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In mathematics, specifically enumerative geometry, the virtual fundamental class[1][2] of a space is a replacement of the classical fundamental class in its Chow ring which has better behavior with respect to the enumerative problems being considered. In this way, there exists a cycle with can be used for answering specific enumerative problems, such as the number of degree rational curves on a quintic threefold. For example, in Gromov–Witten theory, the Kontsevich moduli spaces[3]
for a scheme and a class in , their behavior can be wild at the boundary, such as[4]pg 503 having higher-dimensional components at the boundary than on the main space. One such example is in the moduli space
for the class of a line in . The non-compact "smooth" component is empty, but the boundary contains maps of curves
whose components consist of one degree 3 curve which contracts to a point. There is a virtual fundamental class which can then be used to count the number of curves in this family.
^Pandharipande, R.; Thomas, R. P. (2014). "13/2 ways of counting curves". In Brambila-Paz, Leticia; Newstead, Peter; Thomas, Richard P. W; Garcia-Prada, Oscar (eds.). Moduli Spaces. pp. 282–333. arXiv:1111.1552. doi:10.1017/CBO9781107279544.007. ISBN 9781107279544. S2CID 117183792.
^Battistella, Luca; Carocci, Francesca; Manolache, Cristina (2020-04-09). "Virtual classes for the working mathematician". Symmetry, Integrability and Geometry: Methods and Applications. 16: 026. arXiv:1804.06048. Bibcode:2020SIGMA..16..026B. doi:10.3842/SIGMA.2020.026. S2CID 119167258.
^Kontsevich, M. (1995-06-27). "Enumeration of rational curves via torus actions". arXiv:hep-th/9405035.
^Mirror symmetry. Kentaro Hori. Providence, RI: American Mathematical Society. 2003. ISBN 0-8218-2955-6. OCLC 52374327.{{cite book}}: CS1 maint: others (link)
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In mathematics, specifically enumerative geometry, the virtualfundamentalclass [ X ] E ∙ vir {\displaystyle [X]_{E^{\bullet }}^{\text{vir}}} of a space...
and CX the intrinsic normal cone to X, K. Behrend defines the virtualfundamentalclass of X as [ X ] vir = s 0 ! [ C X ] {\displaystyle [X]^{\text{vir}}=s_{0}^{...
intersection theory on moduli stacks; in particular, to define a virtualfundamentalclass. Consider a regular embedding I : Y → W {\displaystyle I\colon...
_{n\in \mathbb {Z} }n\,\chi (\{\nu _{X}=n\})} is the degree of the virtualfundamentalclass [ X ] vir {\displaystyle [X]^{\text{vir}}} of X, which is an element...
stack X with respect to the Behrend function is the degree of the virtualfundamentalclass of X. Behrend's trace formula Behrend's trace formula generalizes...
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Global Information
![{\displaystyle [X]_{E^{\bullet }}^{\text{vir}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24119e8bd2452b01c832f121caa753ccefb5d848)
![{\displaystyle [X]\in A^{*}(X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d44bb93c1639df0fef2bfc5277dfee01458634a3)
![{\displaystyle {\overline {\mathcal {M}}}_{1,n}(\mathbb {P} ^{2},1[H])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/840523e7be082cbaec14769e0027b2efe8e91951)
