Schedule

We consider moduli spaces M(ß,χ) of one-dimensional semistable sheaves on del Pezzo and K3 surfaces supported on ample curve classes.

Working over a non-archimedean local field F, we define a natural measure on the F-points of such moduli spaces. We prove that the integral of a certain naturally defined gerbe on M(ß,χ) with respect to this measure is independent of the Euler characteristic.

Analogous statements hold for (meromorphic or not) Higgs bundles.

Recent results of Maulik-Shen and Kinjo-Koseki imply that these integrals compute the BPS invariants for the del Pezzo case and for Higgs bundles.

This is a joint work with Giulio Orecchia and Dimitri Wyss.

Maulik and Toda recently gave a mathematical definition of Gopakumar-Vafa(GV) invariants (a.k.a. BPS invariants) for Calabi-Yau threefolds. The GV invariants are expected to be equivalent to other curve counting theories such as Gromov-Witten theory (GW=GV conjecture). One of the mysterious features of GV invariants is their χ-independence, which should be true assuming GW=GV conjecture.

I will explain my joint work with Tasuki Kinjo (Tokyo), where we proved the χ-independence for GV invariants on certain class of non-compact Calabi-Yau threefolds which we call local curves. I will also explain applications to Higgs bundles.