Chern Numbers And Rozansky-witten Invariants Of Compact - download pdf or read online

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Extra resources for Chern Numbers And Rozansky-witten Invariants Of Compact Hyper-kahler Manifolds

Example text

M= p=o YP J ch(Rp,) . %X). SO) x Inverting these relations allows u s to express some characteristic numbers in terms of Hodge numbers. 61) ~ 8x2(X), ~ cg(X) = 7272x0(X)- 1 8 4 - (n = 6) c ~ ( X ) C ~ (=X1368xo(X) ) -2 0 8 ~ 8x2(X), ~ + cg(X) = 36xo(X)- 16x1(X) 4x2(X). For a better readability, we have suppressed the integral signs on the left hand sides. This allows us t o calculate some Chern numbers (all for dimensions less or equal six) of the two example series as their Xy-genus is known. However, this method is not sufficient t o calculate all Chern numbers.

Is any partition of n and cx = A1 A 2 5 3 . . These characteristic numbers are called Chern numbers. c1 c2 cg By results of the previous subsection, the Chern numbers of a holomorphic symplectic manifold X of dimension n can be expressed in terms of Chern numbers of the form c 2 x ( X ) , where X is a partition of n / 2 . 12 (Chern numbers of two-dimensional holomorphic symplectic manifolds) There is only one (non-trivial) Chern number of a holomorphic symplectic manifold X of dimension two, namely cZ(X).

Proof. 110) In this situation, there exist extension classes 4 E Extl(E,Slx @ E l ) and $ E Extl(Etf,R x @€) such that 4 H a,p and $ H a p by the natural maps. Further more, if we denote by 01 the image of 4 under the natural map Extl(E, R x @ C )-+ Ext’(€, R x @ E ) and by a 2 the image of $ under the natural map Extl(E”, R x @ E ) --+ Extl(€, @ E ) , we have QE = a1 a 2 . This can be seen, for example in the description of the extension groups in terms of Cech cohomology. By the naturality of the trace, it follows from this that Tr(az) = Tr(aF,) Tr(az,,),which proves the proposition.

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