arXiv subjects

Thomas G. Leness

Publications and source records attributed to Thomas G. Leness.

At least 19 recordsLinked to original sources

arXiv2026

Virtual Morse-Bott index, moduli spaces of pairs, and applications to topology of smooth four-manifolds

We previously developed an approach to Bialynicki-Birula theory for holomorphic $\mathbb{C}^*$ actions on complex analytic spaces and the concept of virtual Morse-Bott indices for singular critical points of Hamiltonian functions for the induced circle actions (see Feehan, arXiv:2206.14710). For Hamiltonian functions of circle actions on closed, complex Kaehler manifolds, the virtual Morse-Bott index coincides with the classical Morse-Bott index due to Bott (1954) and Frankel (1959). A key principle in our approach is that positivity of the virtual Morse-Bott index at a critical point of the Hamiltonian function implies that the critical point cannot be a local minimum even when that critical point is a singular point in the moduli space. In this monograph, we consider our method in the context of the moduli space of non-Abelian monopoles over a closed, complex, Kaehler surface. We use the Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces) and we prove that these indices are positive in a setting motivated by the conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type obey the Bogomolov-Miyaoka-Yau inequality.

math.DG↗

arXiv2025

Almost Hermitian structures on moduli spaces of non-Abelian monopoles and applications to the topology of symplectic four-manifolds

This work is a sequel to our previous monograph arXiv:2010.15789 (to appear in AMS Memoirs), where we initiated our program to prove that the Bogomolov-Miyaoka-Yau inequality holds for closed, symplectic four-manifolds and, more generally, for closed, smooth four-manifolds with a Seiberg-Witten basic class. This inequality was first proved for compact, complex surfaces of general type independently by Miyaoka and Yau in 1977. Our approach uses a version of Morse theory for a natural Hamiltonian, the square of the $L^2$ norm of the coupled spinors, for the circle action on the moduli space of non-Abelian monopoles over a closed four-manifold. It has the aim of proving the existence of a projectively anti-self-dual connection on a rank-two Hermitian vector bundle over a blow-up of the four-manifold, where the first Pontrjagin number of the vector bundle is negative and greater than or equal to minus the Euler characteristic of the blown-up four-manifold. Our Morse theory argument relies on positivity of virtual Morse-Bott indices for critical points of Hamiltonians for circle actions on complex analytic spaces (or real analytic spaces that, locally, are sufficiently well-approximated by complex analytic model spaces), as developed by the first author in arXiv:2206.14710. In our application to the moduli space of non-Abelian monopoles, the critical points are fixed points of the circle action and thus represented by Seiberg-Witten monopoles.

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arXiv2019

Gluing in geometric analysis via maps of Banach manifolds with corners and applications to gauge theory

We describe a new approach to the problem of constructing gluing parameterizations for open neighborhoods of boundary points of moduli spaces of anti-self-dual connections over closed four-dimensional manifolds. Our approach employs general results from differential topology for $C^1$ maps of smooth Banach manifolds with corners, providing a method that should apply to other problems in geometric analysis involving the gluing construction of solutions to nonlinear partial differential equations.

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arXiv2019

The SO(3) monopole cobordism and superconformal simple type

We show that the SO(3) monopole cobordism formula from Feehan and Leness (2002) implies that all smooth, closed, oriented four-manifolds with $b^1=0$ and $b^+\geq 3$ and odd with Seiberg-Witten simple type satisfy the superconformal simple type condition defined by Marino, Moore, and Peradze (1999) This implies the lower bound, conjectured by Fintushel and Stern (2001) on the number of Seiberg-Witten basic classes in terms of topological data.

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arXiv2019

Superconformal simple type and Witten's conjecture

Let $X$ be a smooth, closed, connected, orientable four-manifold with $b^1(X)=0$ and $b^+(X)\geq 3$ and odd. We show that if $X$ has Seiberg-Witten simple type, then the SO(3)-monopole cobordism formula of Feehan and Leness (2002) implies Witten's Conjecture relating the Donaldson and Seiberg-Witten invariants.

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arXiv2018

An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants

We prove an analogue of the Kotschick-Morgan conjecture in the context of SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the SO(3)-monopole cobordism. The main technical difficulty in the SO(3)-monopole program relating the Seiberg-Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible SO(3) monopoles, namely the moduli spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the moduli space of SO(3) monopoles [arXiv:dg-ga/9710032]. In this monograph, we prove --- modulo a gluing theorem which is an extension of our earlier work in [arXiv:math/9907107] --- that these intersection pairings can be expressed in terms of topological data and Seiberg-Witten invariants of the four-manifold. This conclusion is analogous to the Kotschick-Morgan conjecture concerning the wall-crossing formula for Donaldson invariants of a four-manifold with $b_2^+=1$; that wall-crossing formula and the resulting structure of Donaldson invariants for four-manifolds with $b_2^+=1$ were established, assuming the Kotschick-Morgan conjecture, by Goettsche [arXiv:alg-geom/9506018] and Goettsche and Zagier [arXiv:alg-geom/9612020]. In this monograph, we reduce the proof of the Kotschick-Morgan conjecture to an extension of previously established gluing theorems for anti-self-dual SO(3) connections (see [arXiv:math/9812060] and references therein). Since the first version of our monograph was circulated, applications of our results have appeared in the proof of Property P for knots by Kronheimer and Mrowka [arXiv:math/0311489] and work of Sivek on Donaldson invariants for symplectic four-manifolds [arXiv:1301.0377].

