Global critical chart for local Calabi-Yau threefolds

Abstract

In this paper, we investigate Keller's deformed Calabi-Yau completion of the derived category of coherent sheaves on a smooth variety. In particular, for an n-dimensional smooth variety Y , we describe the derived category of the total space of an ωY -torsor as a certain deformed (n + 1)-Calabi-Yau completion of the derived category of Y .As an application, we investigate the geometry of the derived moduli stack of compactly supported coherent sheaves on a local curve, i.e., a Calabi-Yau threefold of the form TotC (N ), where C is a smooth projective curve and N is a rank two vector bundle on C. We show that the derived moduli stack is equivalent to the derived critical locus of a function on a certain smooth moduli space. This result will be used by the first author and Naoki Koseki in their joint work on Higgs bundles and Gopakumar-Vafa invariants.

On the intersection cohomology of the moduli of SLn$\mathrm{SL}_n$‐Higgs bundles on a curve

“…When gcd(𝑛, 𝑑) = 1, this is verified in [27, Theorem 2.3 (b) and Proposition 2.10], which relies on the support theorem [27, Theorem 2.3 (a)] and a direct calculation over the elliptic locus. Since the moduli space 𝑀 𝑟,𝐿 (𝜋) is nonsingular restricting over the elliptic locus and the calculation of [27] over the elliptic locus does not rely on the coprime assumption, we obtain that the full support property still holds for (22) as a consequence of Theorem 1.1.…”

“…When deg(𝐷) > 2g − 2, the BPS-cohomology coincides with intersection cohomology but for 𝐷 = 𝐾 𝐶 these two are different. Note that combining the recent work [22] and Theorem 0.2 may provide a proof of a version of the Hausel-Thaddeus conjecture for the BPScohomology for 𝐷 = 𝐾 𝐶 ; the approach of Davison [11] further suggests a path line to deduce the 𝐷 = 𝐾 𝐶 case of Theorem 0.2 from the BPS-cohomology.…”