Framed sheaves on projective space and Quot schemes

We prove that, given integers $$m\ge 3$$ m ≥ 3 , $$r\ge 1$$ r ≥ 1 and $$n\ge 0$$ n ≥ 0 , the moduli space of torsion free sheaves on $${\mathbb {P}}^m$$ P m with Chern character $$(r,0,\ldots ,0,-n)$$ ( r , 0 , … , 0 , - n ) that are trivial along a hyperplane $$D \subset {\mathbb {P}}^m$$ D ⊂ P m is isomorphic to the Quot scheme $$\mathrm{Quot}_{{\mathbb {A}}^m}({\mathscr {O}}^{\oplus r},n)$$ Quot A m ( O ⊕ r , n ) of 0-dimensional length n quotients of the free sheaf $${\mathscr {O}}^{\oplus r}$$ O ⊕ r on $${\mathbb {A}}^m$$ A m . The proof goes by comparing the two tangent-obstruction theories on these moduli spaces.

Higher rank K-theoretic Donaldson-Thomas Theory of points

We exploit the critical structure on the Quot scheme $\text {Quot}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$ , in particular the associated symmetric obstruction theory, in order to study rank r K-theoretic Donaldson-Thomas (DT) invariants of the local Calabi-Yau $3$ -fold ${{\mathbb {A}}}^3$ . We compute the associated partition function as a plethystic exponential, proving a conjecture proposed in string theory by Awata-Kanno and Benini-Bonelli-Poggi-Tanzini. A crucial step in the proof is the fact, nontrival if $r>1$ , that the invariants do not depend on the equivariant parameters of the framing torus $({{\mathbb {C}}}^\ast )^r$ . Reducing from K-theoretic to cohomological invariants, we compute the corresponding DT invariants, proving a conjecture of Szabo. Reducing further to enumerative DT invariants, we solve the higher rank DT theory of a pair $(X,F)$ , where F is an equivariant exceptional locally free sheaf on a projective toric $3$ -fold X. As a further refinement of the K-theoretic DT invariants, we formulate a mathematical definition of the chiral elliptic genus studied in physics. This allows us to define elliptic DT invariants of ${{\mathbb {A}}}^3$ in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.

Smoothing Pairs Over Degenerate Calabi–Yau Varieties

We apply the techniques developed in [2] to study smoothings of a pair $(X,\mathfrak{C}^*)$, where $\mathfrak{C}^*$ is a bounded perfect complex of locally free sheaves over a degenerate Calabi–Yau variety $X$. In particular, if $X$ is a projective Calabi–Yau variety admitting the structure of a toroidal crossing space and with the higher tangent sheaf $\mathcal{T}^1_X$ globally generated, and $\mathfrak{F}$ is a locally free sheaf over $X$, then we prove, using the results in [ 8], that the pair $(X,\mathfrak{F})$ is formally smoothable when $\textrm{Ext}^2(\mathfrak{F},\mathfrak{F})_0 = 0$ and $H^2(X,\mathcal{O}_X) = 0$.

Higher rank sheaves on threefolds and functional equations

We consider the moduli space of stable torsion free sheaves of any rank on a smooth projective threefold. The singularity set of a torsion free sheaf is the locus where the sheaf is not locally free. On a threefold it has dimension $\leq 1$. We consider the open subset of moduli space consisting of sheaves with empty or 0-dimensional singularity set. For fixed Chern classes $c_1,c_2$ and summing over $c_3$, we show that the generating function of topological Euler characteristics of these open subsets equals a power of the MacMahon function times a Laurent polynomial. This Laurent polynomial is invariant under $q \leftrightarrow q^{-1}$ (upon replacing $c_1 \leftrightarrow -c_1$). For some choices of $c_1,c_2$ these open subsets equal the entire moduli space. The proof involves wall-crossing from Quot schemes of a higher rank reflexive sheaf to a sublocus of the space of Pandharipande-Thomas pairs. We interpret this sublocus in terms of the singularities of the reflexive sheaf. Comment: 29 pages. Published version

Rost’s Chain Lemma

This chapter states and proves Rost's Chain Lemma. The proof (due to Markus Rost) does not use the inductive assumption that BL(n − 1) holds. Throughout this chapter, 𝓁 is a fixed prime, and 𝑘 is a field containing 1/𝓁 and all 𝓁th roots of unity. It fixes an integer 𝑛 ≥ 2 and an 𝑛-tuple (𝑎1, ..., 𝑎𝑛) of units in 𝑘, such that the symbol ª = {𝑎1, ..., 𝑎𝑛} is nontrivial in the Milnor 𝐾-group 𝐾𝑀 𝑛(𝑘)/𝓁. The chapter produces the statement of the Chain Lemma by first proving the special case 𝑛 = 2. The notion of an 𝓁-form on a locally free sheaf over 𝑆 is then introduced, before the chapter shows how 𝓁-forms may be used to define elements of 𝐾𝑀 𝑛(𝑘(𝑆))/𝓁.

Pencils and nets on curves arising from rank $1$ sheaves on K3 surfaces

Let $S$ be a K3 surface, $C$ a smooth curve on $S$ with $\mathcal{O} _S(C)$ ample, and $A$ a base-point free $g^2_d$ on $C$ of small degree. We use Lazarsfeld-Mukai bundles to prove that $A$ is cut out by the global sections of a rank $1$ torsion-free sheaf $\mathcal{G} $ on $S$. Furthermore, we show that $c_1(\mathcal{G} )$ with one exception is adapted to $\mathcal{O} _S(C)$ and satisfies $\operatorname{Cliff} (c_1(\mathcal{G} )_{|C})\leq \operatorname{Cliff} (A)$, thereby confirming a conjecture posed by Donagi and Morrison. We also show that the same methods can be used to give a simple proof of the conjecture in the $g^1_d$ case.In the final section, we give an example of the mentioned exception where $h^0(C,c_1(\mathcal{G} )_{|C})$ is dependent on the curve $C$ in its linear system, thereby failing to be adapted to $\mathcal{O} _S(C)$.

A pathological case of the C1 conjecture in mixed characteristic

Let K be a field of characteristic 0. Fix integers r, d coprime with r ⩾ 2. Let XK be a smooth, projective, geometrically connected curve of genus g ⩾ 2 defined over K. Assume there exists a line bundle ${\cal L}_K$ on XK of degree d. In this paper we prove the existence of a stable locally free sheaf on XK with rank r and determinant ${\cal L}_K$. This trivially proves the C1 conjecture in mixed characteristic for the moduli space of stable locally free sheaves of fixed rank and determinant over a smooth, projective curve.

Framed symplectic sheaves on surfaces

A framed symplectic sheaf on a smooth projective surface [Formula: see text] is a torsion-free sheaf [Formula: see text] together with a trivialization on a divisor [Formula: see text] and a morphism [Formula: see text] satisfying some additional conditions. We construct a moduli space for framed symplectic sheaves on a surface, and present a detailed study for [Formula: see text]. In this case, the moduli space is irreducible and admits an ADHM-type description and a birational proper map onto the space of framed symplectic ideal instantons.

On Weil homomorphism in locally free sheaves over structured spaces

Inspired by the work of Heller and Sasin [1], we construct in this paper Weil homomorphism in a locally free sheaf W of ϕ-fields [2] over a structured space. We introduce the notion of G-consistent, linear connection on this sheaf, what allows us to clearly define Chern, Pontrjagin and Euler characteristic classes. We also show proper equalities between those classes.