Related papers: Derived invariance of the Albanese relative canoni…

Let $a_X:X\rightarrow \mathrm{Alb}\, X$ be the Albanese map of a smooth complex projective variety. Roughly speaking in this note we prove that for all $i \geq 0$ and $\alpha\in \mathrm{Pic}^0\, X$, the cohomology ranks $h^i(\mathrm{Alb}\,…

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.…

We discuss several conjectures about derived equivalent varieties, defined over fields of arbitrary characteristics, and implications among them. In particular we show that the (conjectural) derived invariance of the Hasse-Weil Zeta…

We prove that smooth projective varieties with equivalent derived categories have isogenous (and sometimes isomorphic) Picard varieties. In particular their irregularity and number of independent vector fields are the same. This is turn…

For every noetherian smooth and separated scheme over an algebraically closed field, we define its derived Albanese in Voevodsky's triangulated category of effective Nisnevich motives. To justify our definition, we relate the derived…

We prove a few cases of a conjecture on the invariance of cohomological support loci under derived equivalence by establishing a concrete connection with the related problem of the invariance of Hodge numbers. We use the main case in order…

In this paper we construct new derived invariants with integral coefficients using the theory of motifs, and give several applications. Specifically, we obtain the following results: For complex algebraic surfaces, we prove that certain…

We discuss $p$-adic unipotent Albanese maps for curves of positive genus, extending the theory of $p$-adic multiple polylogarithms. This construction is then used to relate linear Diophantine conjectures of `Birch and Swinnerton-Dyer type'…

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…

The paper is devoted to study some of the questions arises naturally in connection to the notion of relative derived categories. In particular, we study invariants of recollements involving relative derived categories, generalise two…

We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties. The result is deduced from a generalization of a theorem of M\"oller. Namely, we prove that the image of a certain twisted Abel-Jacobi map lands in…

Let X be a proper smooth variety over the complex numbers. We consider the generalized Albanese variety Alb(X,Y) of X of modulus Y, which is a higher dimensional analogue of the generalized Jacobian variety with modulus of Rosenlicht-Serre.…

We extend the usual projective Abel-Radon transform to the larger context of a smooth complete toric variety X. We define and study toric concavity attached to an algebraic splitting vector bundle on X and we prove a toric version of the…

We study solvable lattice models associated to canonical Grothendieck polynomials and their duals. We derive inversion relations and Cauchy identities.

Originally a technical tool, the derived category of coherent sheaves over an algebraic variety has become over the last twenty years an important invariant in the birational study of algebraic varieties. Problems of birational invariance…

If A is a strongly noetherian graded algebra generated in degree one, then there is a canonically constructed graded ring homomorphism from A to a twisted homogeneous coordinate ring B(X, L, sigma), which is surjective in large degree. This…

We survey recent developments in the study of Hodge theoretic aspects of Alexander-type invariants associated with smooth complex algebraic varieties.

We study the relative Frobenius map associated with a map of derived commutative rings over a field of positive characteristic. As part of this, we examine a relative analog of perfectness and construct a relative inverse limit perfection…

Given a derived equivalence of orbifolds associated to projective varieties with (not necessarily Gorenstein) quotient singularities, we deduce consequences related to the behavior of orbifold Hodge numbers and the Picard variety, extending…

We introduce a canonical isomorphism from the space of pure-type complex differential forms on a compact complex manifold to the one on its infinitesimal deformations. By use of this map, we generalize an extension formula in a recent work…