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In [[mathematics]], a smooth [[algebraic curve]] <math>C</math> in the [[complex projective plane]], of degree <math>d</math>, has [[Genus_(mathematics)#Topology|genus]] given by the formula
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The '''Thom conjecture''', named after French mathematician [[René Thom]], states that if <math>\Sigma</math> is any smoothly embedded connected curve representing the same class in [[homology (mathematics)|homology]] as <math>C</math>, then the genus <math>g</math> of <math>\Sigma</math> satisfies
 
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:<math>g \geq (d-1)(d-2)/2</math>.
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In particular, ''C'' is known as a ''genus minimizing representative'' of its homology class. It was first proved by [[Peter B. Kronheimer|Kronheimer]]–[[Tomasz Mrowka|Mrowka]] in October 1994, using the then-new [[Seiberg–Witten invariant]]s.   
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Assuming that <math>\Sigma</math> has nonnegative self [[intersection number]] this was generalizes to [[Kähler manifold]]s (an example being the complex projective plane) by [[John Morgan (mathematician)|Morgan]]–[[Zoltán Szabó (mathematician)|Szabó]]–[[Clifford Taubes|Taubes]], also using the then-new [[Seiberg–Witten invariant]]s.
 
There is at least one generalization of this conjecture, known as the [[Symplectic geometry|symplectic]] Thom conjecture (which is now a theorem, as proved for example by [[Peter Ozsváth|Ozsváth]] and Szabó in 2000<ref>{{Cite journal |last=Ozsváth |first=Peter |last2=Szabó |first2=Zoltán |year=2000 |title=The symplectic Thom conjecture |journal=[[Annals of Mathematics|Ann. of Math.]] |volume=151 |issue=1 |pages=93–124 |arxiv=math.DG/9811087  }}</ref>). 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.
 
== See also ==
* [[Adjunction formula (algebraic geometry)|Adjunction formula]]
 
== References ==
<references/>
 
{{DEFAULTSORT:Thom Conjecture}}
[[Category:Four-dimensional geometry]]
[[Category:4-manifolds]]
[[Category:Algebraic surfaces]]
[[Category:Conjectures]]

Revision as of 16:18, 27 August 2013

In mathematics, a smooth algebraic curve in the complex projective plane, of degree , has genus given by the formula

.

The Thom conjecture, named after French mathematician René Thom, states that if is any smoothly embedded connected curve representing the same class in homology as , then the genus of satisfies

.

In particular, C is known as a genus minimizing representative of its homology class. It was first proved by Kronheimer–Mrowka in October 1994, using the then-new Seiberg–Witten invariants.

Assuming that has nonnegative self intersection number this was generalizes to Kähler manifolds (an example being the complex projective plane) by Morgan–Szabó–Taubes, also using the then-new 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 Ozsváth and Szabó in 2000[1]). 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.

See also

References

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