Topologically invariant Chern numbers of projective varieties. (English) Zbl 1244.57048
The author proves the following main results:
Theorem 1 A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers.
Theorem 2. In the case of complex dimension \(\geq 3\), a rational linear combination of Chern numbers is a diffeomorphism invariant of smooth complex projective varieties if and only if it is a multiple of the Euler number. In the case of complex dimension 2, both Chern numbers \(c_2\) and \(c_1^2\) are diffeomorphism invariants of complex projective surfaces.
Replacing diffeomorphism by homeomorphism, the author shows that
Theorem 3. A rational combination of Chern numbers is a homeomorphism invariant of smooth complex projective varieties if and only if it is a multiple of the Euler number.
These results solve a long-time problem posed by Hirzebruch. In addition, the author also determine the linear combination of Chern numbers that can be bounded in terms of Betti numbers.
Theorem 1 A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers.
Theorem 2. In the case of complex dimension \(\geq 3\), a rational linear combination of Chern numbers is a diffeomorphism invariant of smooth complex projective varieties if and only if it is a multiple of the Euler number. In the case of complex dimension 2, both Chern numbers \(c_2\) and \(c_1^2\) are diffeomorphism invariants of complex projective surfaces.
Replacing diffeomorphism by homeomorphism, the author shows that
Theorem 3. A rational combination of Chern numbers is a homeomorphism invariant of smooth complex projective varieties if and only if it is a multiple of the Euler number.
These results solve a long-time problem posed by Hirzebruch. In addition, the author also determine the linear combination of Chern numbers that can be bounded in terms of Betti numbers.
Reviewer: Zhi Lü (Shanghai)
MSC:
| 57R20 | Characteristic classes and numbers in differential topology |
| 57R77 | Complex cobordism (\(\mathrm{U}\)- and \(\mathrm{SU}\)-cobordism) |
| 14J99 | Surfaces and higher-dimensional varieties |
| 55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |
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