Giovedì 9 Gennaio alle ore 14.00, Martjin Kool (Utrecht) e Lothar Gottsche (ICTP) terranno due seminari all'interno del seminario di Geometria.
M. Kool: Vafa-Witten invariants and framed sheaves (ore 14)
Abstract: Vafa-Witten invariants are virtual counts of Higgs pairs on a complex smooth projective surface. For positive geometric genus, they are conjecturally governed by universal functions with modular properties. Together with T. Laarakker and L. Göttsche, we conjectured blow-up and symmetry equations for these universal functions. Expressing them in terms of Nekrasov partition functions reduces these to blow-up and symmetry equations for Nekrasov partition functions. The former were proved by Kuhn-Leigh-Tanaka, and we prove the latter using mixed Hodge modules. Joint work with N. Arbesfeld and T. Laarakker.
L. Gottsche: Segre and Verlinde number of surfaces (ore 15:15)
Abstract: Let \(S\) be a smooth projective surface with \(p_g>0\) and \(H^1(S,\mathbb{Z})=0\). We consider the moduli spaces \(M=M_S^H(r,c_1,c_2)\) of \(H\)-semistable sheaves on \(S\) of rank \(r\) and with Chern classes \(c_1,c_2\). Associated a suitable class \(v\) the Grothendieck group of vector bundles on \(S\) there is a deteminant line bundle \(\lambda(v)\in Pic(M)\), and also a tautological sheaf \(\tau(v)\) on M.
We give conjectural formulas for the virtual Verlinde numbers, i.e. the virtual holomorphic Euler characteristics of all determinant bundles \(\lambda(v)\) on M, and for Segre invariants associated to \(\tau(v)\). The argument is based on conjectural blowup formulas and a virtual version of Le Potier's strange duality.
I seminari si svolgeranno in presenza nell'aula C. Per ulteriori informazioni, si può contattare gli organizzatori all'email amos.turchet@uniroma3.it.