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By Daniel Huybrechts

Now again in print, this very hot e-book has been up-to-date to mirror contemporary advances within the idea of semistable coherent sheaves and their moduli areas, which come with moduli areas in confident attribute, moduli areas of central bundles and of complexes, Hilbert schemes of issues on surfaces, derived different types of coherent sheaves, and moduli areas of sheaves on Calabi-Yau threefolds. The authors overview adjustments within the box because the book of the unique version in 1997 and element the reader in the direction of additional literature. References were stated thus far and error got rid of. built from the authors' lectures, this ebook is perfect as a textual content for graduate scholars in addition to a worthwhile source for any mathematician with a history in algebraic geometry who desires to study extra approximately Grothendieck's process.

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F ] := ν ν ν≥0 (−1) [R f∗ F ]. 10: if F is an S-flat family of coherent sheaves on the fibres of f , then [F ] ∈ K 0 (X) and f! [F ] ∈ K 0 (S). Proof. [F ] = i i (−1) [Fi ] i i i (−1) [R f∗ F ] and f! [F ] = n−i [Rn f∗ Fi ].

HN (E) = E, such that the factors griHN = HNi (E)/HNi−1 (E) for i = 1, . . , , are semistable sheaves of dimension d with reduced Hilbert polynomials pi satisfying pmax (E) := p1 > . . > p =: pmin (E). 3 The Harder-Narasimhan Filtration 17 Obviously, E is semistable if and only if E is pure and pmax (E) = pmin (E). A priori, the definition of the maximal and minimal p of a sheaf E depends on the filtration. We will see in the next theorem, that the Harder-Narasimhan filtration is uniquely determined, so that there is no ambiguity in the notation.

If μ(M ) = 1, then necessarily M = L1 and M would provide a splitting of the extension in contrast to the assumption. Hence again μ(M ) ≤ 0 < 1/2. On the other hand, a direct sum L0 ⊕ L1 of line bundles of different degree is not even semistable. By a similar technique, one can also construct semistable bundles which are not stable, but simple: let X be a projective curve of genus g ≥ 2 over an algebraically closed field k and let E1 and E2 be two nonisomorphic stable vector bundles of rank r1 and r2 , respectively, with μ(E1 ) = μ(E2 ).

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