Watanabe, Kiwamu

Associate Professor,Graduate School of Science and Engineering, Chuo University*Profile is at the time of the award.

2020Inamori Research GrantsScience & Engineering

Research topics
Characterization of homogeneous varieties among Fano varieties
In this research, we study characterizations of highly symmetric rational homogeneous manifolds among Fano manifolds. The starting point of this research is the solution of the Hartshorne conjecture due to S. Mori: the only projective manifold with ample tangent bundle is the projective space. It is known that this conjecture is a generalization of the Frankel conjecture in complex geometry. After that, the generalized Frankel conjecture was solved by N. Mok. Based on this historical background, the final goal of our research is to solve the following conjecture due to F. Campana and T. Peternell, which is a generalization of the above conjectures including the Hartshorne conjecture: (CP conjecture) any Fano manifold with nef tangent bundle is rational homogeneous.

Outline of Research Achievments

A figure represented locally as a set of zeros of some polynomials is called an algebraic variety. Lines, circles, and parabolas that we learned in middle school and high school are basic examples of algebraic varieties. When we think of a circle or a parabola, we sometimes think of its tangent lines, but as a generalization of that, the tangent bundle of the manifold is determined by collecting the tangent spaces of the manifold. I studied the structure of algebraic varieties in terms of positivity of tangent bundles. As a starting point for our research, we can cite Mori’s solution to the Hartshorn conjecture, which states that ”the only smooth projective variety with ample tangent bundle is a projective space”. Geometrically, we argue that all-rounded (smooth projective) varieties have only a basic shape called the projective space. As a generalization of this result, I conducted joint research with Akihiro Kanemitsu of Saitama University, and succeeded in obtaining a structural theorem for manifolds with nef tangent bundle. Demailly-Peternell-Schneider (hereinafter referred to as DPS) showed that the previously known results over algebraically closed fields of characteristic 0 also hold over algebraically closed fields of positive characteristic. In positive characteristic, various obstacles that do not appear in characteristic 0 appear, but we overcame them and established a truly new method to solve the problem. The results are being submitted to an academic journal. I also succeeded in obtaining the structural theorem for manifolds whose 2nd exterior power of the tangent bundle is nef when the characteristic is 0. This result can also be viewed as a generalization of the DPS theorem above. This result has been published in Advances in Mathematics.

Watanabe K (2021)  Positivity of the second exterior power of the tangent bundles. Advances in Mathematics 385: 107757. https://doi.org/10.1016/j.aim.2021.107757

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