Donaldson Type Invariants for Algebraic Surfaces: Transition by Takuro Mochizuki PDF

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Extra info for Donaldson Type Invariants for Algebraic Surfaces: Transition of Moduli Stacks

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We have the following commutative diagram: (m) F (m) ∗ ΩY (m) /S −−−−→ F (m) ∗ ΩY (m) /YG ⏐ ⏐ ⏐ ⏐ = b F (m) ∗ ΩY (m) /S −−−−→ ΩP (F )(m) /Z Here, b is the composite of the differential F (m) ∗ ΩY (m) /S −→ ΩP (F )(m) /S and the natural projection ΩP (F )(m) /S −→ ΩP (F )(m) /Z . Let qi : Y ×S Gm −→ Y (i = 0, 1, . . , m) be the morphism given by qi (y, g1 , . . , gm ) = y · g1 · · · · · gi . They induce an isomorphism Y ×S Gm −→ Y (m) . Under the identification, qi is the projection onto the i-th component.

2 for the general case. 5 The author expects that Dξ Φ(y) can be described in a more beautiful way like the wall crossing formula in the rank 2 case ([50], [53]). That is the reason why “weak” is added. 6 Master Space As mentioned in Preface, we use master spaces due to M. Thaddeus as one of the most important ingredient in this study. We explain how to utilize it in our situation. 3). To begin with, we give a remark. It is known that coarse moduli schemes of semistable torsion-free sheaves are obtained as the categorical quotient of the sets of the semistable points of some projective varieties provided with actions of reductive groups.

N • Each L≥ X /Y is an object in Dqcoh (X ). • If f is smooth and representable, then LX /Y is quasi-isomorphic to its 0-th cohomology sheaf, which is isomorphic to the locally free sheaves of Kahler dif−n ferentials ΩX /Y . In general, if f is smooth, any L≥ X /Y is of perfect amplitude [−n,1] 0 contained in [0, 1]. In particular, they are isomorphic to L≥ X /Y . 2 M. 1 for Artin stacks [2]. 2 Quotient Stacks Let S be a variety. Let G be a group scheme smooth over S. Let Y be a smooth S-scheme with a G-action.

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