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David Baraglia

Publications and source records attributed to David Baraglia.

At least 37 records · Page 2Linked to original sources
arXiv2022

Knot concordance invariants from Seiberg-Witten theory and slice genus bounds in 4-manifolds

We construct a new family of knot concordance invariants $θ^{(q)}(K)$, where $q$ is a prime number. Our invariants are obtained from the equivariant Seiberg-Witten-Floer cohomology, constructed by the author and Hekmati, applied to the degree $q$ cyclic cover of $S^3$ branched over $K$. In the case $q=2$, our invariant $θ^{(2)}(K)$ shares many similarities with the knot Floer homology invariant $ν^+(K)$ defined by Hom and Wu. Our invariants $θ^{(q)}(K)$ give lower bounds on the genus of any smooth, properly embedded, homologically trivial surface bounding $K$ in a definite $4$-manifold with boundary $S^3$.

math.GT↗
arXiv2021

On the Bauer-Furuta and Seiberg-Witten invariants of families of $4$-manifolds

We show how the families Seiberg-Witten invariants of a family of smooth $4$-manifolds can be recovered from the families Bauer-Furuta invariant via a cohomological formula. We use this formula to deduce several properties of the families Seiberg-Witten invariants. We give a formula for the Steenrod squares of the families Seiberg-Witten invariants leading to a series of mod $2$ relations between these invariants and the Chern classes of the spin$^c$ index bundle of the family. As a result we discover a new aspect of the ordinary Seiberg-Witten invariants of a $4$-manifold $X$: they obstruct the existence of certain families of $4$-manifolds with fibres diffeomorphic to $X$. As a concrete geometric application, we shall detect a non-smoothable family of $K3$ surfaces. Our formalism also leads to a simple new proof of the families wall crossing formula. Lastly, we introduce $K$-theoretic Seiberg-Witten invariants and give a formula expressing the Chern character of the $K$-theoretic Seiberg-Witten invariants in terms of the cohomological Seiberg-Witten invariants. This leads to new divisibility properties of the families Seiberg-Witten invariants.

math.DG↗
arXiv2021

Tautological classes of definite 4-manifolds

We prove a diagonalisation theorem for the tautological, or generalised Miller-Morita-Mumford classes of compact, smooth, simply-connected definite $4$-manifolds. Our result can be thought of as a families version of Donaldson's diagonalisation theorem. We prove our result using a families version of the Bauer-Furuta cohomotopy refinement of Seiberg-Witten theory. We use our main result to deduce various results concerning the tautological classes of such $4$-manifolds. In particular, we completely determine the tautological rings of $\mathbb{CP}^2$ and $\mathbb{CP}^2 \# \mathbb{CP}^2$. We also derive a series of linear relations in the tautological ring which are universal in the sense that they hold for all compact, smooth, simply-connected definite $4$-manifolds.

math.DG↗
arXiv2020

Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants

We obtain constraints on the topology of families of smooth $4$-manifolds arising from a finite dimensional approximation of the families Seiberg-Witten monopole map. Amongst other results these constraints include a families generalisation of Donaldson's diagonalisation theorem and Furuta's $10/8$ theorem. As an application we construct examples of continuous $\mathbb{Z}_p$-actions for any odd prime $p$, which can not be realised smoothly. As a second application we show that the inclusion of the group of diffeomorphisms into the group of homeomorphisms is not a weak homotopy equivalence for any compact, smooth, simply-connected indefinite $4$-manifold with signature of absolute value greater than $8$.

math.DG↗
arXiv2020

The alpha invariant of complete intersections

We compute the alpha invariant of any smooth complex projective spin complete intersection of complex dimension $1 \; ({\rm mod} \; 4)$. We prove that the alpha invariant depends only on the total degree and Pontryagin classes. Our findings are consistent with a long-standing conjecture, often called the Sullivan Conjecture, which states that two complete intersections with the same dimensions, total degrees, Pontryagin and Euler classes are diffeomorphic.

math.DG↗
arXiv2019

A note on the Nielsen realization problem for K3 surfaces

We will show the following three theorems on the diffeomorphism and homeomorphism groups of a $K3$ surface. The first theorem is that the natural map $π_{0}(Diff(K3)) \to Aut(H^{2}(K3;\mathbb{Z}))$ has a section over its image. The second is that, there exists a subgroup $G$ of $π_{0}(Diff(K3))$ of order two over which there is no splitting of the map $Diff(K3) \to π_{0}(Diff(K3))$, but there is a splitting of $Homeo(K3) \to π_{0}(Homeo(K3))$ over the image of $G$ in $π_{0}(Homeo(K3))$, which is non-trivial. The third is that the map $π_{1}(Diff(K3)) \to π_{1}(Homeo(K3))$ is not surjective. Our proof of these results is based on Seiberg-Witten theory and the global Torelli theorem for $K3$ surfaces.

