By Piotr Pragacz

ISBN-10: 3764385367

ISBN-13: 9783764385361

ISBN-10: 3764385375

ISBN-13: 9783764385378

The articles during this quantity are dedicated to:

- moduli of coherent sheaves;

- valuable 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 goal is to provide "friendly" introductions to the above issues via a chain of finished texts ranging from a really straightforward point and finishing with a dialogue of present examine. In those texts, the reader will locate classical effects and strategies in addition to new ones. The booklet is addressed to researchers and graduate scholars in algebraic geometry, algebraic topology and singularity conception. many of the fabric awarded within the quantity has no longer seemed in books before.

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

**Example text**

1. 1. Locally free resolutions of vector bundles on C. Let F be a vector bundle on C. 1 we ﬁnd a locally free sheaf F on C2 such that F|C = F , and a free resolution of F on C2 : · · · F ⊗ O2 (−2C) / F ⊗ O2 (−C) /F /F / 0. From this it follows that for every vector bundle E on C we have ExtiO2 (F, E) Hom(F ⊗ Li , E) for i ≥ 1. 2. Construction of quasi locally free sheaves. Let F be a quasi locally free coherent sheaf on C2 . Let E = EF , F = FF . We have an exact sequence (∗) 0 −→ E −→ F −→ F −→ 0 and E, F are vector bundles on C2 .

18 (1978), 577–614. , Trautmann, G. 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.

A closed point z ∈ R is called GIT-semistable if, for some m > 0, there is a G-invariant section s of OR (m) such that s(z) = 0. If, moreover, the orbit of z is closed in the open set of all GIT-semistable points, it is called GIT-polystable, and, if furthermore, this closed orbit has the same dimension 52 T. , if z has ﬁnite stabilizer), then z is called a GIT-stable point. We say that a closed point of R is GIT-unstable if it is not GIT-semistable. We will use the following characterization in [Mu1] of GIT-(semi)stability.

### Algebraic cycles, sheaves, shtukas, and moduli by Piotr Pragacz

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