arXiv subjects

Miguel Moreira

Publications and source records attributed to Miguel Moreira.

14 recordsLinked to original sources

arXiv2026

On the Feyzbakhsh-Thomas programme for Fano $3$-folds

Let $X$ be a Fano $3$-fold with even canonical class which satisfies the generalized Bogomolov-Gieseker inequality, such as $\mathbb P^3$. We express Donaldson-Thomas invariants counting Gieseker semistable sheaves of rank $r$, where $r > 0$, on $X$ in terms of those counting sheaves of rank $0$ and pure dimension $2$. This implements an analogue of the programme initiated by S. Feyzbakhsh and R. Thomas in the case of Calabi-Yau varieties. The methods include $K$-theoretic Donaldson-Thomas theory and, quite unexpectedly, the use of certain combinatorial properties of vertex algebras.

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arXiv2026

Rationality and symmetry of stable pairs generating series of Fano 3-folds

The generating series of descendent invariants of stable pairs on 3-folds is conjectured to be rational and to satisfy a $q\leftrightarrow q^{-1}$ symmetry. We prove this conjecture for Fano 3-folds. We utilize the same path of stability conditions that Toda used in his proof of the Calabi--Yau version of the conjecture, relating stable pairs and $L$ invariants, and work of the two authors that allows an extension of Joyce's descendent wall-crossing formula to non-standard hearts of $D^b(X)$. We use Ehrhart theory to deal with the combinatorics coming out of the wall-crossing formula. Furthermore, we specialize the wall-crossing formula to primary insertions and prove a strong rationality result predicted by the Pandharipande--Thomas/Gopakumar--Vafa correspondence.

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arXiv2025

Generalized K-theoretic invariants and wall-crossing via non-abelian localization

Given an abelian category and a stability condition satisfying appropriate conditions, we define generalized $K$-theoretic invariants and prove that they satisfy wall-crossing formulas. For this, we introduce a new associative algebra structure on the $K$-homology of the stack of objects of an abelian category, which we call the $K$-Hall algebra. We first define $\delta$-invariants directly coming from the stack of semistable objects and use the $K$-Hall algebra to take a formal logarithm and construct $\varepsilon$-invariants. We prove that these satisfy appropriate wall-crossing formulas using the non-abelian localization theorem. Based on work of Joyce in the cohomological setting, Liu had previously defined similar invariants assuming the existence of a framing functor; we show that when their definition of invariants makes sense it agrees with ours. Our results extend Joyce--Liu wall-crossing to non-standard hearts of $D^b(X)$, for which framing functors are not known to exist.

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arXiv2025

On the intersection theory of moduli spaces of parabolic bundles

This paper concerns the intersection numbers of tautological classes on moduli spaces of parabolic bundles on a smooth projective curve. We show that such intersection numbers are completely determined by wall-crossing formulas, Hecke isomorphisms, and flag bundle structures and resulting Weyl symmetry. As applications of these ideas, we prove the Newstead--Earl--Kirwan vanishing -- and a natural strengthening in terms of Chern filtrations -- and the Virasoro constraints for parabolic bundles. Both of these results were already known for moduli of stable bundles without parabolic structure, but even in those cases our proofs are new and independent of the existing ones. We use a Joyce style vertex algebra formulation of wall-crossing, and define intersection numbers even in the presence of strictly semistable parabolic bundles; all of our results hold in that setting as well.

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arXiv2024

On the Chern filtration for the moduli of bundles on curves

We introduce and study the Chern filtration on the cohomology of the moduli of bundles on curves. This can be viewed as a natural cohomological invariant defined via tautological classes that interpolates between additive Betti numbers and the multiplicative ring structure. In the rank two case, we fully compute the Chern filtration for moduli of stable bundles and all intermediate stacks in the Harder--Narasimhan stratification. We observe a curious symmetry of the Chern filtration on the moduli of rank two stable bundles, and construct $\mathfrak{sl}_2$-actions that categorify this symmetry. Our study of the Chern filtration is motivated by the $P=C$ phenomena in several related geometries.

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arXiv2024

Cohomology rings of the moduli of one-dimensional sheaves on the projective plane

We initiate a systematic study on the cohomology rings of the moduli stack $\mathfrak{M}_{d,χ}$ of semistable one-dimensional sheaves on the projective plane. We introduce a set of tautological relations of geometric origin, including Mumford-type relations, and prove that their ideal is generated by certain primitive relations via the Virasoro operators. Using BPS integrality and the computational efficiency of Virasoro operators, we show that our geometric relations completely determine the cohomology rings of the moduli stacks up to degree 5. As an application, we verify the refined Gopakumar--Vafa/Pandharipande--Thomas correspondence for local $\mathbb{P}^2$ in degree 5. Furthermore, we propose a substantially strengthened version of the $P=C$ conjecture, originally introduced by Shen and two of the authors. This can be viewed as an analogue of the $P=W$ conjecture in a compact and Fano setting.

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arXiv2024

Virasoro constraints and representations for quiver moduli spaces

We study the Virasoro constraints for moduli spaces of representations of quiver with relations by Joyce's vertex algebras. Using the framed Virasoro constraints, we construct a representation of half of the Virasoro algebra on the cohomology of moduli stacks of quiver representations under smoothness assumption. By exploiting the non-commutative nature of the Virasoro operators, we apply our theory for quivers to del Pezzo surfaces using exceptional collections. In particular, the Virasoro constraints and representations are proven for moduli of sheaves on $\mathbb{P}^2$, $\mathbb{P}^1\times \mathbb{P}^1$ and $\text{Bl}_{\mathsf{pt}}(\mathbb{P}^2)$. Lastly, we unravel the Virasoro constraints for Grassmannians in terms of symmetric polynomials and Hecke operators.

