The enumerative geometry of r-th roots of line bundles is the subject of Witten's conjecture and occurs in the calculation of Gromov-Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the generalization of the standard techniques from the theory of moduli of stable curves. In math.AG/0603687, we construct a compact stack by describing the notion of stability in the context of twisted curves. In this paper, by working with stable twisted curves, we extend Mumford's formula for the Chern character of the Hodge bundle to the direct image of the universal r-th root in K-theory.
The $P = W$ conjecture identifies the perverse filtration of the Hitchin system on the cohomology of the moduli space of Higgs bundles with the weight filtration of the corresponding character variety. In this paper, we introduce an enumerative approach to to this problem; our technique only uses the structure of the equivariant intersection numbers on the moduli space of Higgs bundles, and little information about the topology of the Hitchin map. In the rank $2$ case, starting from the known intersection numbers of the moduli of stable bundles, we derive the equivariant intersection numbers o…
We apply virtual localization to the problem of finding blowup formulae for virtual sheaf-theoretic invariants on a smooth projective surface. This leads to a general procedure that can be used to express virtual enumerative invariants on the blowup in terms of those on the original surface. We use an enhanced master space construction over the moduli spaces of $m$-stable sheaves introduced by Nakajima and Yoshioka. Our work extends their analogous results for the equivariant moduli spaces of framed sheaves on $\mathbb{P}^2$. In contrast to their work, we make no use of GIT methods and work wi…
We explore the geometric interpretation of the twisted index of 3d ${\mathcal N} =4$ gauge theories on $S^1\times Σ$ where $Σ$ is a closed Riemann surface. We focus on a rich class of supersymmetric quiver gauge theories that have isolated vacua under generic mass and FI parameter deformations. We show that the path integral localises to a moduli space of generalised vortex equations on $Σ$, which can be understood algebraically as quasi-maps to the Higgs branch. We show that the twisted index reproduces the virtual Euler characteristic of the moduli spaces of twisted quasi-maps and demonstrat…
Motivated by the relation between (twisted) K3 surfaces and special cubic fourfolds, we construct moduli spaces of polarized twisted K3 surfaces of any fixed degree and order. We do this by mimicking the construction of the moduli space of untwisted polarized K3 surfaces as a quotient of a bounded symmetric domain.
For $i\geq2$, we compute the first coefficients of the class $[\overline{D}(μ;3)]$ in the rational Picard group of the moduli of Prym curves $\overline{\mathcal{R}}_{2i}$, where $D(μ;3)$ is the divisor parametrizing pairs $[C,η]$ for which there exists a degree $2i$ map $π\colon C\rightarrow \mathbb{P}^1$ having ramification profile $(2,\ldots,2)$ above two points $q_1, q_2$, a triple ramification somewhere else and satisfying $\mathcal{O}_C(\frac{π^{*}(q_1)-π^{*}(q_2)}{2})\cong η$. Furthermore, we provide several new Prym enumerative results related to this situation.
We prove a canonical identifications of the spaces of generalized theta functions on the moduli spaces of anti-invariant vector bundles in the ramified case and the conformal blocks associated to twisted Kac-Moody affine algebras.
We study the BPS invariants for local del Pezzo surfaces, which can be obtained as the signed Euler characteristic of the moduli spaces of stable one-dimensional sheaves on the surface $S$. We calculate the Poincare polynomials of the moduli spaces for the curve classes $β$ having arithmetic genus at most 2. We formulate a conjecture that these Poincare polynomials are divisible by the Poincare polynomials of $((-K_S).β-1)$-dimensional projective space. This conjecture motivates upcoming work on log BPS numbers.
Enumerative invariants in Algebraic Geometry 'count' $τ$-(semi)stable objects $E$ with fixed topological invariants $[E]=a$ in some geometric problem, using a virtual class $[{\cal M}_a^{\rm ss}(τ)]_{\rm virt}$ in homology, for the moduli spaces ${\cal M}_a^{\rm st}(τ)\subseteq{\cal M}_a^{\rm ss}(τ)$ of $τ$-(semi)stable objects. We get numbers by taking integrals $\int_{[{\cal M}_a^{\rm ss}(τ)]_{\rm virt}}Υ$ for cohomology classes $Υ$. Let $\cal A$ be a $\mathbb C$-linear abelian category in Algebraic Geometry. There are two moduli stacks of objects in $\cal A$: the usual moduli stack $\cal M$…
We construct a smooth Artin stack parameterizing the stable weighted curves of genus one with twisted fields and prove that it is isomorphic to the blowup stack of the moduli of genus one weighted curves studied by Hu and Li. This leads to a blowup-free construction of Vakil-Zinger's desingularization of the moduli of genus one stable maps to projective spaces. This construction provides the cornerstone of the theory of stacks with twisted fields, which is thoroughly studied in arXiv:2005.03384 and leads to a blowup-free resolution of the stable map moduli of genus two.
In this paper we introduce the stack of polarized twisted conics and we use it to give a new point of view on $\overline{\mathcal{M}}_2$. In particular, we present a new and independent approach to the computation of the integral Chow ring of $\overline{\mathcal{M}}_2$, previously determined by Eric Larson.
An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $τ$-(semi)stable objects $E$ with fixed topological invariants $[E]=α$ in some geometric problem, using a virtual class $[{\cal M}_α^{\rm ss}(τ)]_{\rm virt}$ in some homology theory for the moduli spaces ${\cal M}_α^{\rm st}(τ)\subseteq{\cal M}_α^{\rm ss}(τ)$ of $τ$-(semi)stable objects. Examples include Mochizuki's invariants counting coherent sheaves on surfaces, Donaldson-Thomas type invariants counting coherent sheaves on Calabi-Yau 3- and 4-folds…
In order to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points, we construct a compactification of their moduli space. We term the latter a $1$--\textit{complete variety of cubic hypersurfaces} in analogy to the space of complete quadrics. Paolo Aluffi explored the case of plane cubic curves. Starting from his work, we construct such a space in arbitrary dimension by a sequence of five blow-ups. The counting problem is then reduced to the computation of five Chern classes, climbing the sequence of blow-ups. Comput…
The first part of this work constructs positive-genus real Gromov-Witten invariants of real-orientable symplectic manifolds of odd "complex" dimensions; the present part focuses on their properties that are essential for actually working with these invariants. We determine the compatibility of the orientations on the moduli spaces of real maps constructed in the first part with the standard node-identifying immersion of Gromov-Witten theory. We also compare these orientations with alternative ways of orienting the moduli spaces of real maps that are available in special cases. In a sequel, we …
We provide a definition of Tanaka-Thomas's Vafa-Witten invariants for étale gerbes over smooth projective surfaces using the moduli spaces of $μ_r$-gerbe twisted sheaves and Higgs sheaves. Twisted sheaves and their moduli are naturally used to study the period-index theorem for the corresponding $μ_r$-gerbe in the Brauer group of the surfaces. Deformation and obstruction theory of the twisted sheaves and Higgs sheaves behave like general sheaves and Higgs sheaves. We define virtual fundamental classes on the moduli spaces and define the twisted Vafa-Witten invariants using virtual localization…
