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Classification of generalized Kähler-Ricci solitons on complex surfaces. (English) Zbl 1501.53099

In complex geometry, the Enriques-Kodaira classification is one of the most celebrated results of the 20th century. Analogous to the uniformization theorem for Riemann surfaces, this result gives a classification of compact complex surfaces (i.e., real four-dimensional) into ten classes. Many of these families are fairly well-understood, but several ones are still somewhat mysterious.
In order to better understand the geometry of these more complicated families, one possibility is to find a geometric flow which (hopefully) converges to a canonical metric. This strategy was most famously used by Perelman in his proof of the Poincaré and Geometrization theorems, and a similar approach seems promising for understanding complex surfaces.
However, there is an immediate issue of finding a suitable flow. For non-Kähler manifolds, Ricci flow is not well-suited because it doesn’t preserve the Hermitian structure. Instead, it is necessary to find other flows which smooth the metric while respecting the complex geometry. J. Streets and G. Tian [Int. Math. Res. Not. 2010, No. 16, 3101–3133 (2010; Zbl 1198.53077)] identified one flow which is particularly useful for studying pluriclosed Hermitian manifolds, which are Hermitian manifolds whose Kähler form \(\omega=g(J \cdot, \cdot)\) satisfies \[ \partial \bar{\partial} \omega=0. \] The flow they studied became known as pluriclosed flow, and it appears particularly helpful for understanding the geometry of Kodaira’s Class VII surfaces.
Before it is possible to prove any classification theorem using flows, it is necessary to first understand the fixed point of the flow, which are known as solitons. Such solitons appear as limit points or singularity models, so it is crucial to understand their geometry. For pluriclosed flow on surfaces, the steady soliton equations reduce to the following coupled system of equations: \begin{align*} \mathrm{Ric}-\frac{1}{4} H^{2}+\nabla^{2} f=0 \\ \frac{1}{2} d^{*} H+i_{\frac{1}{2} \nabla f} H=0 \end{align*} where \(H=d^{c} \omega\) is a closed 3-form, \(H_{i j}^{2}=H_{i p q} H_{j}^{p q}\), and \(f\) is a smooth function.
In this paper, the authors construct compact steady solitons for the pluriclosed flow explicitly. By doing so, they find that these solitons admit very rich structure, which is not at all apparent from the soliton equations alone. In particular, these steady solitons admit a generalized Kähler structure in two distinct ways, with vanishing and nonvanishing Poisson structure.
To explain this result, it is helpful to consider a simpler analogy. (This analogy was given by Professor Ustinovsky when discussing this work.) If one considers the Ricci flow on a Riemann surface which is topologically a sphere but whose metric has no non-trivial isometries, for all positive time the symmetry group will remain trivial. However, in the limit, the metric will converge to a round sphere, whose symmetry group is \(\mathrm{O}(3)\), which is much larger. The results of this paper suggest that a similar phenomenon occurs with the pluriclosed flow, in which the limits of the geometric flow admit very rich structure, even when the structure is not present earlier along the flow.
In the latter half of the paper, the authors provide several applications of their results. In particular, they provide generalized Kähler structures with non-vanishing Poisson structure on non-standard Hopf surfaces, which answers the question of whether such structures always exist. In the setting of generalized Kähler geometry with vanishing Poisson structure, they show that these solitons are unique and are global attractors for the generalized Kähler-Ricci flow among metrics with maximal symmetry.

MSC:

53E30 Flows related to complex manifolds (e.g., Kähler-Ricci flows, Chern-Ricci flows)
35C08 Soliton solutions

Citations:

Zbl 1198.53077
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