Abstract:Abstract. We study the equivariant version of the genus zero BPS invariants of the total space of a rank 2 bundle on P 1 whose determinant is O P 1 (−2). We define the equivariant genus zero BPS invariants by the residue integrals on the moduli space of stable sheaves of dimension one as proposed by Sheldon Katz [11]. We compute these invariants for low degrees by counting the torus fixed stable sheaves. The results agree with the prediction in local Gromov-Witten theory studied in [3].
“…He used his methods to calculate (1) for X = P 2 and r = 2 obtaining the holomorphic part of a quasi-modular form of weight 3/2 [Kly2,VW]. Other localization calculations on (stacky/ordinary) toric surfaces appear in [Per2,Cho,GJK,Koo2].…”
Abstract. Let M be the moduli space of rank 2 stable torsion free sheaves with Chern classes c i on a smooth 3-fold X. When X is toric with torus T , we describe the T -fixed locus of the moduli space. Connected components of M T with constant reflexive hulls are isomorphic to products of P 1 . We mainly consider such connected components, which typically arise for any c 1 , "low values" of c 2 , and arbitrary c 3 .In the classical part of the paper, we introduce a new type of combinatorics called double box configurations, which can be used to compute the generating function Z(q) of topological Euler characteristics of M (summing over all c 3 ). The combinatorics is solved using the double dimer model in a companion paper. This leads to explicit formulae for Z(q) involving the MacMahon function.In the virtual part of the paper, we define Donaldson-Thomas type invariants of toric Calabi-Yau 3-folds by virtual localization. The contribution to the invariant of an individual connected component of the T -fixed locus is in general not equal to its signed Euler characteristic due to T -fixed obstructions. Nevertheless, the generating function of all invariants is given by Z(q) up to signs.
“…He used his methods to calculate (1) for X = P 2 and r = 2 obtaining the holomorphic part of a quasi-modular form of weight 3/2 [Kly2,VW]. Other localization calculations on (stacky/ordinary) toric surfaces appear in [Per2,Cho,GJK,Koo2].…”
Abstract. Let M be the moduli space of rank 2 stable torsion free sheaves with Chern classes c i on a smooth 3-fold X. When X is toric with torus T , we describe the T -fixed locus of the moduli space. Connected components of M T with constant reflexive hulls are isomorphic to products of P 1 . We mainly consider such connected components, which typically arise for any c 1 , "low values" of c 2 , and arbitrary c 3 .In the classical part of the paper, we introduce a new type of combinatorics called double box configurations, which can be used to compute the generating function Z(q) of topological Euler characteristics of M (summing over all c 3 ). The combinatorics is solved using the double dimer model in a companion paper. This leads to explicit formulae for Z(q) involving the MacMahon function.In the virtual part of the paper, we define Donaldson-Thomas type invariants of toric Calabi-Yau 3-folds by virtual localization. The contribution to the invariant of an individual connected component of the T -fixed locus is in general not equal to its signed Euler characteristic due to T -fixed obstructions. Nevertheless, the generating function of all invariants is given by Z(q) up to signs.
“…Sheaves supported on a line. If the sheaf is supported on a line, the problem is the same as the problem on local P 1 with k = 1 studied in [6]. By the discussion in [6, Section 5.4 and ( 16)], we have 7 equivariant sheaves supported on a fixed line.…”
Section: Torus Fixed Locus Imentioning
confidence: 99%
“…XY, XZ, YZ)14,11,12,8,4,3,13,7,5,10,9,6 (XY, XZ, Y 2 ) 12, 15, 11, 13, 8, 4, 14, 7, 5, 10, 9, 6 (YZ, XZ, Y 2 ) X 2 , Y 2 ) 13,17,9,14,12,16,6,8,15,7,10,11 …”
Abstract. We describe the torus fixed locus of the moduli space of stable sheaves with Hilbert polynomial 4m + 1 on P 2 . We determine the torus representation of the tangent spaces at the fixed points, which leads to the computation of the Betti and Hodge numbers of the moduli space.Acknowledgements.
“…By computing slopes, we see that F is semistable. For more details on the torus equivariant sheaves, see [1, §2.3] and [2].…”
Section: Example 28mentioning
confidence: 99%
“…Hence, part (1) of Proposition 2.4 must be satisfied. For part (2), we use the semistability of F. Suppose the generalized spectrum of F does not satisfy the condition in part (2). Then we can write π * F = G ′ ⊕G ′′ such that Hom(G ′ , G ′′ (1)) = 0.…”
Section: Upper Bound For a Projective Varietymentioning
Abstract. We find the sharp bounds on h 0 (F) for one-dimensional semistable sheaves F on a projective variety X by using the spectrum of semistable sheaves. The result generalizes the Clifford theorem. When X is the projective plane P 2 , we study the stratification of the moduli space by the spectrum of sheaves. We show that the deepest stratum is isomorphic to a subscheme of a relative Hilbert scheme. This provides an example of a family of semistable sheaves having the biggest dimensional global section space.
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