Logarithmic Donaldson-Thomas theory

Abstract

Let $X$ be a smooth threefold with a simple normal crossings divisor $D$. We construct the Donaldson-Thomas theory of the pair $(X|D)$ enumerating ideal sheaves on $X$ relative to $D$. These moduli spaces are compactified by studying subschemes in expansions of the target geometry, and the moduli space carries a virtual fundamental class leading to numerical invariants with expected properties. We formulate punctual evaluation, rationality and wall-crossing conjectures, in parallel with the standard theory. Our formalism specializes to the Li-Wu theory of relative ideal sheaves when the divisor is smooth, and is parallel to recent work on logarithmic Gromov-Witten theory with expansions.

Figures (6)

• Figure 1: This captures how the authors visualize expansions and subschemes in them. The two pictures on the left are typical tropicalizations of subschemes. The top left is the tropicalization of a subscheme in the interior of the moduli problem, while the bottom left is a more complicated one. The pictures on the left are outputs of the Tevelev theorem, and determine expansions of $X$ -- a gluing together of projective bundles over the strata of $X$. The reader can notice, for example in the second row, a bijection between components of the expansion and the vertices of the graph on the left. The circled component on the right, in both pictures, is the main component, i.e. $X$ itself. The subschemes drawn, indicated in red, can have singularities, or embedded components -- three embedded points can be seen on the bottom right. However (i) the subscheme must be disjoint from the codimension $2$ strata, i.e. the corners, and (ii) the embedded points must lie in the interiors of irreducible components.
• Figure 2: On the left we sketch a $1$-complex in the cone complex $\mathbb R_{\geq 0}^2$. The cone complex $\Sigma_X$ is ${\mathbb R}_{\geq 0}^2$ drawn with a dashed arrow while the $1$-complex is solid. On the right is a cartoon of the corresponding expansion. It is obtained by performing deformation to the normal cone of a codimension $2$ stratum and then passing to an open. The wavy lines indicate the divisors where the logarithmic structure is nontrivial. The holes indicate codimension $2$ strata that have been removed from the expansion. The red curve in the middle is a subscheme, of the type that we will soon introduce.
• Figure 3: The $1$-complex depicted in solid arrows with the fan of $\mathbb P^2$ in dashed arrows.
• Figure 4: The figure shows a family of embedded $1$-complexes in $\mathbb R^3$ over a $1$-dimensional base. The $1$-complex on the left corresponds to a generic point but at $t = 0$ the two skew edges cross forming a new vertex. As a consequence, the total space has two $2$-dimensional cones meeting at a point in their interior. In order to obtain a total space that, after taking cones, is a cone complex, vertices have to be added to $e_1$ and $e_2$.
• Figure 5: An $1$-complex with a tube vertex on the left and the corresponding tube component on the right. The subscheme depicted in that tube component is a tube subscheme -- it is attempting to depict the preimage of a $0$-dimensional subscheme of length $2$, along the projection. On the figure on the right, the circle in the bottom left indicates the main component of the expansion, while the rest of the components are bundles over strata. The tube component contains a tube subscheme, and the picture is meant to depict a subscheme satisfying DT stability.

Theorems & Definitions (129)

• Theorem A
• Theorem B
• Definition 1.1.1: Cone complexes
• Definition 1.1.2: Flat maps and reduced fibers
• Definition 1.3.1
• Definition 1.3.2
• Remark 1.3.3
• Remark 1.3.4
• Theorem 1.5.1
• Theorem 1.5.2