Amin Gholampour

On Donaldson-Thomas invariants of threefold stacks and gerbes

We present a construction of Donaldson-Thomas invariants for three-dimensional projective Calabi-Yau Deligne-Mumford stacks. We also study the structure of these invariants for etale gerbes over such stacks.

The quantum Lefschetz hyperplane principle can fail for positive orbifold hypersurfaces

We show that the Quantum Lefschetz Hyperplane Principle can fail for certain orbifold hypersurfaces and complete intersections. It can fail even for orbifold hypersurfaces defined by a section of an ample line bundle.

Sheaves on weighted projective planes and modular forms

We give an explicit description of toric sheaves on the weighted projective plane P(a,b,c) viewed as a toric Deligne-Mumford stack. The integers (a,b,c) are not necessarily chosen coprime or mutually coprime allowing for gerbe and root stack structures. As an application, we describe the fixed point locus of the moduli scheme of stable rank 1 and 2 torsion free sheaves on P(a,b,c) with fixed K-group class. Summing over all K-group classes, we obtain explicit formulae for generating functions of the topological Euler characteristics. In the case of stable rank 2 locally free sheaves on P(a,b,c) with a,b,c≤2 the generating functions can be expressed in terms of Hurwitz class numbers and give rise to modular forms of weight 3/2. This generalizes Klyachko’s computation on P2 and is consistent with S-duality predictions from physics.

Counting curves on surfaces in Calabi–Yau 3-folds

Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of two-dimensional torsion sheaves, enumerating pairs

Rank 2 wall-crossing and the Serre correspondence

Z\subset H

Counting invariants for the ADE McKay quivers

Z⊂H in a Calabi–Yau threefold

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