By Rick Miranda

ISBN-10: 0821802682

ISBN-13: 9780821802687

During this ebook, Miranda takes the procedure that algebraic curves are most sensible encountered for the 1st time over the complicated numbers, the place the reader's classical instinct approximately surfaces, integration, and different innovations will be introduced into play. accordingly, many examples of algebraic curves are awarded within the first chapters. during this means, the e-book starts as a primer on Riemann surfaces, with advanced charts and meromorphic services taking middle degree. however the major examples come from projective curves, and slowly yet without doubt the textual content strikes towards the algebraic class. Proofs of the Riemann-Roch and Serre Duality Theorems are awarded in an algebraic demeanour, through an variation of the adelic evidence, expressed thoroughly when it comes to fixing a Mittag-Leffler challenge. Sheaves and cohomology are brought as a unifying equipment within the latter chapters, in order that their software and naturalness are instantly visible. Requiring a historical past of a one semester of advanced variable! idea and a yr of summary algebra, this is often an exceptional graduate textbook for a second-semester path in complicated variables or a year-long direction in algebraic geometry.

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**Extra resources for Algebraic Curves and Riemann Surfaces **

**Example text**

Let U C C n be an open subset. Thus, U can in particular be considered as a 2n-dimensional real manifold. For x 6 U we have its real tangent space TXU at the point x which is of real dimension In. , zn = xn + iyn are the standard coordinates on C™. Moreover, the vectors ^ - , . . , -^— are global trivializing sections of TU. Each tangent space TXU admits a natural almost complex structure defined by T . rp TT l . lxu rp JJ *-±xv, J) Qx. _d_ I *• 9 y . , _d_ 9 y . d i 9 x i , which is compatible with the global trivialization.

Xn,yn = I(xn) as above, a straightforward calculation yields n\ • ujn = vol, where ui is the associated fundamental form. 9 for a far reaching generalization of this. e. A{/\ V*) C /\ ~ V*. Moreover, one has A = *~1 o L o *. Proof. The first assertion follows from the fact that L is of degree two and that f\* V* = 0 f\k V* is orthogonal. By definition of the Hodge ^-operator one has (a, L(3) -vol = (L(3, a) -vol = L/3 A *a = oj A j3 A *a = j3 A (UJ A *a) = (/3, *" 1 (L(*a))) • vol. Recall that ( , )c had been defined as the hermitian extension to V<£ of the scalar product ( , ) on V*.

2 / / / is an almost complex structure on a real vector space V, then V admits in a natural way the structure of a complex vector space. Proof. The C-module structure on V is defined by (a + ib) • v = a • v + b • I(v), where a, b £ M. The M-linearity of I and the assumption I2 = —id yield ((a + ib){c + id)) • v = (a + ib)((c + id) • v) and in particular i(i • v) = —v. • Thus, almost complex structures and complex structures are equivalent notions for vector spaces. In particular, an almost complex structure can only exist on an even dimensional real vector space.

### Algebraic Curves and Riemann Surfaces by Rick Miranda

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