Virtual fundamental class explained
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] \overline{l{M}}g,n(X,\beta)
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
\overline{l{M}}1,n(P2,1[H])
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.
Geometric motivation
We can understand the motivation for the definition of the virtual fundamental class[5] pg 10 by considering what situation should be emulated for a simple case (such as a smooth complete intersection). Suppose we have a variety
(representing the coarse space of some moduli problem
) which is cut out from an ambient smooth space
by a section
of a rank-
vector bundle
. Then
has "virtual dimension"
(where
is the dimension of
). This is the case if
is a transverse section, but if
is not, and it lies within a sub-bundle
where it is transverse, then we can get a homology cycle by looking at the
Euler class of the cokernel bundle
over
. This bundle acts as the normal bundle of
in
.
Now, this situation dealt with in Fulton-MacPherson intersection theory by looking at the induced cone
and looking at the intersection of the induced section
on the induced cone and the zero section, giving a cycle on
. If there is no obvious ambient space
for which there is an embedding, we must rely upon deformation theory techniques to construct this cycle on the moduli space representing the fundamental class. Now in the case where we have the section
cutting out
, there is a four term exact sequence
0\toTX\toTY|X\xrightarrow{ds}E|X\toob\to0
where the last term represents the "obstruction sheaf". For the general case there is an exact sequence
0\tol{T}1\toE1\toE2\tol{T}2\to0
where
act similarly to
and
act as the tangent and obstruction sheaves. Note the construction of Behrend-Fantechi is a dualization of the exact sequence given from the concrete example above
[6] pg 44.
Remark on definitions and special cases
There are multiple definitions of virtual fundamental classes,[7] [8] [9] 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 classpg 10 is for the following case: suppose we have an embedding of a scheme
into a smooth scheme
and a vector bundle (called the
obstruction bundle)
such that the normal cone
embeds into
over
. One natural candidate for such an obstruction bundle if given by
for the divisors associated to a non-zero set of generators
for the ideal
. Then, we can construct the virtual fundamental class of
using the generalized Gysin morphism given by the composition
A*(Y)\xrightarrow{\sigma}A*(CX/Y)\xrightarrow{i*}A*(EX/Y)
\xrightarrow{
} A_(X)
denoted
, where
is the map given by
and
is the inverse of the flat pullback isomorphism
.
Here we use the
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
which is just the generalized Gysin morphism of the fundamental class of
.
Remarks on the construction
The first map in the definition of the Gysin morphism corresponds to specializing to the normal cone[10] pg 89, which is essentially the first part of the standard Gysin morphism, as defined in Fultonpg 90. But, because we are not working with smooth varieties, Fulton's cone construction doesn't work, since it would give
, 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
relevant to the variety
.
See also
References
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- Battistella. Luca. Carocci. Francesca. Manolache. Cristina. Cristina Manolache . 2020-04-09. Virtual classes for the working mathematician. 1804.06048. Symmetry, Integrability and Geometry: Methods and Applications. 16. 026. 10.3842/SIGMA.2020.026. 2020SIGMA..16..026B. 119167258.
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- Book: Virtual fundamental cycles in symplectic topology. 2019. John, March 21- Morgan, Dusa McDuff, Mohammad Tehrani, Kenji Fukaya, Dominic D. Joyce, Simons Center for Geometry and Physics. 978-1-4704-5014-4. Providence, Rhode Island. 1080251406.
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- Book: Fulton, William. Intersection Theory.. 1998. Springer New York. 978-1-4612-1700-8. N. New York. 958523758.
- Virtual fundamental classes, global normal cones and Fulton's canonical classes