Over the past 5 years, my fellow students in Vanderbilt's math department have helped me too many times to count—for their help and camaraderie I am very grateful. I would like to thank Zach Edwards for accidentally trapping me in Shields Market's fridge – many of the final ideas in this thesis happened unexpectedly there. First, ALE manifolds are notable in the field of physics in the study of gravitational instantons – 4-dimensional complete Riemannian manifolds that satisfy the vacuum Einstein equations.
One can see this tameness greatly exploited in the work of Arezzo and Pacard [Arezzo and Pacard, 2006] where they provide a method to construct cscK metrics on inflation of compact cscK manifolds that have no non-zero holomorphic vector allow. fields that disappear somewhere.
Summary of Results
Concretely speaking, we have a family of extreme metrics onΣk, a space where the existence of cscK metrics is hindered, and by taking a limit we are able to obtain a scalar planar metric onO(−k). This opens the possibility of analyzing the moduli space of Kühler metrics at these magnifications by representing classes for which the existence/non-existence of cscK metrics is known. Apart from our construction above, we also address a problem of non-existence of cscK metrics on ruled surfaces.
The existence of cscK metrics on certain inflations can be shown using parabolic stability due to Rollin-Singer [Rollin and Singer, 2009b].
General background
Extremal Metrics
Since the first term in (II.1) is non-negative and the second is constant within a K¨ahler class, cscK metrics minimize the Calabi function since S(ω) = ˆS. Therefore, the task of determining whether a metric is extremal becomes a problem of differential equations.
ALE Manifolds
ADM Mass
In [Hein and LeBrun, 2016], Hein and LeBrun prove very remarkable results about ALE K¨ahler manifolds and their mass. This allows them to identify a compactly supported representative of the first Chern class that plays a key role in their theorem below. Furthermore, since there is only one end, the mass becomes an invariant of the manifold itself rather than simply an invariant of an end.
In §II.2 we will be able to see the mass of our constructed metrics as the coefficient of the log term of our potential.
Construction of Scalar-flat, ALE Metrics
Asymptotically Euclidean Case
- Construction
- K¨ahler Class of ω m
- The Burns Metric
As mentioned above, we will look at metrics of the formω =i∂∂f(s)where f is a strictly convex function. To determine whether or not ω is extreme, we want to look at the second derivative of the scalar curvature of ω. We will use the Legendre transform to write our metric in terms of the Hwang-Singer momentum profile [Hwang and Singer, 2002].
The S1 symmetry of the z component of s tells us that we only need to look in the fiber. Second, we will later show that the K¨ahler class of the resulting metric is determined by m. Now that we have constructed our family of metrics (and verified that they are indeed metrics), one may be curious about some of their properties.
We can see that we actually have an extremal metric for every K¨ahler class Σ1. Burns then completed this metric by appending aCP1 to the origin to obtain the Burns metric. It is clear to see that the completed metric is then the metric on the increase of C2 at the origin, or, in other words, on the entire O(−1) space.
For our purposes, we will take a = 1 to remain consistent with our normalization of the area of the CP1 to be 2π. To see that the Burns metric is ALE, one must first convert to the standard Euclidean coordinates and look at the asymptotics as |z| gets large. However, there is a very attractive connection between the family we constructed in II.2.1.1 and the Burns metric that has so far gone unnoticed.
One way we can intuitively understand this limiting procedure is to look at how it affects areas D0, D∞, and F.
Asymptotically Locally Euclidean Case
- Construction
- K¨ahler Class of ω k,m
- Limit Metrics
- Ricci-flat case (k = 2)
- Non Ricci-flat case (k > 2)
Therefore, this limiting process can be understood as simply allowing the infinite divisor to grow to infinity. To determine whether the metric is extremal or not, we want to look at the second derivative of the scalar curvature. Specifically, we will shift our attention from perfect metrics on Σk to imperfect metrics on Σk\D∞.
The annihilation of C2,∞ affects the process of solving the potential, so we will consider the cases k= 2 and k >2 separately. Note that when checking the ALE condition, we are free to work in the diffeomorphism at infinity as we choose. In LeBrun's calculation [LeBrun, 1988] he showed that the sign of the logterm is aligned with the sign of the mass.
In this chapter we will look at the existence and non-existence of cscK metrics on rational surfaces. Although this approach can be applied to various regularized surfaces, we will demonstrate the approach on a specific 6-fold blow-up of CP1×CP1. Although we will not provide a complete classification of the K¨ahler cone, we will begin to paint a clearer picture of existence and non-existence.
