Abstract
Abstract Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and K-theoretic invariants for many moduli stacks of interest, including K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the K-theory and Gysin maps of sheaf stacks. In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory. To this end, we further investigate the K-theory of sheaf stacks and its functoriality properties. As applications of the localization formulas, we establish a K-theoretic wall-crossing formula for simple $\mathbb{C} ^\ast $ -wall crossings and define K-theoretic invariants refining the Jiang-Thomas virtual signed Euler characteristics.
Highlights
Enumerative geometry is the study of counts of geometric objects subject to a set of given conditions
Li-Tian [20] and Behrend-Fantechi [4] developed the theory of virtual fundamental cycles, which have been instrumental in defining and investigating several algebro-geometric enumerative invariants of great importance, such as Gromov-Witten [2], Donaldson-Thomas [29] and Pandharipande-Thomas [25] invariants, and are still one of the major components in modern enumerative geometry
A perfect obstruction theory : → L gives an embedding of C into the vector bundle stack E = h1/h0 ( ∨), and the virtual fundamental cycle is defined as the intersection of the zero section 0E with C
Summary
Enumerative geometry is the study of counts of geometric objects subject to a set of given conditions. In [16], we introduced the notion of coherent sheaves on a sheaf stack F, defined the -theory of F and constructed a Gysin map 0!F of F, which enabled us to construct the virtual structure sheaf [ vir] ∈ 0 ( ) for a Deligne-Mumford stack equipped with an almost perfect obstruction theory. In this preliminary section, we collect necessary ingredients from [16].
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