By Piotr Pragacz

The articles during this quantity are committed to:

- moduli of coherent sheaves;

- crucial bundles and sheaves and their moduli;

- new insights into Geometric Invariant Theory;

- stacks of shtukas and their compactifications;

- algebraic cycles vs. commutative algebra;

- Thom polynomials of singularities;

- 0 schemes of sections of vector bundles.

The major objective is to offer "friendly" introductions to the above themes via a chain of accomplished texts ranging from a truly user-friendly point and finishing with a dialogue of present study. In those texts, the reader will locate classical effects and techniques in addition to new ones. The publication is addressed to researchers and graduate scholars in algebraic geometry, algebraic topology and singularity thought. lots of the fabric awarded within the quantity has no longer seemed in books before.

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**Extra info for Algebraic cycles, sheaves, shtukas, and moduli**

**Example text**

Let E = IZ ⊗ L. Then Z is uniquely determined by E, but L need not be unique. The integer i0 (E) = h0 (OZ ) is called the index of E (in particular the index of a line bundle on C2 is 0). It is invariant by deformation of E. 1. Let p ≥ 0, d be integers and E a rank 2 vector bundle of degree d on C. Let q = 12 (d + deg(L) + p). Then 1. If E can be deformed in torsion free sheaves on C2 that are not concentrated on C and of index p, then there exist a line bundle V on C of degree q and a nonzero morphism α : V → E such that Hom((E/ im(α)) ⊗ L, V ) = {0}.

A good quotient is a scheme M with a G-invariant morphism p : R → M such that 1. p is surjective and aﬃne G G ) = OM , where OR is the sheaf of G-invariant functions on R. 2. p∗ (OR 3. If Z is a closed G-invariant subset of R, then p(Z) is closed in M . Furthermore, if Z1 and Z2 are two closed G-invariant subsets of R with Z1 ∩ Z2 = ∅, then f (Z1 ) ∩ f (Z2 ) = ∅. 3 (Geometric quotient). A geometric quotient p : R → M is a good quotient such that p(x1 ) = p(x2 ) if and only if the orbit of x1 is equal to the orbit of x2 .

Limits of instantons. Intern. Journ. of Math. 3 (1992), 213–276. , Spindler, H. Vector bundles on complex projective spaces. Progress in Math. 3, Birkh¨ auser (1980). [18] Ramanan, S. The moduli spaces of vector bundles over an algebraic curve. Math. Ann. 200 (1973), 69–84. T. Moduli of representations of the fundamental group of a smooth projective variety I. Publ. Math. IHES 79 (1994), 47–129. -M. , Trautmann, G. Deformations of coherent analytic sheaves with compact supports. Memoirs of the Amer.