Derived invariants of irregular varieties and Hochschild homology

Abstract

We study the behavior of cohomological support loci of the canonical bundle under derived equivalence of smooth projective varieties. This is achieved by investigating the derived invariance of a generalized version of Hochschild homology. Furthermore, using techniques coming from birational geometry, we establish the derived invariance of the Albanese dimension for varieties having non-negative Kodaira dimension. We apply our machinery to study the derived invariance of the holomorphic Euler characteristic and of certain Hodge numbers for special classes of varieties. Further applications concern the behavior of particular types of fibrations under derived equivalence.Comment: Small fixes and substantial expository improvements. 22 page

Derived invariants arising from the Albanese map

“…Roughly speaking in this note we prove that, for all i ≥ 0 and for all α ∈ Pic 0 X the cohomology ranks h i (Alb X, a X * ω X ⊗ P α ) are derived invariants. In the case of varieties of maximal Albanese dimension 1 this settles in the affirmative (a strengthened version of) a conjecture of Lombardi and Popa ([LoPo] Conjecture 11) -proved by Lombardi for i = 0 and partially for i = 1 ( [Lo1]) -and proves a weaker version of it for arbitrary varieties. For varieties of maximal Albanese dimension this implies the derived invariance of the Hodge numbers h 0,j for all j ≥ 0 and of all canonical cohomological support loci, proving in this case another conjecture of Popa.…”

“…Finally we provide an application to derived invariance of certain irregular fibrations.1 this means that dim aX(X) = dim X. To be of maximal Albanese dimension is a derived invariant property [Lo1]…”

“…1 this means that dim aX(X) = dim X. To be of maximal Albanese dimension is a derived invariant property [Lo1]…”

Derived equivalence and fibrations over curves and surfaces

“…On the other hand, the isomorphism between the base curves of the fibrations is constructed by manipulating the support of the kernel of an equivalence in the style of Kawamata ([Kaw02]). Earlier attempts to proving Theorem 1 appear in [Lom14,Remark 7.4] and [LP15, Theorems 6(i) and 14]. Moreover, Theorem 1 answers affirmatively a question posed in [LP15, Question 13] (cf.…”

Derived equivalence and non-vanishing loci II

“…A more general statement has already been proved in [10,Theorem 3.2]. We extract the argument we need here in order to keep the proof self-contained, following [12, §3] as well.…”