in any slice is a closed set with Hausdorff codimension at least three. In particular for arguments which involve slicing down to dimension $1$ or $2$ the method is successful. The arguments can be viewed as an extension of the minimal hypersurface regularity theory to this setting of minimal slicings.