We'll describe the details of the inflation in the next section, but it's worth noting some particularly useful properties of our choice of inflation before we begin. Second, this burst falls into an interesting location in the theory of canonical metrics, as it has a non-definitive first Chern class and is therefore outside the scope of the Kähler-Einstein program.
Parabolic Stability
Namely, existence and non-existence are often treated separately in the literature—that is, it is rare that the existence and non-existence of cscK metrics are both described on a given manifold. The positive scalar curvature setting is often more rigid than the zero and negative scalar curvature cases, making the results more surprising. Rollin-Singer explores other related notions of semi-/polystability [Rollin and Singer, 2009b], but we only need stable for our purposes.
Rollin-Singer shows in [Rollin and Singer, 2005] that there is a unique choice of inflation points to obtain the above diagram. If the parabolic structure has multiple points, this process is repeated for the remaining parabolic points. Rollin-Singer then shows [Rollin and Singer, 2009b, Theorem D] that such an iterative enlargement of a parabolic stable ruled surface admits a cscK metric.
Example of Parabolically Stable Ruled Surfaces
These singularities can be resolved leading to the Hirzebruch-Jung string corresponding to the weight 1/2 as described above. The resolution of the singularities along the 3-distinct fibers allows a cscK metric, provided the areas of the singular divisors of each resolution are small and the proper transformation of the singular fibers is large (relative to the singular divisors). Note that the fiber F′ has half the area of a generic fiber due to the quotienting procedure.
The diagram on the right depicts the resolution where EenE′ denotes the exceptional divisors introduced by the resolution and F˜′ is the right transformation of F′. It will be easier for our calculation later to use a basis for homology resulting from the iterative blow-up, so we introduce it here. We let Ei, Fi and Gi denote the ith exceptional divisor of the repeated inflations at K1, K2 and K3 respectively.
Note that in this basis, the exceptional divisors from the desingularizations correspond to the curves labeled with index1 and the proper fiber transformations (for example, H˜ in the figure above), while the proper fiber transformation in the desingularization corresponds to the curves labeled with index2.
Stability
Slope Stability
The last part needed to understand the constraint given by slope stability is the slope of a submanifoldZ ⊂X. Letπ: ˆX→ X be the inflation of X togetherZ with extraordinary divisorE and let JZ denote the ideal bundle of Z. -Thomas refers to this as a kind of correction term which should account for the difference between the Hilbert polynomial of a normal crossing variety with 2 components and the sum of the Hilbert polynomials of its components.
In the above definition, the fact that the global sectionsL⊗k⊗ JZϵ(Z,X,L)k saturateJZϵ(Z,X,L)k means that the global sections generate the line bundleπ∗L⊗O(−ϵ(Z, X, L)kE). Therefore, slope stability provides an obstacle to the existence of cscK metrics in the compact setting. However, it is often easier to work with the related notion of quotient slope instead.
These formulas are particularly easy to work with, and therefore, will be the primary calculation tool we use for slope stability.
The Destabilizing Curve
The procedure above is a quick and painless method of showing positivity, but it comes with the price that the coefficients indexed by 2 depend on those indexed by 1. In our later argument, this dependence will not be a problem for the coefficients of [Fi] and [Gi]. Remember that for surfaces, the Nakai-Moishezon criterion says thatΩ > 0if and only ifΩ2 > 0andΩ·C > 0for every curve C ⊆ X.
Since C intersects at least one of E1 and E2, at least one of C·E1 and C·E2 is in fact greater than or equal to 1. Note that the above lemma tells us information about the Seshadri constantϵ(E1, ); namely thatϵ(E1, X, L)≥min{4a5 ,αa}. Now that we know which classes are K¨ahler, we can start finding classes where cscK metrics are hindered.
To achieve this, we will show that the curve E1 is destabilizer(M, L) where L is a line bundle such that c1(L) = Ω as defined above (III.1) with rational coefficients. Note that from a technical point of view it must be an integral cohomology class for Ω to represent c1(L), which it is not. Then kΩ admits cscK metrics if and only ifΩ does, so we can assume that our classΩ has integer coefficients.
Recall that E1 destabilizes if, relative to the quotient slope:. who are satisfied because of our choice of fi's and gi's.
