Wikipedia / Genus of a multiplicative sequence
Wikipedia / Genus of a multiplicative sequence
In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifoldss up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the rational numbers, having the property that they are constructed from a sequence of polynomials in characteristic classes that arise as coefficients in formal power series with good multiplicative properties.
Definition
A genus to each manifold X such that
is the disjoint union);
;
if X is the boundary of a manifold with boundary.
The manifolds and manifolds with boundary may be required to have additional structure; for example, they might be oriented, spin, stably complex, and so on (see list of cobordism theories or the ring of modular forms.
is a ring homomorphism from the cobordism ring of manifolds (with additional structure) to another ring.
is the signature of the oriented manifold X is a genus from oriented manifolds to the ring of integers.
The genus associated to a formal power series
is called multiplicative if
implies that
is a formal power series in z with constant term 1, we can define a multiplicative sequence
by
,
is the kth elementary symmetric function will often in practice be Pontryagin classes.)
of compact, connected, smooth, oriented manifolds corresponding to Q is given by
are the Pontryagin classes of X. The power series Q is called the characteristic power series. A theorem of René Thom, which states that the rationals tensored with the cobordism ring is a polynomial algebra in generators of degree 4k for positive integers k, implies that this gives a bijection between formal power series Q with rational coefficients and leading coefficient 1, and genera from oriented manifolds to the rational numbers.
L genus
The L genus is the genus of the formal power series
are the Bernoulli numbers. The first few values are:
(for further L-polynomials see or ). Now let M be a closed smooth oriented manifold of dimension 4n with Pontrjagin classes. Friedrich Hirzebruch showed that the L genus of M in dimension 4n evaluated on the fundamental class, the signature of M (i.e., the signature of the intersection form on the 2nth cohomology group of M):
.
This is now known as the Hirzebruch signature theorem (or sometimes the Hirzebruch index theorem).
is always integral for a smooth manifold was used by John Milnor to give an example of an 8-dimensional piecewise linear (PL) manifold with no smooth structure, and so was not smoothable.
Application on K3 surfaces
Since projective K3 surfaces are smooth complex manifolds of dimension two, their only non-trivial Pontryagin class. It can be computed as -48 using the tangent sequence and comparisons with complex Chern classes, we have its signature. This can be used to compute its intersection form as a unimodular lattice, and using the classification of unimodular lattices.
Todd genus
The Todd genus is the genus of the formal power series
as before, Bernoulli numbers. The first few values are
), and this suffices to show that the Todd genus agrees with the arithmetic genus for algebraic varieties as the arithmetic genus is also 1 for complex projective spaces. This observation is a consequence of the Hirzebruch–Riemann–Roch theorem, and in fact is one of the key developments that led to the formulation of that theorem.
 genus
The  genus is the genus associated to the characteristic power series
.) The first few values are
The  genus of a spin manifold is an integer, and an even integer if the dimension is 4 mod 8 (which in dimension 4 implies Rochlin's theorem) – for general manifolds, the  genus is not always an integer. This was proven by Hirzebruch and Armand Borel; this result both motivated and was later explained by the Atiyah–Singer index theorem, which showed that the  genus of a spin manifold is equal to the index of its Dirac operator.
By combining this index result with a Weitzenbock formula for the Dirac Laplacian, André Lichnerowicz deduced that if a compact spin manifold admits a metric with positive scalar curvature, its  genus must vanish. This only gives an obstruction to positive scalar curvature when the dimension is a multiple of 4, but Nigel Hitchin-valued obstruction in dimensions 1 or 2 mod 8. These results are essentially sharp. Indeed, Mikhail Gromov, H. Blaine Lawson-valued analog are the only obstructions to the existence of positive-scalar-curvature metrics on simply-connected spin manifolds of dimension greater than or equal to 5.
Elliptic genus
A genus is called an elliptic genus satisfies the condition
. (As usual, Q is the characteristic power series of the genus.)
One explicit expression for f(z) is
where
and sn is the Jacobi elliptic function.
Examples:
. This is the L-genus.
. This is the  genus.
. This is a generalization of the L-genus.
The first few values of such genera are:
Example (elliptic genus for quaternionic projective plane) :
Example (elliptic genus for octonionic projective plane, or Cayley plane):
Witten genus
The Witten genus is the genus associated to the characteristic power series
where σL is the Weierstrass sigma function for the lattice L, and G is a multiple of an Eisenstein series.
The Witten genus of a 4k dimensional compact oriented smooth spin manifold with vanishing first Pontryagin class is a modular form of weight 2k, with integral Fourier coefficients.
See also
Notes
References
Friedrich Hirzebruch Topological Methods in Algebraic Geometry Text of the original German version: http://hirzebruch.mpim-bonn.mpg.de/120/6/NeueTopologischeMethoden_2.Aufl.pdf
Friedrich Hirzebruch, Thomas Berger, Rainer Jung Manifolds and Modular Forms