Virtual fundamental class

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In mathematics, specifically enumerative geometry, the virtual fundamental class ${\displaystyle [X]_{E^{\bullet }}^{\text{vir}}}$^[1][2] of a space ${\displaystyle X}$ is a replacement of the classical fundamental class ${\displaystyle [X]\in A^{*}(X)}$ 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 ${\displaystyle d}$ rational curves on a quintic threefold. For example, in Gromov–Witten theory, the ^[3]

${\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,\beta )}$

for ${\displaystyle X}$ a scheme and ${\displaystyle \beta }$ a class in ${\displaystyle A_{1}(X)}$, 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

${\displaystyle {\overline {\mathcal {M}}}_{1,n}(\mathbb {P} ^{2},1[H])}$

for ${\displaystyle H}$ the class of a line in ${\displaystyle \mathbb {P} ^{2}}$. The non-compact "smooth" component is empty, but the boundary contains maps of curves

${\displaystyle f:C\to \mathbb {P} ^{2}}$

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.

Remark on definitions and special cases[]

There are multiple definitions of virtual fundamental classes,^[2][5][6][7] all of which are subsumed by the definition for morphisms of Deligne-Mumford stacks using the intrinsic normal cone and a perfect obstruction theory, but the first definitions are more amenable for constructing lower-brow examples for certain kinds of schemes, such as ones with components of varying dimension. In this way, the structure of the virtual fundamental classes becomes more transparent, giving more intuition for their behavior and structure.

Virtual fundamental class of an embedding into a smooth scheme[]

One of the first definitions of a virtual fundamental class^{[2]pg 10} is for the following case: suppose we have an embedding of a scheme ${\displaystyle X}$ into a smooth scheme ${\displaystyle Y}$

${\displaystyle i:X\hookrightarrow Y}$

and a vector bundle (called the obstruction bundle)

${\displaystyle \pi :E_{X/Y}\to X}$

such that the normal cone ${\displaystyle C_{X/Y}}$ embeds into ${\displaystyle E_{X/Y}}$ over ${\displaystyle X}$. One natural candidate for such an obstruction bundle if given by

${\displaystyle E_{X/Y}=\bigoplus _{j=1}^{r}i^{*}{\mathcal {O}}_{Y}(-D_{j})}$

for the divisors associated to a non-zero set of generators ${\displaystyle f_{1},\ldots ,f_{r}}$ for the ideal ${\displaystyle {\mathcal {I}}_{X/Y}}$. Then, we can construct the virtual fundamental class of ${\displaystyle X}$ using the generalized Gysin morphism given by the composition

${\displaystyle A_{*}(Y)\xrightarrow {\sigma } A_{*}(C_{X/Y})\xrightarrow {i_{*}} A_{*}(E_{X/Y})\xrightarrow {0_{E_{X/Y}}^{!}} A_{*-r}(X)}$

denoted ${\displaystyle f_{E_{X/Y}}^{!}}$, where ${\displaystyle \sigma }$ is the map given by

${\displaystyle \sigma ([V])=[C_{V\cap X}V]}$

and ${\displaystyle 0_{E_{X/Y}}^{!}}$is the inverse of the flat pullback isomorphism

${\displaystyle \pi ^{*}:A_{k-r}(X)\to A_{k}(E_{X/Y})}$.

Here we use the ${\displaystyle 0}$ in the map since it corresponds to the zero section of vector bundle. Then, the virtual fundamental class of the previous setup is defined as

${\displaystyle [X]_{E_{X/Y}}^{\text{vir}}:=f_{E_{X/Y}}^{!}([Y])}$

which is just the generalized Gysin morphism of the fundamental class of ${\displaystyle Y}$.

Remarks on the construction[]

The first map in the definition of the Gysin morphism corresponds to specializing to the normal cone^{[8]pg 89}, which is essentially the first part of the standard Gysin morphism, as defined in Fulton^{[8]pg 90}. But, because we are not working with smooth varieties, Fulton's cone construction doesn't work, since it would give ${\displaystyle C_{X/Y}\cong N_{X/Y}}$, hence the normal bundle could act as the obstruction bundle. In this way, the intermediate step of using the specialization of the normal cone only keeps the intersection-theoretic data of ${\displaystyle Y}$ relevant to the variety ${\displaystyle X}$.

See also[]

• Chow group of a stack

References[]

• Virtual fundamental classes, global normal cones and Fulton's canonical classes