math.DG↗
arXiv2018

A gluing formula for families Seiberg-Witten invariants

We prove a gluing formula for the families Seiberg-Witten invariants of families of $4$-manifolds obtained by fibrewise connected sum. Our formula expresses the families Seiberg-Witten invariants of such a connected sum family in terms of the ordinary Seiberg-Witten invariants of one of the summands, under certain assumptions on the families. We construct some variants of the families Seiberg-Witten invariants and prove the gluing formula also for these variants. One variant incorporates a twist of the families moduli space using the charge conjugation symmetry of the Seiberg-Witten equations. The other variant is an equivariant Seiberg-Witten invariant of smooth group actions. We consider several applications of the gluing formula including: obstructions to smooth isotopy of diffeomorpihsms, computation of the mod $2$ Seiberg-Witten invariants of spin structures, relations between mod $2$ Seiberg-Witten invariants of $4$-manifolds and obstructions to the existence of invariant metrics of positive scalar curvature for smooth group actions on $4$-manifolds.

math.DG↗
arXiv2018

Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory

Let $X$ be a smooth, compact, oriented $4$-manifold. Building upon work of Li-Liu, Ruberman, Nakamura and Konno, we consider a families version of Seiberg-Witten theory and obtain obstructions to the existence of certain group actions on $X$ by diffeomorphisms. The obstructions show that certain group actions on $H^2(X , \mathbb{Z})$ preserving the intersection form can not be lifted an action of the same group on $X$ by diffeomorphisms. Using our obstructions, we construct numerous examples of group actions which can be realised continuously but can not be realised smoothly for any differentiable structure. For example, we construct compact simply-connected $4$-manifolds $X$ and involutions $f : H^2(X , \mathbb{Z}) \to H^2(X , \mathbb{Z})$ such that $f$ can be realised by a continuous involution on $X$ or by a diffeomorphism, but not by an involutive diffeomorphism for any smooth structure on $X$.

math.DG↗
arXiv2018

A foliated Hitchin-Kobayashi correspondence

We prove an analogue of the Hitchin-Kobayashi correspondence for compact, oriented, taut Riemannian foliated manifolds with transverse Hermitian structure. In particular, our Hitchin-Kobayashi theorem holds on any compact Sasakian manifold. We define the notion of stability for foliated Hermitian vector bundles with transverse holomorphic structure and prove that such bundles admit a basic Hermitian-Einstein connection if and only if they are polystable. Our proof is obtained by adapting the proof by Uhlenbeck and Yau to the foliated setting. We relate the transverse Hermitian-Einstein equations to higher dimensional instanton equations and in particular we look at the relation to higher contact instantons on Sasaki manifolds. For foliations of complex codimension 1, we obtain a transverse Narasimhan-Seshadri theorem. We also demonstrate that the weak Uhlenbeck compactness theorem fails in general for basic connections on a foliated bundle. This shows that not every result in gauge theory carries over to the foliated setting.

math.DG↗
arXiv2017

Cayley and Langlands type correspondences for orthogonal Higgs bundles

Through Cayley and Langlands type correspondences, we give a geometric description of the moduli spaces of real orthogonal and symplectic Higgs bundles of any signature in the regular fibres of the Hitchin fibration. As applications of our methods, we complete the concrete abelianization of real slices corresponding to all quasi-split real forms, and describe how extra components emerge naturally from the spectral data point of view.

math.DG↗
arXiv2017

Special Kähler geometry of the Hitchin system and topological recursion

We investigate the special Kähler geometry of the base of the Hitchin integrable system in terms of spectral curves and topological recursion. The Taylor expansion of the special Kähler metric about any point in the base may be computed by integrating the $g = 0$ Eynard-Orantin invariants of the corresponding spectral curve over cycles. In particular, we show that the Donagi-Markman cubic is computed by the invariant $W^{(0)}_3$. We use topological recursion to go one step beyond this and compute the symmetric quartic of second derivatives of the period matrix.

