Virtual fundamental class explained

In mathematics, specifically enumerative geometry, the virtual fundamental class

vir
[X]
E\bullet
[1] [2] of a space

X

is a replacement of the classical fundamental class

[X]\inA*(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

d

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

X

a scheme and

\beta

a class in

A1(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

\overline{l{M}}1,n(P2,1[H])

for

H

the class of a line in

P2

. The non-compact "smooth" component is empty, but the boundary contains maps of curves

f:C\toP2

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

X

(representing the coarse space of some moduli problem

l{X}

) which is cut out from an ambient smooth space

Y

by a section

s

of a rank-

r

vector bundle

E\toY

. Then

X

has "virtual dimension"

(n-r)

(where

n

is the dimension of

Y

). This is the case if

s

is a transverse section, but if

s

is not, and it lies within a sub-bundle

E'\subsetE

where it is transverse, then we can get a homology cycle by looking at the Euler class of the cokernel bundle

E/E'

over

X

. This bundle acts as the normal bundle of

X

in

Y

.

Now, this situation dealt with in Fulton-MacPherson intersection theory by looking at the induced cone

E|X

and looking at the intersection of the induced section

s

on the induced cone and the zero section, giving a cycle on

X

. If there is no obvious ambient space

Y

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

s:Y\toE

cutting out

X

, 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

E1,E2

act similarly to

TY|X,E|X

and

l{T}1,l{T}2

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

X

into a smooth scheme

Y

i:X\hookrightarrowY

and a vector bundle (called the obstruction bundle)

\pi:EX/Y\toX

such that the normal cone

CX/Y

embeds into

EX/Y

over

X

. One natural candidate for such an obstruction bundle if given by

EX/Y=

r
oplus
j=1
*l{O}
i
Y(-D

j)

for the divisors associated to a non-zero set of generators

f1,\ldots,fr

for the ideal

l{I}X/Y

. Then, we can construct the virtual fundamental class of

X

using the generalized Gysin morphism given by the composition

A*(Y)\xrightarrow{\sigma}A*(CX/Y)\xrightarrow{i*}A*(EX/Y) \xrightarrow{

!
0
EX/Y
} A_(X)
denoted
!
f
EX/Y
, where

\sigma

is the map given by

\sigma([V])=[CV\capV]

and
!
0
EX/Y
is the inverse of the flat pullback isomorphism
*:A
\pi
k-r

(X)\toAk(EX/Y)

.
Here we use the

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
vir
[X]
EX/Y

:=

!([Y])
f
EX/Y
which is just the generalized Gysin morphism of the fundamental class of

Y

.

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

CX/Y\congNX/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

Y

relevant to the variety

X

.

See also

References

  1. Book: Pandharipande. R.. Thomas. R. P.. Leticia. Brambila-Paz. Leticia Brambila Paz . Peter. Newstead. Richard P. W. Thomas. Oscar. Garcia-Prada. Moduli Spaces. 13/2 ways of counting curves. 2014. 282–333. 10.1017/CBO9781107279544.007. 1111.1552. 9781107279544. 117183792.
  2. 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.
  3. Kontsevich. M.. 1995-06-27. Enumeration of rational curves via torus actions. hep-th/9405035.
  4. Book: Mirror symmetry. 2003. American Mathematical Society. Kentaro Hori. 0-8218-2955-6. Providence, RI. 52374327.
  5. Thomas . R. P. . 2001-06-11 . A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations . math/9806111 .
  6. Book: Pandharipande . R. . Thomas . R. P. . Moduli Spaces . 13/2 ways of counting curves . 2014 . 1111.1552 . 282–333 . 10.1017/CBO9781107279544.007. 9781107636385 . 117183792 .
  7. Siebert. Bernd. 2005-09-04. Virtual fundamental classes, global normal cones and Fulton's canonical classes. math/0509076.
  8. 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.
  9. Li. Jun. Tian. Gang. 1998-02-13. Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. alg-geom/9602007.
  10. Book: Fulton, William. Intersection Theory.. 1998. Springer New York. 978-1-4612-1700-8. N. New York. 958523758.