As a consistency check: an explicit calculation using these expressions leads to the sign predicted by Main Conjecture 6. We should therefore not separate these three terms. This motivates the following definition. The proof is similar to that of Proposition 6. The degenerate cases can be done similarly. In this section, we explain where the sign in Main Conjecture 6. The argument is a variation on [ 39 , Sect.

Examples 6. In this section, we prove Main Conjecture 6.

Pandharipande and Thomas need two conjectures for their argument [ 47 , Conj. We need completely similar analogs of their conjectures in our setting. The second will be described below. The analog of the second conjecture of Pandharipande and Thomas [ 47 , Conj. Their definition and equations 45 and 46 hold under the assumptions of Conjecture 6.

Suppose the setting is as in Conjecture 6. See Examples 6. We would like to thank K. Behrend, J. Bryan, J. Manschot, D. Maulik, and R. Thomas for useful discussions. We would also like to thank the anonymous referees whose comments led to an immense improvement of the exposition, e. Sign In. Advanced Search.

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Part I: Classical. Part II: Virtual. Oxford Academic. Google Scholar. Martijn Kool. Correspondence to be sent to: e-mail: m. Benjamin Young. Cite Citation. Permissions Icon Permissions. See [ 24 ] for the construction of the moduli space and its history. In this article, we do not consider strictly semi-stable sheaves so this moduli space may be non-compact.

In this case, the moduli space is compact. When the moduli space is compact, this leads to a virtual cycle, which can be used to define deformation invariants.

Its degree is known as a Donaldsonâ€”Thomas DT invariant. Equivariant torsion free sheaves are much harder to enumerate than equivariant reflexive sheaves. This can be viewed as the degree 0 part of rank 2 DT theory on smooth toric 3-folds.

Mentioned to us by Maulik and Manschot. Note that unlike the rank 1 case, there are no signs in this formula essentially because rank 2 is even. The following is strong evidence for Conjecture C. This formula is due to Klyachko [ 28 , 29 ]. Concretely, this works as follows. Let the situation be as in the previous definition. It is not hard to show this is a morphism [ 32 ]. The map 8 is therefore surjective but not injective. It can be made injective as follows. As discussed in the introduction, double duals of members of a flat family of torsion free sheaves do not need to form a flat family.

These data are represented by Figure 1. Open in new tab Download slide. We start with some definitions. Analogous to Definition 2. In Definition 2. The following generating function was introduced in Remark 2. Geometric proof. The geometric proof proceeds as follows. We first realize the left hand sides of Theorems 2. Proposition 2. Combining 17 , Prop. This problem is discussed in [ 10 , Ch. Theorem 3. Using formula 17 , Proposition 2. By the proof of [Kol, Thm. Consider the short exact sequence of Lemma 4.

Verschoren, electronic resource Resource Information. The item Compatibility, stability, and sheaves, J. Verschoren, electronic resource represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University Of Pikeville. This item is available to borrow from 1 library branch. Creator Bueso, J.

Contributor Jara, P. Language eng. Publication New York, Marcel Dekker, c Extent xiv, p. Isbn Label Compatibility, stability, and sheaves Title Compatibility, stability, and sheaves Statement of responsibility J. Verschoren Creator Bueso, J. Verschoren, A. Label Compatibility, stability, and sheaves, J. Verschoren, electronic resource Instantiates Compatibility, stability, and sheaves Publication New York, Marcel Dekker, c Bibliography note Includes bibliographical references p.

Preliminaries: There are lots of variations on the settings in which we define sheaves. I am concerned with the details linking the general definition below to the Grothendieck topology it generates; particularly, the assertion that sheaves with respect to a coverage are the same as sheaves with respect to the generated Grothendieck coverage. Descent can be phrased concisely in terms of subfunctors of Yoneda embeddings.

What is not clear to me is how, "One can then show that for every coverage, there is a unique Grothendieck coverage having the same sheaves. Sign up to join this community.

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Why are sheaves of a coverage the same as those on its generated Grothendieck coverage? Ask Question. Asked 1 year, 5 months ago.