A Theory of Generalized Donaldson-Thomas Invariants

作者：Joyce Dominic^*; Song Yinan

来源：Memoirs of the American Mathematical Society, 2012, 217(1020): 1-+.

DOI：10.1090/s0065-9266-2011-00630-1

• Donaldson-Thomas invariant
• Calabi-Yau 3-fold
• coherent sheaf
• vector bundle
• stability condition
• semistable
• Gieseker stability
• moduli space
• Artin stack

Donaldson-Thomas invariants DT alpha(tau) are integers which 'count' tau-stable coherent sheaves with Chern character a on a Calabi-Yau 3-fold X, where tau denotes Gieseker stability for some ample line bundle on X. They are unchanged under deformations of X. The conventional definition works only for classes a containing no strictly tau-semistable sheaves. Behrend showed that DT alpha(tau) can be written as a weighted Euler characteristic chi(M-st(alpha)(tau), nu(Mst alpha(tau))) of the stable moduli scheme M-st(alpha)(tau) by a constructible function nu(Mst alpha(tau)) we call the 'Behrend function'.
This book studies generalized Donaldson-Thomas invariants (DT) over bar (alpha) (tau). They are rational numbers which 'count' both tau-stable and tau-semistable coherent sheaves with Chern character alpha on X; strictly tau-semistable sheaves must be counted with complicated rational weights. The (DT) over bar (alpha)(tau) are defined for all classes alpha, and are equal to DT alpha(tau) when it is defined. They are unchanged under deformations of X, and transform by a wall-crossing formula under change of stability condition tau.
To prove all this we study the local structure of the moduli stack M of coherent sheaves on X. We show that an atlas for 931 may be written locally as Crit(f) for f : U --> C holomorphic and U smooth, and use this to deduce identities on the Behrend function nu(m). We compute our invariants (DT) over bar (alpha)(tau) in examples, and make a conjecture about their integrality properties. We also extend the theory to abelian categories mod-CQ/I of representations of a quiver Q with relations I coming from a superpotential W on Q, and connect our ideas with Szendroi's non-commutative Donaldson-Thomas invariants, and work by Reineke and others on invariants counting quiver representations. Our book is closely related to Kontseyich and Soibelman's independent paper Stability structures, motivic Donaldson-Thomas invariants and cluster transformations.

• 出版日期2012-5