Elliptic stable envelope for Hilbert scheme of points in the plane. (English) Zbl 1471.14012
Stable envelopes are special classes in the cohomology or \(K\)-theory or elliptic cohomology of geometrically relevant moduli spaces. They govern the enumerative geometry of the space, the associated representation theory of a quantum group, and are related to solutions of some associated differential equations. In all these areas concrete formulas for stable envelopes are wanted.
The paper under review gives concrete formulas for the elliptic stable envelopes when the moduli space is the Hilbert scheme of points on the space.
Stable envelopes are associated to the torus fixed points of the space. The torus fixed points on the Hilbert scheme are parametrized by certain tuples of partitions. The author describes some trees subordinate to those partitions. To each tree a product of elliptic functions (and their inverses) is assigned, and the formula is a sum of such functions for all trees.
The proof technique is abelianization: an appropriate resolution of the attracting set of the fixed point, calculating the relevant class in the resolution, and pushing it forward.
As a byproduct \(K\)-theoretic and cohomological stable envelopes are also obtained, as the trigonometric and rational limits of the elliptic stable envelope formula.
The paper under review gives concrete formulas for the elliptic stable envelopes when the moduli space is the Hilbert scheme of points on the space.
Stable envelopes are associated to the torus fixed points of the space. The torus fixed points on the Hilbert scheme are parametrized by certain tuples of partitions. The author describes some trees subordinate to those partitions. To each tree a product of elliptic functions (and their inverses) is assigned, and the formula is a sum of such functions for all trees.
The proof technique is abelianization: an appropriate resolution of the attracting set of the fixed point, calculating the relevant class in the resolution, and pushing it forward.
As a byproduct \(K\)-theoretic and cohomological stable envelopes are also obtained, as the trigonometric and rational limits of the elliptic stable envelope formula.
Reviewer: Richárd Rimányi (Chapel Hill)
MSC:
| 14C05 | Parametrization (Chow and Hilbert schemes) |
| 14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |
| 05E10 | Combinatorial aspects of representation theory |
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 19L47 | Equivariant \(K\)-theory |
| 32C99 | Analytic spaces |
Keywords:
quiver variety; stable envelope; elliptic cohomology; Hilbert scheme of points in the planeReferences:
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