The intrinsic normal cone

• Published in: Inventiones mathematicae Vol. 128; no. 1; pp. 45 - 88
• Main Authors: Behrend, K., Fantechi, B.
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.04.1997
• ISSN: 0020-9910; 1432-1297
• DOI: 10.1007/s002220050136

Let (ProQuest: Formulae and/or non-USASCII text omitted; see image) be an algebraic stack in the sense of Deligne-Mumford. We construct a purely (ProQuest: Formulae and/or non-USASCII text omitted; see image) -dimensional algebraic stack over (ProQuest: Formulae and/or non-USASCII text omitted; see image) (in the sense of Artin), the intrinsic normal cone (ProQuest: Formulae and/or non-USASCII text omitted; see image) . The notion of (perfect) obstruction theory for (ProQuest: Formulae and/or non-USASCII text omitted; see image) is introduced, and it is shown how to construct, given a perfect obstruction theory for (ProQuest: Formulae and/or non-USASCII text omitted; see image) , a pure-dimensional virtual fundamental class in the Chow group of (ProQuest: Formulae and/or non-USASCII text omitted; see image) . We then prove some properties of such classes, both in the absolute and in the relative context. Via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory, and sometimes a perfect one.[PUBLICATION ABSTRACT]