Homological Units

Abstract

We define and study the invariance properties of homological units. Some applications are given to the derived invariance of Hodge numbers. In particular, we prove that if X and Y are derived equivalent smooth projective varieties of dimension 4 having the same h 1,1 , then they have the same Hodge numbers. We also give a geometric interpretation of the conjectural invariance of homological units in terms of derived Jacobians.Let X and Y be smooth projective varieties over the field of complex numbers. If X and Y are derived equivalent, there Hochschild homology groups are isomorphic [Orl03]. The Hochschild-Kostant-Rosenberg Theorem [Mar09] then implies the isomorphism of vector spaces:It is still unknown if all the graded pieces which appear in the above decomposition are in fact isomorphic. Namely, the following conjecture is folklore:Conjecture 1.0.1 Let X and Y be smooth projective varieties. Assume that X and Y are derived equivalent, then X and Y have the same Hodge numbers.Note that if X (or Y ) has ample or anti-ample canonical bundle, a famous result of Bondal and Orlov states that derived equivalence of X and Y implies that both varieties are actually isomorphic [BO02]. Hence, conjecture 1.0.1 is interesting when the sign of the canonical bundle of X (and Y ) is not definite. For instance if X and Y have trivial canonical bundle.In dimension 1 and 2 (and also in dimension 3, if X and Y have trivial canonical bundle), this conjecture is an obvious consequence of the derived invariance of Hochschild homology, the Hochschild-Kostant-Rosenberg isomorphism and the Hodge symmetry. In dimension 3, without any assumption on the canonical bundle, the conjecture is still true and has been settled by Popa and Schnell [PS11]. If one assumes that the derived equivalence comes from a birational correspondence then the conjecture has been proved by Orlov [Orl05]. In dimension 4 or higher (and without the birational assumption), nothing is known, even if the canonical bundles of X and Y are trivial.Leaving aside this hard problem, one could be interested in a slightly less ambitious question: Question 1.0.2 Let X and Y be two smooth projective varieties. Assume that X and Y are derived equivalent. Denote by HH • (X) (resp. HH • (X)), the Hochschild homology graded vector space (resp. Hochschild cohomology graded algebra) of X.• Are there non-trivial, canonically defined, sub-vector spaces of HH • (X) and HH • (Y ) which are isomorphic?• Are there non-trivial, canonically defined, sub-algebras of HH • (X) and HH • (Y ) which are isomorphic?1. The Fourier-Mukai kernel representing Φ is a (possibly shifted) generically pure vector bundle 1 on X × Y ,

Formality of -objects

“…Note that the graded endomorphism algebra coincides with the cohomology algebra where the multiplication is given by the cup product. This algebra, sometimes called the homological unit of , is conjectured to be a derived invariant of the variety ; see [Abu17]. In this section we will show that the generated thick triangulated category only depends on this graded algebra.…”

“…All triangulated categories are assumed to be k-linear, and subcategories thereof to be triangulated and full. The shift functor will be denoted by [1]. All triangles are meant to be distinguished and denoted by A → B → C, hiding the morphism C → A [1].…”

“…The shift functor will be denoted by [1]. All triangles are meant to be distinguished and denoted by A → B → C, hiding the morphism C → A [1]. We write Hom * (A, B) = i∈Z Hom(A, B[i])[−i] for the derived homomorphisms in a triangulated category: this is a complex equipped with the zero differential.…”

A note on derived equivalences and birational geometry

“…Finally, Abuaf gave an alternative proof of the invariance of the isogeny class of , linking it to his conjecture on homological units .…”

Varieties with $\mathbb {P}$-units

“…Given a variety X, the graded algebra H * (O X ) := ⊕ dim X i=0 H i (X, O X )[−i] is considered an important invariant; see, in particular, Abuaf [Abu15] who calls H * (O X ) the homological unit of X and conjectures that it is stable under derived equivalences. In this paper, we want to study varieties which have trivial canonical bundle and the property that the algebra H * (O X ) is generated by one element.…”