557 research outputs found
Modular forms and Donaldson invariants for 4-manifolds with
We study the Donaldson invariants of simply connected -manifolds with
, and in particular the change of the invariants under wall-crossing. We
assume the conjecture of Kotschick and Morgan about the shape of the
wall-crossing terms (which Oszva\'th and Morgan are now able to prove), and are
determine a generating function for the wall-crossing terms in terms of modular
forms. As an application we determine all the Donaldson invariants of the
projective plane in terms of modular forms. The main tool are the blowup
formulas, which are used to obtain recursive relations.Comment: I correct a number of missing attributions and citations. In
particular this applies to the cited paper of Kotschick and Lisca "Instanton
invariants via topology", which contains some ideas which have been important
for this work. AMSLaTe
Riemann-Roch theorems and elliptic genus for virtually smooth Schemes
For a proper scheme X with a fixed 1-perfect obstruction theory, we define
virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic
genus; they are deformation invariant, and extend the usual definition in the
smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and
Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a
polynomial, and use this to define a virtual topological Euler characteristic.
We prove that the virtual elliptic genus satisfies a Jacobi modularity
property; we state and prove a localization theorem in the toric equivariant
case. We show how some of our results apply to moduli spaces of stable sheaves.Comment: 31 page
Virtual refinements of the Vafa-Witten formula
We conjecture a formula for the generating function of virtual
-genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with
holomorphic 2-form. Specializing the conjecture to minimal surfaces of general
type and to virtual Euler characteristics, we recover (part of) a formula of C.
Vafa and E. Witten.
These virtual -genera can be written in terms of descendent Donaldson
invariants. Using T. Mochizuki's formula, the latter can be expressed in terms
of Seiberg-Witten invariants and certain explicit integrals over Hilbert
schemes of points. These integrals are governed by seven universal functions,
which are determined by their values on and . Using localization we calculate these functions up to some
order, which allows us to check our conjecture in many cases.
In an appendix by H. Nakajima and the first named author, the virtual Euler
characteristic specialization of our conjecture is extended to include
-classes, thereby interpolating between Vafa-Witten's formula and Witten's
conjecture for Donaldson invariants.Comment: 44 pages. Published version. Appendix C by first named author and H.
Nakajim
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