By Daniel Huybrechts

Now again in print, this extremely popular ebook has been up to date to mirror fresh advances within the conception of semistable coherent sheaves and their moduli areas, which come with moduli areas in confident attribute, moduli areas of significant 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 evaluation alterations within the box because the book of the unique variation in 1997 and aspect the reader in the direction of extra literature. References were stated thus far and mistakes got rid of. built from the authors' lectures, this booklet is perfect as a textual content for graduate scholars in addition to a necessary source for any mathematician with a history in algebraic geometry who desires to examine extra approximately Grothendieck's method.

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Then it is a topological sphere and z: K → P1 is a conformal equivalence between K and the projective line. Case 2. K is noncompact and p is unbounded. Then z: K → C is a conformal equivalence between K and the whole complex plane. Case 3. K is noncompact and p tends to the ﬁnite number p(∞). Then z maps K 1:1 onto a disk of radius r = exp [ p(∞)]. This is Riemann’s case. Idea of the proof. Springer [1981] explains the actual construction of p: Let ∧1 be the class of smooth 1-forms ω = ω1 d x1 + ω2 d x2 on K and let ∗ω = −ω2 d x1 + ω1 d x2 .

Punctured Spheres. The once-punctured sphere is the plane, so the twicepunctured sphere is a punctured plane, and its universal cover K may be viewed as the Riemann surface of the logarithm; in short, K is a plane. The thricepunctured sphere (= the doubly punctured plane) is different: Its universal cover is the disk. 14 Examples 47 Proof. ), so only K = C needs to be ruled out. But if K = C, then the covering group is populated by substitutions of P S L(2, C) ﬁxing ∞; see Section 5. These are of the form z → az + b and have ﬁxed points if a = 1, whereas covering maps do not; see ex.

Check all that by means of pictures. Think of an example in which the lifted curve is not closed. Exercise 6. Prove that every covering map arises in this way. Exercise 7. Prove that covering maps have no ﬁxed points, the identity excepted. Fundamental Group. The loops of M, starting and ending at o, fall into deformation classes; and it is easy to see that these classes form a group: The formation of classes respects the composition of loops effected by passing ﬁrst about loop 1 and then about loop 2; the identity is the class of the trivial loop = the point o itself; the inverse is the class of the loop run backward; and so on.