Ricci flow on compact Kahler manifolds

PI提出了一个任意的紧凑的Kahler流形上的Ricci流的行为阐明。这是一个重大问题，在该领域的复杂的微分几何，这已受到广泛关注，近年来，也感谢佩雷尔曼的里程碑式的工作里奇流紧凑的真实的3流形。在Kahler情况下，已知Ricci流保持Kahler条件，因此也称为Kahler-Ricci流（KRF）。它的最大存在时间可以根据初始度量和流形的复结构上同调地计算，特别是它是无限的当且仅当流形的正则丛是数值有效的。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相关联来理解流动的行为，至少当Kahler流形是投影代数时。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。早期的工作PI和柯林斯证明，在这种情况下，流动形式的奇点沿着一个解析子品种V，证明猜想的费尔德曼-Ilmanen-克诺夫。PI提出要走得更远：首先，他提出要证明演化度量的直径在奇异时间上保持一致有界。第二，证明了当且仅当流形允许Fano纤维化结构（并且极限上同调类从这个纤维化的基础上拉回来）时，总体积在奇点处为零。PI早些时候与Zhang一起证明了这一点，当时复维数至多为3。第三，他提出证明，如果总体积不趋于零，那么上面的子簇V可以复解析地收缩，并且流可以在一个新的紧致解析空间上重新开始，正如Song-Tian所建议的那样。这一过程预计将终止在许多步骤，无论是与法诺纤维化或与解决方案，存在的所有积极的时间。该提案的第二个主要目标将是了解所有正时间存在的解决方案的长期行为。如前所述，当正则丛在数值上有效时，它们正好存在。代数几何中一个长期存在的猜想（丰度）预言，在这种情况下，正则丛是半满的，因此流形被低维基上的卡-丘流形纤维化，通常具有奇异纤维。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所述，并证明远离奇异纤维的更高正则性。最后，在不假设规范丛是半样本的情况下，PI提出证明KRF弱收敛到规范类中的闭正流（对应于规范丛上的奇异厄米特度量），这与流的初始度量无关，并且具有最小奇点。

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.