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Register a new userCounting sheaves on Calabi-Yau 4-folds, I
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Borisov-Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived differential geometry. We construct an algebraic virtual cycle. A key step is a localisation of Edidin-Grahams square root Euler class for $SO(r,mathbb C)$ bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localisation formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We give a $K$-theoretic refinement by defining $K$-theoretic square root Euler classes and their localis
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Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of 2-dimensional torsion sheaves, enumerating pairs $Zsubset H$ in a Calabi-Yau threefold X. Here H is a member of a sufficiently positive linear system and Z is a 1-dimensional subscheme of it. The associated sheaf is the ideal sheaf of $Zsubset H$, pushed forward to X and considered as a certain Joyce-Song pair in the derived category of X. We express these invariants in terms of the MNOP invariants of X.
We study Hilbert schemes of points on a smooth projective Calabi-Yau 4-fold $X$. We define $mathrm{DT}_4$ invariants by integrating the Euler class of a tautological vector bundle $L^{[n]}$ against the virtual class. We conjecture a formula for their generating series, which we prove in certain cases when $L$ corresponds to a smooth divisor on $X$. A parallel equivariant conjecture for toric Calabi-Yau 4-folds is proposed. This conjecture is proved for smooth toric divisors and verified for more general toric divisors in many examples. Combining the equivariant conjecture with a vertex calculation, we find explicit positive rational weights, which can be assigned to solid partitions. The weighted generating function of solid partitions is given by $exp(M(q)-1)$, where $M(q)$ denotes the MacMahon function.
Let $X$ be a compact Calabi-Yau 3-fold, and write $mathcal M,bar{mathcal M}$ for the moduli stacks of objects in coh$(X),D^b$coh$(X)$. There are natural line bundles $K_{mathcal M}tomathcal M$, $K_{bar{mathcal M}}tobar{mathcal M}$, analogues of canonical bundles. Orientation data on $mathcal M,bar{mathcal M}$ is an isomorphism class of square root line bundles $K_{mathcal M}^{1/2},K_{bar{mathcal M}}^{1/2}$, satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman arXiv:1006.270 in their theory of motivic Donaldson-Thomas invariants, and is important in categorifying Donaldson-Thomas theory using perverse sheaves. We show that natural orientation data can be constructed for all compact Calabi-Yau 3-folds, and also for compactly-supported coherent sheaves and perfect complexes on noncompact Calabi-Yau 3-folds $X$ with a spin smooth projective compactification $Xhookrightarrow Y$. This proves a long-standing conjecture in Donaldson-Thomas theory. These are special cases of a more general result. Let $X$ be a spin smooth projective 3-fold. Using the spin structure we construct line bundles $K_{mathcal M}tomathcal M$, $K_{bar{mathcal M}}tobar{mathcal M}$. We define spin structures on $mathcal M,bar{mathcal M}$ to be isomorphism classes of square roots $K_{mathcal M}^{1/2},K_{bar{mathcal M}}^{1/2}$. We prove that natural spin structures exist on $mathcal M,bar{mathcal M}$. They are equivalent to orientation data when $X$ is a Calabi-Yau 3-fold with the trivial spin structure. We prove this using our previous paper arXiv:1908.03524, which constructs spin structures (square roots of a certain complex line bundle $K_Ptomathcal B_P$) on differential-geometric moduli stacks $mathcal B_P$ of connections on a principal U$(m)$-bundle $Pto X$ over a compact spin 6-manifold $X$.
Let X be an n-dimensional Calabi-Yau with ordinary double points, where n is odd. Friedman showed that for n=3 the existence of a smoothing of X implies a specific type of relation between homology classes on a resolution of X. (The converse is also true, due to work of Friedman, Kawamata and Tian.) We sketch a more topological proof of this result, and then extend it to higher dimensions. For n>3 the Yukawa product on the middle dimensional (co)homology plays an unexpected role. We also discuss a converse, proving it for nodal Calabi-Yau hypersurfaces in projective space.
We first construct a derived equivalence between a small crepant resolution of an affine toric Calabi-Yau 3-fold and a certain quiver with a superpotential. Under this derived equivalence we establish a wall-crossing formula for the generating function of the counting invariants of perverse coherent systems. As an application we provide certain equations on Donaldson-Thomas, Pandeharipande-Thomas and Szendrois invariants. Finally, we show that moduli spaces associated with a quiver given by successive mutations are realized as the moduli spaces associated the original quiver by changing the stability conditions.
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