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arXiv2014

Witten's conjecture for many four-manifolds of simple type

We prove that Witten's Conjecture [arXiv:hep-th/9411102] on the relationship between the Donaldson and Seiberg-Witten series for a four-manifold of Seiberg-Witten simple type with $b_1=0$ and odd $b_2^+\geq 3$ follows from our $\mathrm{SO}(3)$-monopole cobordism formula [arXiv:math/0203047] when the four-manifold has $c_1^2\geq χ_h-3$ or is abundant.

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arXiv2012

SO(3)-monopoles: The overlap problem

The SO(3)-monopole program, initiated by Pidstrigatch and Tyurin [arXiv:dg-ga/9507004], yields a relationship between the Donaldson and Seiberg-Witten invariants through a cobordism between the moduli spaces defining these invariants. The main technical difficulty in this program lies in describing the links of singularities in this cobordism arising from the Seiberg-Witten moduli subspaces. In related articles, we defined maps which, essentially, define normal bundles of strata of these singularities. The link in question is then the boundary of the union of the tubular neighborhoods associated with these normal bundles. However, the SO(3)-monopole program requires the computation of intersection numbers with links where more than one stratum appears in the family of singularities and thus more than one tubular neighborhood appears in the definition of the link. Computations of intersection numbers in unions of open sets have proved difficult for even two open sets, as early work of Leness [arXiv:dg-ga/9603016] and Ozsvath [1994] demonstrated. In this note, we give a brief introduction to our monograph [arXiv:math/0203047], in which we implement these computations.

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arXiv2008

Degeneracy loci of families of Dirac operators

Generalizing some results from R. Leung's thesis, we compute, in rational cohomology, the Poincare dual of the degeneracy locus of the family of Dirac operators parameterized by the moduli space of projectively anti-self-dual $\SO(3)$ connections. This is the first step in a program to derive a relation between the Donaldson and spin invariants.

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arXiv2002

On Donaldson and Seiberg-Witten invariants

This article is based on a lecture by the first author at the International Georgia Topology Conference 2001 (Athens, Georgia) and the Mathematische Arbeitstagung 2001 (Bonn, Germany). We sketch a proof of Witten's formula relating the Donaldson and Seiberg-Witten series modulo powers of degree c+2, with c = -{1/4}(7 chi + 11 sigma), for four-manifolds obeying some mild conditions, where chi and sigma are their Euler characteristic and signature. We use the moduli space of SO(3) monopoles as a cobordism between a link of the Donaldson moduli space of anti-self-dual SO(3) connections and links of the moduli spaces of Seiberg-Witten monopoles. Gluing techniques allow us to compute contributions from Seiberg-Witten moduli spaces lying in the first (or `one-bubble') level of the Uhlenbeck compactification of the moduli space of SO(3) monopoles.

math.DG↗

arXiv2001

SO(3) Monopoles, Level-One Seiberg-Witten Moduli Spaces, and Witten's Conjecture in Low Degrees

We prove Witten's formula relating the Donaldson and Seiberg-Witten series modulo powers of degree $c+2$, with $c=-{1/4}(7χ+11σ)$, for four-manifolds obeying some mild conditions, where $χ$ and $σ$ are their Euler characteristic and signature. We use the moduli space of SO(3) monopoles as a cobordism between a link of the Donaldson moduli space of anti-self-dual SO(3) connections and links of the moduli spaces of Seiberg-Witten monopoles. Gluing techniques allow us to compute contributions from Seiberg-Witten moduli spaces lying in the first (or `one-bubble') level of the Uhlenbeck compactification of the moduli space of SO(3) monopoles.

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arXiv2001

PU(2) monopoles. II: Top-level Seiberg-Witten moduli spaces and Witten's conjecture in low degrees

In this article we complete the proof---for a broad class of four-manifolds---of Witten's conjecture that the Donaldson and Seiberg-Witten series coincide, at least through terms of degree less than or equal to c-2, where c is a linear combination of the Euler characteristic and signature of the four-manifold. This article is a revision of sections 4--7 of an earlier version, while a revision of sections 1--3 of that earlier version now appear in a separate companion article (math.DG/0007190). Here, we use our computations of Chern classes for the virtual normal bundles for the Seiberg-Witten strata from the companion article (math.DG/0007190), a comparison of all the orientations, and the PU(2) monopole cobordism to compute pairings with the links of level-zero Seiberg-Witten moduli subspaces of the moduli space of PU(2) monopoles. These calculations then allow us to compute low-degree Donaldson invariants in terms of Seiberg-Witten invariants and provide a partial verification of Witten's conjecture.

dg-ga↗

arXiv2001

PU(2) monopoles and links of top-level Seiberg-Witten moduli spaces

This is the first of two articles in which we give a proof - for a broad class of four-manifolds - of Witten's conjecture that the Donaldson and Seiberg-Witten series coincide, at least through terms of degree less than or equal to c-2, where c is a linear combination of the Euler characteristic and signature of the four-manifold. This article is a revision of sections 1-3 of an earlier version of the article dg-ga/9712005, now split into two parts, while a revision of sections 4-7 of that earlier version appears in a recently updated dg-ga/9712005. In the present article, we construct virtual normal bundles for the Seiberg-Witten strata of the moduli space of PU(2) monopoles and compute their Chern classes.