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arXiv2024

Cohomological $χ$-dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb{P}^2$

We prove that the cohomology rings of the moduli space $M_{d,χ}$ of one-dimensional sheaves on the projective plane are not isomorphic for general different choices of the Euler characteristics. This stands in contrast to the $χ$-independence of the Betti numbers of these moduli spaces. As a corollary, we deduce that $M_{d,χ}$ are topologically different unless they are related by obvious symmetries, strengthening a previous result of Woolf distinguishing them as algebraic varieties.

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arXiv2024

Virasoro constraints on moduli of sheaves and vertex algebras

In enumerative geometry, Virasoro constraints were first conjectured in Gromov-Witten theory with many new recent developments in the sheaf theoretic context. In this paper, we rephrase the sheaf-theoretic Virasoro constraints in terms of primary states coming from a natural conformal vector in Joyce's vertex algebra. This shows that Virasoro constraints are preserved under wall-crossing. As an application, we prove the conjectural Virasoro constraints for moduli spaces of torsion-free sheaves on any curve and on surfaces with only $(p,p)$ cohomology classes by reducing the statements to the rank 1 case.

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arXiv2023

Contact invariants of Q-Gorenstein toric contact manifolds, the Ehrhart polynomial and Chen-Ruan cohomology

Q-Gorenstein toric contact manifolds provide an interesting class of examples of contact manifolds with torsion first Chern class. They are completely determined by certain rational convex polytopes, called toric diagrams, and arise both as links of toric isolated singularities and as prequantizations of monotone toric symplectic orbifolds. In this paper we show how the cylindrical contact homology invariants of a Q-Gorenstein toric contact manifold are related to (i) the Ehrhart (quasi-)polynomial of its toric diagram; (ii) the Chen-Ruan cohomology of any crepant toric orbifold resolution of its corresponding toric isolated singularity; (iii) the Chen-Ruan cohomology of any monotone toric symplectic orbifold base that gives rise to it through prequantization.

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arXiv2022

Weyl symmetry for curve counting invariants via spherical twists

We study the curve counting invariants of Calabi--Yau 3-folds via the Weyl reflection along a ruled divisor. We obtain a new rationality result and functional equation for the generating functions of Pandharipande--Thomas invariants. When the divisor arises as resolution of a curve of $A_1$-singularities, our results match the rationality of the associated Calabi--Yau orbifold. The symmetry on generating functions descends from the action of an infinite dihedral group of derived auto-equivalences, which is generated by the derived dual and a spherical twist. Our techniques involve wall-crossing formulas and generalized DT invariants for surface-like objects.

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arXiv2022

On contact invariants of non-simply connected Gorenstein toric contact manifolds

The first two authors showed in~\cite{AM1} how the Conley-Zehnder index of any contractible periodic Reeb orbit of a non-degenerate toric contact form on a good toric contact manifold with zero first Chern class, i.e. a Gorenstein toric contact manifold, can be explicitly computed using moment map data. In this paper we show that the same explicit method can be used to compute Conley-Zehnder indices of non-contractible periodic Reeb orbits. Under appropriate conditions, the (finite) number of such orbits in a given free homotopy class and with a given index is a contact invariant of the underlying contact manifold. We compute these invariants for two sets of examples that satisfy these conditions: $5$-dimensional contact manifolds that arise as unit cosphere bundles of $3$-dimensional lens spaces, and $2n+1$-dimensional Gorenstein contact lens spaces. As applications, we will see that these invariants can be used to show that diffeomorphic lens spaces might not be contactomorphic and that there are homotopy classes of diffeomorphisms of some lens spaces that do not contain any contactomorphism. Following a suggestion by one referee, we will also see that this type of applications can be proved alternatively by looking at the total Chern class of these canonical contact structures on lens spaces.

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arXiv2021

Virasoro conjecture for the stable pairs descendent theory of simply connected 3-folds (with applications to the Hilbert scheme of points of a surface)

This paper concerns the recent Virasoro conjecture for the theory of stable pairs on a 3-fold proposed by Oblomkov, Okounkov, Pandharipande and the author in arXiv:2008.12514. Here we extend the conjecture to 3-folds with non-$(p,p)$-cohomology and we prove it in two specializations. For the first specialization, we let $S$ be a simply-connected surface and consider the moduli space $P_n(S\times \mathbb{P}^1, n[\mathbb{P}^1])$, which happens to be isomorphic to the Hilbert scheme $S^{[n]}$ of $n$ points on $S$. The Virasoro constraints for stable pairs, in this case, can be formulated entirely in terms of descendents in the Hilbert scheme of points. The two main ingredients of the proof are the toric case and the existence of universal formulas for integrals of descendents on $S^{[n]}$. The second specialization consists in taking the 3-fold $X$ to be a cubic and the curve class $β$ to be the line class. In this case we compute the full theory of stable pairs using the geometry of the Fano variety of lines.

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arXiv2017

IFOHAM-an iterative algorithm based on the first-order equation of HAM: exploratory preliminary results

In this work we present and study an iterative algorithm used to asymptotically solve nonlinear differential equations. This algorithm (Iterative First Order HAM or IFOHAM) is based on the first order equation of the Homotopy Analysis Method, HAM. We show that IFOHAM generalizes Picard-Lindeloff's iteration algorithm. Moreover, IFOHAM shares with HAM the possibility of ensuring convergence by adequately choosing c0, a convergence control parameter. Preliminary results show that IFOHAM exhibits a very good performance both in aspects related to the speed of convergence and in aspects related to the CPU calculation time. It should also be noted that the IFOHAM is a very low complexity algorithm easily programmable in a symbolic computing environment.

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