Volume 13, issue 4 (2009)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Polynomial Bridgeland stability conditions and the large volume limit

Arend Bayer

Geometry & Topology 13 (2009) 2389–2425
Abstract

We introduce the notion of a polynomial stability condition, generalizing Bridgeland stability conditions on triangulated categories. We construct and study a family of polynomial stability conditions for any normal projective variety. This family includes both Simpson-stability and large volume limits of Bridgeland stability conditions.

We show that the PT/DT–correspondence relating stable pairs to Donaldson–Thomas invariants (conjectured by Pandharipande and Thomas) can be understood as a wall-crossing in our family of polynomial stability conditions. Similarly, we show that the relation between stable pairs and invariants of one-dimensional torsion sheaves (proven recently by the same authors) is a wall-crossing formula.

Keywords
stability condition, derived category, counting invariant, wall crossing, Donaldson–Thomas invariant
Mathematical Subject Classification 2000
Primary: 14F05, 18E30
Secondary: 14J32, 14D20, 14N35
References
Publication
Received: 28 March 2009
Accepted: 8 May 2009
Published: 12 June 2009
Proposed: Jim Bryan
Seconded: Simon Donaldson, Frances Kirwan
Authors
Arend Bayer
Department of Mathematics
University of Utah
155 South 1400 East
Room 233
Salt Lake City, UT 84112
http://www.math.utah.edu/~bayer/