Categorical absorptions of singularities and degenerations (Q6543023)

For a singular proper scheme \(X\) over a field, there are two fundamental triangulated categories: the category \(D^{\text{perf}}(X)\) of perfect complexes, which is proper but not smooth, and the bounded derived category \(D^b(X)\) of coherent sheaves, which is smooth but not proper. In the literature there are many works on ``categorical resolution'' of these categories where they are replaced by a common ``enlargement'' which is smooth and proper.\N\NThis paper gives a different direction of study: It seeks a ``smaller'' smooth and proper triangulated category. The paper introduces the notion of a categorical absorption of singularities (Definition 1.1): It is an admissible subcategory \(\mathcal{P} \subset D^b(X)\) which is responsible for singularity and the remaining part \(\mathcal{D} := {}^\perp\mathcal{P}\) is smooth and proper.\N\NThe idea is elucidated in two situations in Example 1.2, which lead to the notion of \(\mathbb{P}^{\infty,q}\)-objects (Definition 1.7). \S2 and \S3 are devoted to the study of the category of ordinary double point, the category generated by a \(\mathbb{P}^{\infty,q}\)-object. This paper mainly studies \(\mathbb{P}^{\infty,1}\)- and \(\mathbb{P}^{\infty,2}\)-objects. It turns out that semi-orthogonal collection of these objects provide nice categorical absorption of singularities in family (Theorems 1.8, 1.9; proved in \S4).\N\NAnother main result of the paper is the construction of a categorical absorption for a projective variety with isolated ordinary double points (Theorem 6.1). It is based on the results in \S4 and the crepant categorical resolution discussed in \S5. The obtained \(\mathcal{P}\) has some nice properties, discussed in \S6.2. The final \S6.3 gives some geometric applications.