arXiv subjects
Publications and source records attributed to Simon Schirren.
We define a perfect obstruction theory for a moduli of symplectic Higgs sheaves $(E,ϕ)$ on projective surfaces $S$. Key to this is a minimality assumption on $\textrm{ch}(E)$ that forces all $E$ to be locally free. This might have implications to define a virtual count and $Sp(r)$-Vafa-Witten invariants.
We present a new method for constructing virtual cycles for rank-2 Higgs sheaves $(E,ϕ)$ on a smooth projective surface $S$. Using this, we redefine the $\mathbf{SU}(2)$-perfect obstruction theory previously constructed by Tanaka-Thomas. The key step in our construction involves modifying the $\mathbf{C}^\times$-localisation formula of Graber-Pandharipande by replacing the torus action with an involution $(E,ϕ) \mapsto (E^*,-ϕ^*)$.