Daigo Ito (Berkeley)
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PDL C-38
Title pretalk: Reconstruction and birational geometry in the Matsui spectrum of derived categories
Abstract pretalk: The bounded derived category of coherent sheaves captures various properties of smooth projective varieties. One of the most important results is the Bondal--Orlov reconstruction theorem, which states that if the (anti-)canonical bundle is ample, then the variety can be reconstructed from its derived category. In this talk, I will introduce a new and conceptually straightforward proof using the Matsui spectrum, a locally ringed space constructed from the derived category in the spirit of tensor triangular geometry. I will then explain how this perspective naturally leads to considering tensor-ample line bundles and extends to capture birational geometric phenomena.
Abstract pretalk: The bounded derived category of coherent sheaves captures various properties of smooth projective varieties. One of the most important results is the Bondal--Orlov reconstruction theorem, which states that if the (anti-)canonical bundle is ample, then the variety can be reconstructed from its derived category. In this talk, I will introduce a new and conceptually straightforward proof using the Matsui spectrum, a locally ringed space constructed from the derived category in the spirit of tensor triangular geometry. I will then explain how this perspective naturally leads to considering tensor-ample line bundles and extends to capture birational geometric phenomena.
Ttitle: A derived category analogue of the Nakai--Moishezon criterion
Abstract: In the study of derived categories of coherent sheaves, ample line bundles play a fundamental role; their tensor powers generate the derived category. This raises a natural question: does this generation property characterize ampleness? The answer is negative, but we show that this categorical property, tensor-ampleness, admits a classical numerical criterion naturally extending the Nakai–Moishezon criterion. Moreover, the cone of tensor-ample divisors lies between the big cone and the ample cone. In this talk I will focus on explaining the case of surfaces, where the geometry becomes especially clear. This is a joint work with Noah Olander.