Ricci flow on compact Kahler manifolds

PI建议阐明任意紧致Kahler流形上的Ricci流的行为。这是复微分几何领域的一个主要问题，近年来受到了极大的关注，这也要归功于佩雷尔曼在紧致实3-流形上关于Ricci流的里程碑式的工作。在Kahler情形中，众所周知，Ricci流保持Kahler条件，因此也称为Kahler-Ricci流(KRF)。它的最大生存时间可以根据流形的初始度量和复结构上同调来计算，特别地，它是无限的当且仅当流形的标准丛是数值有效的。Song-Tian的一个有影响力的程序旨在通过将其与代数几何中的最小模型程序联系起来，至少在Kahler流形是射影代数的情况下理解流的行为。该提案的第一个主要目标将是研究在有限时间内形成的奇点。PI和Collins的早期工作证明了在这种情况下，流沿着解析亚簇V形成奇点，从而证明了Feldman-Ilmanen-Knopf猜想。PI建议走得更远：首先，他建议证明不断发展的指标的直径在单一时间保持一致有界。第二，证明当且仅当流形允许Fano纤维结构时，总体积在奇点处归零(极限上同调类从该纤维结构的基拉回)。当复数维至多为3时，Pi与Zhang较早地证明了这一点。第三，他建议证明，如果总体积不为零，则上面的亚簇V可以被复解析压缩，并且流可以在一个新的紧致解析空间上重新开始，正如宋田所建议的那样。这一过程预计将以有限多个步骤终止，要么使用Fano纤维，要么使用所有正时间存在的解决方案。该提案的第二个主要目标将是了解所有积极时期存在的解决方案的长期行为。如上所述，当正则束在数值上有效时，它们正好存在。代数几何(丰度)中一个由来已久的猜想预测，在这种情况下，正则丛是半充分的，所以流形是由Calabi-Yau流形在较低维基上纤化的，通常是用奇异纤维。在这种情况下，PI建议识别(归一化)流的全局Gromov-Hausdorff极限，如Song-Tian猜想的那样，并证明远离奇异纤维的更高的正则性。最后，在不假设正则丛是半采样的情况下，PI建议证明KRF弱收敛于正则类中的闭正电流(对应于正则丛上的奇异厄米度规)，它与流的初始度量无关，并且具有极小的奇性。

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.