ON THE BOUNDED DERIVED CATEGORY OF IGr(3, 7)
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Springer Science+Business Media New York (2020)
ON THE BOUNDED DERIVED CATEGORY OF IGr(3, 7) A. FONAREV∗ Algebraic Geometry Section Steklov Mathematical Institute of Russian Academy of Sciences 8 Gubkin str. Moscow 119991, Russia Laboratory of Mirror Symmetry NRU HSE 6 Usacheva str. Moscow 119048, Russia [email protected] Abstract. We construct a minimal Lefschetz decomposition of the bounded derived category of coherent sheaves on the isotropic Grassmannian IGr(3, 7). Moreover, we show that IGr(3, 7) admits a full exceptional collection consisting of equivariant vector bundles.
1. Introduction One of the most important invariants of a smooth projective variety X is the bounded derived category Db (X) of coherent sheaves on it. As it often happens, the bounded derived category (from now on we will drop the word bounded) is rather easy to define, but quite difficult to describe explicitly. A rather fruitful approach to the latter problem is to split the derived category into smaller pieces, which is precisely the notion of a semiorthogonal decomposition. Then one can study the components of a given decomposition on their own and, finally, how they are glued together. In the best case scenario one can decompose the derived category in such a way that the pieces are as simple as they can get: equivalent to the derived category of a point, which is in turn equivalent to the category of graded finite-dimensional vector spaces over the base field. Such a decomposition corresponds to what is called a full exceptional collection. The study of exceptional collections goes back to seminal works of A. Beilinson ([1]) and M. Kapranov ([7]), where it was shown that projective spaces and, more generally, Grassmannians admit full exceptional collections consisting of vector bundles. Having a full exceptional collection is a very strong condition on the variety. First of all, it is very easy to give an example of a variety whose derived category does not admit any nontrivial semiorthogonal decompositions at all: any smooth DOI: 10.1007/S00031-020-09614-z The author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. No 14.641.31.0001. The author is a “Young Russian Mathematics” award winner and a Simons-IUM fellow and would like to thank its sponsors and jury. Received May 17, 2019. Accepted June 9, 2020. Corresponding Author: A. Fonarev, e-mail: [email protected] ∗
A. FONAREV
projective curve of positive genus would suffice (see [15]). Next, the Grothendieck group of the variety, K0 (X), must necessarily be free. Also, it was recently shown that the integral Chow motive of a smooth projective variety of dimension at most 3 which admits a full exceptional collection is of Lefschetz type (see [6]). Finally, a conjecture by D. Orlov predicts that having a full exceptional collection implies rationality. It is worth mentioning that even though varieties with a full exceptional collection are rare, it was recently proved that any triangulated category with a full exceptional collection
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