math.DG↗
arXiv2017

On the image of the parabolic Hitchin map

We determine the image of the (strongly) parabolic Hitchin map for all parabolics in classical groups and $G_2$. Surprisingly, we find that the image is isomorphic to an affine space in all cases, except for certain "bad parabolics" in type $D$, where the image can be singular.

math.AG↗
arXiv2017

Complete integrability of the parahoric Hitchin system

Let $\mathcal{G}$ be a parahoric group scheme over a complex projective curve $X$ of genus greater than one. Let $\mathrm{Bun}_{\mathcal{G}}$ denote the moduli stack of $\mathcal{G}$-torsors on $X$. We prove several results concerning the Hitchin map on $T^*\!\mathrm{Bun}_{\mathcal{G}}$. We first show that the parahoric analogue of the global nilpotent cone is isotropic and use this to prove that $\mathrm{Bun}_{\mathcal{G}}$ is "very good" in the sense of Beilinson-Drinfeld. We then prove that the parahoric Hitchin map is a Poisson map whose generic fibres are abelian varieties. Together, these results imply that the parahoric Hitchin map is a completely integrable system.

math.AG↗
arXiv2016

Monodromy of the $SL(n)$ and $GL(n)$ Hitchin fibrations

We compute the monodromy of the Hitchin fibration for the moduli space of $L$-twisted $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$-Higgs bundles for any $n$, on a compact Riemann surface of genus $g>1$. We require the line bundle $L$ to either be the canonical bundle or satisfy $deg(L) > 2g-2$. The monodromy group is generated by Picard-Lefschetz transformations associated to vanishing cycles of singular spectral curves. We construct such vanishing cycles explicitly and use this to show that the $SL(n,\mathbb{C})$ monodromy group is a {\em skew-symmetric vanishing lattice} in the sense of Janssen. Using the classification of vanishing lattices over $\mathbb{Z}$, we completely determine the structure of the monodromy groups of the $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$ Hitchin fibrations. As an application we determine the image of the restriction map from the cohomology of the moduli space of Higgs bundles to the cohomology of a non-singular fibre of the Hitchin fibration.

math.AG↗
arXiv2016

Brauer group of moduli of Higgs bundles and connections

Given a compact Riemann surface $X$ and a semisimple affine algebraic group $G$ defined over $\mathbb C$, there are moduli spaces of Higgs bundles and of connections associated to $(X,\, G)$. We compute the Brauer group of the smooth locus of these varieties.

math.AG↗
arXiv2016

Arithmetic of singular character varieties and their $E$-polynomials

We calculate the $E$-polynomials of the $SL_3(\mathbb{C})$ and $GL_3(\mathbb{C})$-character varieties of compact oriented surfaces of any genus and the $E$-polynomials of the $SL_2(\mathbb{C})$ and $GL_2(\mathbb{C})$-character varieties of compact non-orientable surfaces of any Euler characteristic. Our methods also give a new and significantly simpler computation of the $E$-polynomials of the $SL_2(\mathbb{C})$-character varieties of compact orientable surfaces, which were computed by Logares, Muñoz and Newstead for genus $g=1,2$ and by Martinez and Muñoz for $g \ge 3$. Our technique is based on the arithmetic of character varieties over finite fields. More specifically, we show how to extend the approach of Hausel and Rodriguez-Villegas used for non-singular (twisted) character varieties to the singular (untwisted) case.

math.AG↗
arXiv2015

Monodromy of rank 2 twisted Hitchin systems and real character varieties

We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of $L$-twisted $G$-Higgs bundles, for the groups $G = GL(2,\mathbb{C})$, $SL(2,\mathbb{C})$ and $PSL(2,\mathbb{C})$. We also determine the twisted Chern class of the regular locus, which obstructs the existence of a section of the moduli space of $L$-twisted Higgs bundles of rank $2$ and degree $deg(L)+1$. By counting orbits of the monodromy action with $\mathbb{Z}_2$-coefficients, we obtain in a unified manner the number of components of the character varieties for the real groups $G = GL(2,\mathbb{R})$, $SL(2,\mathbb{R})$, $PGL(2,\mathbb{R})$, $PSL(2,\mathbb{R})$, as well as the number of components of the $Sp(4,\mathbb{R})$-character variety with maximal Toledo invariant. We also use our results for $GL(2,\mathbb{R})$ to compute the monodromy of the $SO(2,2)$ Hitchin map and determine the components of the $SO(2,2)$ character variety.

math.DG↗