Ricci flow on compact Kahler manifolds

PI提出了在任意紧化Kahler流形上解释Ricci流的行为。这是复杂微分几何领域的一个主要问题，近年来受到了广泛关注，这也要归功于佩雷尔曼关于紧实3流形上的里奇流的里程碑式的工作。在Kahler的情况下，已知Ricci流保留了Kahler条件，因此也被称为Kahler-Ricci流（KRF）。它的最大存在时间可以用流形的初始度量和复杂结构上同调地计算，特别是当且仅当流形的正则束是数值有效时，它是无限的。Song-Tian的一个有影响力的程序旨在通过将Kahler流形与代数几何中的最小模型程序联系起来，了解流的行为，至少当Kahler流形是射影代数时。该方案的第一个主要目标是研究在有限时间内形成的奇点。PI和Collins的早期工作证明，在这种情况下，流沿着解析子变种V形成奇点，证明了Feldman-Ilmanen-Knopf的一个猜想。PI提议更进一步：首先，他提议证明演化度量的直径在奇异时间保持均匀的有界。第二，证明当且仅当流形允许法诺振动结构（并且极限上同调类从该振动的基部拉回）时，总体积在奇点处趋于零。PI先前与Zhang证明了这一点，当复维数最多为3时。第三，他提出，如果总体积不趋近于零，那么上面的子变量V可以解析地收缩，流可以在一个新的紧化解析空间上重新开始，如Song-Tian所建议的那样。这一过程预计将在有限多个步骤中终止，要么是法诺纤维，要么是存在所有正时间的溶液。该建议的第二个主要目标将是理解存在于所有正时间的解决方案的长期行为。如上所述，当规范包在数值上有效时，这些特性就存在。代数几何中的一个长期猜想（丰度）预测，在这种情况下，规范束是半样本的，因此流形是由Calabi-Yau流形在较低维的基础上编织的，通常是奇异纤维。在这种情况下，PI建议确定（归一化）流的全局Gromov-Hausdorff极限，正如Song-Tian推测的那样，并证明远离奇异纤维的更高规律性。最后，在不假设正则束是半样本的情况下，PI提出证明KRF弱收敛于正则类中的闭正电流（对应于正则束上的奇异厄米度规），它与流的初始度规无关，并且具有极小的奇异性。

The PI proposes to elucidate the behavior of the Ricci flow on an arbitrary compact Kahler manifold. This is a major question in the field of complex differential geometry, which has received much attention in recent years, thanks also to Perelman's landmark work on the Ricci flow on compact real 3-manifolds. In the Kahler case, it is known that the Ricci flow preserves the Kahler condition, and it is thus also known as the Kahler-Ricci flow (KRF). Its maximal existence time can be computed cohomologically in terms of the initial metric and of the complex structure of the manifold, and in particular it is infinite if and only if the canonical bundle of the manifold is numerically effective. An influential program of Song-Tian aims to understand the behavior of the flow, at least when the Kahler manifold is projective algebraic, by relating it to the Minimal Model Program in algebraic geometry. The first main goal of the proposal will be to study singularities which form in finite time. Earlier work of the PI and Collins proved that in this case the flow forms singularities along an analytic subvariety V, proving a conjecture of Feldman-Ilmanen-Knopf. The PI proposes to go substantially further: first, he proposes to show that the diameter of the evolving metrics remains uniformly bounded above at the singular time. Second, to show that the total volume goes to zero at the singularity if and only if the manifold admits a Fano fibration structure (and the limiting cohomology class is pulled back from the base of this fibration). The PI proved this earlier with Zhang when the complex dimension is at most 3. And third, he proposes to show that if the total volume does not go to zero, then the subvariety V above can be contracted complex analytically and the flow can restart on a new compact analytic space, as suggested by Song-Tian. This process is expected to terminate in finitely many steps, either with a Fano fibration or with a solution that exists for all positive time. The second main goal of the proposal will be to understand the long-time behavior of solutions that exist for all positive time. As mentioned above, these exist precisely when the canonical bundle is numerically effective. A long-standing conjecture in algebraic geometry (abundance) predicts that in this case the canonical bundle is semiample, so the manifold is fibered by Calabi-Yau manifolds over a lower-dimensional base, in general with singular fibers. In this case, the PI proposes to identify the global Gromov-Hausdorff limit of the (normalized) flow, as conjectured by Song-Tian, and to prove higher regularity away from the singular fibers. Lastly, without assuming that the canonical bundle is semiample, the PI proposes to show that the KRF converges weakly to a closed positive current in the canonical class (which corresponds to a singular Hermitian metric on the canonical bundle), which is independent of the initial metric of the flow, and has minimal singularities.