mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a smooth algebraic curve

C

in the complex projective plane, of degree

d

, has

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

given by the genus–degree formula :

g = (d-1)(d-2)/2

. The Thom conjecture, named after French mathematician

René Thom René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became w ...

, states that if

\Sigma

is any smoothly embedded connected curve representing the same class in

homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...

C

, then the genus

g

\Sigma

satisfies the inequality :

g \geq (d-1)(d-2)/2

. In particular, ''C'' is known as a ''genus minimizing representative'' of its homology class. It was first proved by

Peter Kronheimer Peter Benedict Kronheimer (born 1963) is a British mathematician, known for his work on gauge theory and its applications to 3- and 4-dimensional topology. He is William Caspar Graustein Professor of Mathematics at Harvard University and former ...

and

Tomasz Mrowka Tomasz Mrowka (born September 8, 1961) is an American mathematician specializing in differential geometry and gauge theory. He is the Singer Professor of Mathematics and former head of the Department of Mathematics at the Massachusetts Institu ...

in October 1994, using the then-new Seiberg–Witten invariants. Assuming that

\Sigma

has nonnegative self

intersection number In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for ta ...

this was generalized to

Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...

s (an example being the complex projective plane) by John Morgan, Zoltán Szabó, and

Clifford Taubes Clifford Henry Taubes (born February 21, 1954) is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology. His brother is the journalist Gary Taubes. ...

, also using the Seiberg–Witten invariants. There is at least one generalization of this conjecture, known as the symplectic Thom conjecture (which is now a theorem, as proved for example by

Peter Ozsváth Peter Steven Ozsváth (born October 20, 1967) is a professor of mathematics at Princeton University. He created, along with Zoltán Szabó, Heegaard Floer homology, a homology theory for 3-manifolds. Education Ozsváth received his Ph.D. from P ...

and Szabó in 2000). It states that a symplectic surface of a symplectic 4-manifold is genus minimizing within its homology class. This would imply the previous result because algebraic curves (complex dimension 1, real dimension 2) are symplectic surfaces within the complex projective plane, which is a symplectic 4-manifold.

References

{{DEFAULTSORT:Thom Conjecture Four-dimensional geometry 4-manifolds Algebraic surfaces Conjectures that have been proved Theorems in geometry

See also

References