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arXiv1999

PU(2) monopoles. III: Existence of gluing and obstruction maps

This is the third installment in our series of articles (dg-ga/9712005, dg-ga/9710032) on the application of the PU(2) monopole equations to prove Witten's conjecture (hep-th/9411102) concerning the relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. The moduli space of solutions to the PU(2) monopole equations provides a noncompact cobordism between links of compact moduli spaces of U(1) monopoles of Seiberg-Witten type and the moduli space of anti-self-dual SO(3) connections, which appear as singularities in this larger moduli space. In this paper we prove the first part of a general gluing theorem for PU(2) monopoles, with the proof of the second half to appear in a companion article. The ultimate purpose of the gluing theorem is to provide topological models for neighborhoods of ideal Seiberg-Witten moduli spaces appearing in lower levels of the Uhlenbeck compactification of the moduli space of PU(2) monopoles and thus permit calculations of their contributions to Donaldson invariants using the PU(2)-monopole cobordism.

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arXiv1999

PU(2) monopoles and a conjecture of Marino, Moore, and Peradze

In this article we show that some of the recent results of Marino, Moore, and Peradze (math.DG/9812042, hep-th/9812055) -- in particular their conjecture that all closed, smooth four-manifolds with b_2^+ > 1 (and Seiberg-Witten simple type) are of `superconformal simple type' -- can be understood using a simple mathematical argument via the PU(2)-monopole cobordism of Pidstrigach and Tyurin (dg-ga/9507004) and results of the first and third authors (dg-ga/9712005, dg-ga/9709022).

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arXiv1999

Donaldson invariants and wall-crossing formulas. I: Continuity of gluing maps

The present article is the first in a series whose ultimate goal is to prove the Kotschick-Morgan conjecture concerning the wall-crossing formula for the Donaldson invariants of a four-manifold with b^+ = 1. The conjecture asserts that the wall-crossing terms due to changes in the metric depend at most on the homotopy type of the four-manifold and the degree of the invariant. Our principal interest in this conjecture is due to the fact that its proof is expected to resemble that of an important intermediate step towards a proof of Witten's conjecture concerning the relation between Donaldson and Seiberg-Witten invariants (hep-th/9411102, hep-th/9709193), using PU(2) monopoles as described in (dg-ga/9709022, dg-ga/9712005). Moreover, it affords us another venue in which to address some of the technical difficulties arising in our work on Witten's conjecture. The additional difficulties in the case of PU(2) monopoles are due to the more complicated gluing theory, the presence of obstructions to deformation and gluing, and the need to consider links of positive-dimensional families of `reducibles' even in the presence of `simple type' assumptions. Witten's conjecture should then follow from a final step analogous to Goettsche's computation of the wall-crossing terms, assuming that the Kotschick-Morgan conjecture holds (alg-geom/9506018).

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arXiv1997

PU(2) Monopoles, I: Regularity, Uhlenbeck Compactness, and Transversality

We prove the existence of perturbations for the PU(2) monopole equations, yielding transversality on the complement of the anti-self-dual or reducible solutions, and the existence of an Uhlenbeck compactification for the moduli space of solutions to these perturbed PU(2) monopole equations. In December 1994, V. Pidstrigach and A. Tyurin and then others proposed a method to prove Witten's conjecture concerning the relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. Their proposal uses a moduli space of solutions to the PU(2) monopole equations, which are a natural generalization of the U(1) monopole equations of Seiberg and Witten and the equation for anti-self-dual SO(3) connections, to construct a cobordism between links of compact moduli spaces of U(1) monopoles of Seiberg-Witten type and the moduli space of anti-self-dual connections, which appear as singularities in this larger moduli space. A basic requirement of this cobordism technique is the existence of an Uhlenbeck compactification for the moduli space of PU(2) monopoles and of generic-parameter transversality results for all the moduli spaces of PU(2) monopoles which appear in this compactification, on the complement of the anti-self-dual and U(1) solutions.

dg-ga↗

arXiv1997

Uhlenbeck compactness and transversality for the moduli space of PU(2) monopoles

This research announcement gives a brief report of the main results in our paper "PU(2) monopoles, I: Regularity, Uhlenbeck compactness, and transversality" (Journal of Differential Geometry, to appear). We describe the existence of perturbations for the PU(2) monopole equations, yielding both useful transversality properties and an Uhlenbeck compactification for this perturbed moduli space.

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