Higher-page Bott–Chern and Aeppli cohomologies and applications

For every positive integer r, we introduce two new cohomologies, that we call E r {E_{r}} -Bott–Chern and E r {E_{r}} -Aeppli, on compact complex manifolds. When r = 1 {r\kern-1.0pt=\kern-1.0pt1} , they coincide with the usual Bott–Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when r ≥ 2 {r\geq 2} . They provide analogues in the Bott–Chern–Aeppli context of the E r {E_{r}} -cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page- ( r - 1 ) {(r-1)} - ∂ ⁡ ∂ ¯ {\partial\bar{\partial}} -manifolds that we introduced very recently. We also prove analogues of the Serre duality for these higher-page Bott–Chern and Aeppli cohomologies and for the spaces featuring in the Frölicher spectral sequence. We obtain a further group of applications of our cohomologies to the study of Hermitian-symplectic and strongly Gauduchon metrics for which we show that they provide the natural cohomological framework.

LOCAL DUALITY FOR THE SINGULARITY CATEGORY OF A FINITE DIMENSIONAL GORENSTEIN ALGEBRA

A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the $\mathfrak{p}$ -local and $\mathfrak{p}$ -torsion subcategories of the derived category, for each homogeneous prime ideal $\mathfrak{p}$ arising from the action of a commutative ring via Hochschild cohomology.

The construction problem for Hodge numbers modulo an integer in positive characteristic

Let k be an algebraically closed field of positive characteristic. For any integer $m\ge 2$ , we show that the Hodge numbers of a smooth projective k-variety can take on any combination of values modulo m, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers.

The $${\bar{\partial }}$$-Equation, Duality, and Holomorphic Forms on a Reduced Complex Space

We solve the $${\bar{\partial }}$$∂¯-equation for (p, q)-forms locally on any reduced pure-dimensional complex space and we prove an explicit version of Serre duality by introducing suitable concrete fine sheaves of certain (p, q)-currents. In particular this gives a condition for the $${\bar{\partial }}$$∂¯-equation to be globally solvable. Our results also give information about holomorphic p-forms on singular spaces.

Grothendieck-Serre duality and theta-invariants on arithmetic surfaces

In the paper, a description of the Grothendieck-Serre duality on an arithmetic surface by means of fixing a horizontal divisor is given and this description is applied to the generalization of theta-invariants.

Local duality for representations of finite group schemes

A duality theorem for the stable module category of representations of a finite group scheme is proved. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the $\mathfrak{p}$-local and $\mathfrak{p}$-torsion subcategories of the stable category, for each homogeneous prime ideal $\mathfrak{p}$ in the cohomology ring of the group scheme.