599 research outputs found
Donaldson-Thomas Invariants of 2-Dimensional sheaves inside threefolds and modular forms
Motivated by the S-duality conjecture, we study the Donaldson-Thomas
invariants of the 2 dimensional Gieseker stable sheaves on a threefold. These
sheaves are supported on the fibers of a nonsingular threefold X fibered over a
nonsingular curve. In the case where X is a K3 fibration, we express these
invariants in terms of the Euler characteristic of the Hilbert scheme of points
on the K3 fiber and the Noether-Lefschetz numbers of the fibration. We prove
that a certain generating function of these invariants is a vector modular form
of weight -3/2 as predicted in S-duality.Comment: Some corrections were made and some arguments were extended. Many
thanks to the referee's helpful comments. 22 pages, to Appear in Adv. Math.
(2018). arXiv admin note: text overlap with arXiv:1305.133
Stable reflexive sheaves and localization
We study moduli spaces of rank 2 stable reflexive sheaves on
. Fixing Chern classes , , and summing over , we
consider the generating function of Euler
characteristics of such moduli spaces. The action of the torus on
lifts to and we classify all sheaves in
. This leads to an explicit expression for
. Since is bounded below and above,
is a polynomial. We find a simple formula for
its leading term when .
Next, we study moduli spaces of rank 2 stable torsion free sheaves on
and consider the generating function of Euler characteristics of
such moduli spaces. We give an expression for this generating function in terms
of and Euler characteristics of Quot schemes of
certain -equivariant reflexive sheaves, which are studied elsewhere. Many
techniques of this paper apply to any toric 3-fold. In general,
depends on the choice of polarization which
leads to wall-crossing phenomena. We briefly illustrate this in the case of
.Comment: 27 pages. Published version. Typo's correcte
Higher rank sheaves on threefolds and functional equations
We consider the moduli space of stable torsion free sheaves of any rank on a
smooth projective threefold. The singularity set of a torsion free sheaf is the
locus where the sheaf is not locally free. On a threefold it has dimension
. We consider the open subset of moduli space consisting of sheaves
with empty or 0-dimensional singularity set.
For fixed Chern classes and summing over , we show that the
generating function of topological Euler characteristics of these open subsets
equals a power of the MacMahon function times a Laurent polynomial. This
Laurent polynomial is invariant under (upon
replacing ). For some choices of these open
subsets equal the entire moduli space.
The proof involves wall-crossing from Quot schemes of a higher rank reflexive
sheaf to a sublocus of the space of Pandharipande-Thomas pairs. We interpret
this sublocus in terms of the singularities of the reflexive sheaf.Comment: 29 pages. Published versio
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