arXiv subjects

Andrei Teleman

Publications and source records attributed to Andrei Teleman.

At least 19 recordsLinked to original sources

arXiv2026

An extension theorem for bundles with respect to strictly pseudoconvex extensions

Let $\mathcal{X}$ be a complex manifold and $X$ a compact complex manifold with boundary in $\mathcal{X}$. For a complex Lie group $G$ and a regularity class $$\mathfrak{r}\in \big\{\mathcal{C}^k|\ k\in\mathbb{N}\cup\{\infty\}\big\}\cup\big\{\Lambda^r_{\rm loc}|\ r\in (0,\infty)\big\} $$ we define the sheaf of groups $\mathcal{O}^{\mathfrak{r}\,G}_X$ on $X$ by \begin{align*} \mathcal{O}^{\mathfrak{r}\,G}_X(V):=\{u\in \mathcal{C}(V,G)|\ &u \hbox{ has regularity class $\mathfrak{r}$ on $V$}, \\ &u|_{V\cap\mathrm{int}(X)} \hbox{ is holomorphic}\}. \end{align*} Let $X_0\Subset Z_0\Subset \mathcal{X}$ be a strictly pseudoconvex extension in $\mathcal{X}$, and let $X:= \bar X_0$, $Z:= \bar Z_0$ be the corresponding compact manifolds with boundary in $\mathcal{X}$. Let $K\subset X_0$ be a compact set and $\mathcal{P}$ a holomorphic principal $G$-bundle of class $\mathfrak{r}$ (i.e. an $\mathcal{O}^{\mathfrak{r}\,G}_X$-torsor) on $X\setminus K$. Assuming that (H1) the given strictly pseudoconvex extension $X_0\Subset Z_0$ is non-critical, or that (H2) the underlying topological bundle of $\mathcal{P}$ extends to $Z\setminus K$, we prove that $\mathcal{P}$ admits an extension to $Z\setminus K$. This gives a new proof and a new generalisation of the extension problem stated in the article S. Donaldson, Boundary value problems for Yang-Mills fields, Journal of Geometry and Physics 8, (1992) and studied with different methods in a previous article of the author.

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arXiv2026

A gauge theoretical generalization of Bryant's correspondence

A classical theorem in the theory of minimal surfaces establishes a correspondence between minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$. A hyperbolic version of this correspondence is due to Bryant: null holomorphic curves in ${\rm SL}(2,\mathbb{C})$ correspond to CMC-1 surfaces in the hyperbolic space $\mathbb{H}^3$. We also have a relativistic Bryant type correspondence: CMC-1 immersions in the hyperbolic space are replaced by space-like CMC-1 immersion in the de Sitter space. We prove a mutual generalisation of all these results: let $H$ be a real Lie group, $\pi:P \to M$ a principal $H$-bundle, $A$ a connection on $P$ and $\alpha\in A^1_{\rm Ad}(P,\mathfrak{h})$ a tensorial 1-form of type ${\rm Ad}$ which induces isomorphisms $A_\xi \to \mathfrak{h}$. Such a pair $(\alpha,A)$ defines an almost complex structure $J^\alpha_A$ on $P$, which is integrable if and only $(\alpha,A)$ solves a gauge-invariant first order differential system. A non-degenerate symmetric ${\rm Ad}_H$-invariant bilinear form $g$ on $\mathfrak{h}$ defines pseudo-Riemannian metrics $g^\alpha_M$, $\mathfrak{g}^\alpha_A$ on $M$, respectively $P$, and a non-degenerate bilinear form $\omega^{\alpha,g}_A:T_P\times_P T_P\to \mathbb{C}$ which is holomorphic when $J^\alpha_A$ is integrable. Assuming that this is the case, we have a Bryant type correspondence between space-like, $\omega^{\alpha,g}_A$-isotropic holomorphic immersions $Y\to P$ and space-like conformal immersions $Y\to (M,g^\alpha_M)$ whose mean curvature vector field is given by a simple explicit formula. In particular, one obtains such a correspondence for any principal bundle of the form $G\to G/H$, where $G$ is a complex Lie group, and $H$ is a real form of $G$ endowed with a non-degenerate, ${\rm Ad}_H$-invariant, symmetric bilinear form $g$ on its Lie-algebra $\mathfrak{h}$.

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arXiv2025

Reductive homogeneous spaces associated with real forms. A gauge-theoretical generalisation

Let $G$ be a connected complex Lie group. A real form of $G$ is a closed subgroup $H\subset G$ whose Lie algebra $\mathfrak{h}$ is a real form of the Lie algebra $\mathfrak{g}$ of $G$. A pair $(G,H)$ of this type is reductive, and the corresponding quotient $G/H$ is a reductive homogeneous space whose canonical connection is torsion free. Regarded as a principal $H$-bundle over $G/H$, $G$ comes with tensorial 1-form $\alpha$ of type $\mathrm{Ad}$ and a natural left invariant connection $A$. This remark suggests the following gauge theoretical generalisation of the class of reductive pairs of the form $(G,H)$ as above: Let $H$ be an arbitrary Lie group. A triple $(P\stackrel{\pi}{\to}M,\alpha,A)$, where $P\stackrel{\pi}{\to}M$ is a principal $H$-bundle, $\alpha$ a tensorial 1-form of type $\mathrm{Ad}$ on $P$ and $A$ a connection on $P$ will be called admissible if the induced linear maps $A_y\to \mathfrak{h}$, $y\in P$, are all isomorphisms. If this is the case one obtains a canonical linear connection $\nabla^\alpha_A$ on $M$ and a canonical almost complex structure $J^\alpha_A$ on $P$ which, by a result of R. Zentner, is integrable if an only if the pair $(\alpha,A)$ satisfies a gauge invariant first order differential system. A triple $(P\stackrel{\pi}{\to}M,\alpha,A)$ as above will be called integrable if this integrability condition is satisfied. Any integrable triple $(P\stackrel{\pi}{\to}M,\alpha,A)$ with $M$ simply connected and $\nabla^\alpha_A$ complete can be identified with the triple associated with a real form of a complex Lie group. In this article we explain the strategy of the proof of this classification result and we prove in detail a theorem which plays an important role in this strategy and is of independent interest. In the last section we introduce the moduli spaces of integrable pairs on a principal bundle, and we give explicit examples.

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arXiv2024

On the classification of Inoue surfaces

We prove that any Inoue surface admits a unique holomorphic connection. Using this result we show that two Inoue surfaces $S=H\times\mathbb{C}/G$, $S'=H\times\mathbb{C}/G'$ are biholomorphic if and only if $G$, $G'$ are conjugate in the group of affine transformations of $H\times\mathbb{C}$. This result allows us to prove explicit classification theorems for Inoue surfaces: Let $\mathcal{M}$ be the set of ${\rm SL}(3,\mathbb{Z})$-matrices $M$ with a real eigenvalue $\alpha>1$ and two non-real eigenvalues, and $\mathcal{N}^\pm $ the set of ${\rm GL}(2,\mathbb{Z})$-matrices $N$ with a real eigenvalue $\alpha>1$ and $\det(N)=\pm 1$. We prove that: For any ${\rm GL}(3,\mathbb{Z})$-similarity class $\mathfrak{M}\in \mathcal{M}/\sim$, there exists exactly two biholomorphism classes of type I Inoue surfaces. For any ${\rm GL}(2,\mathbb{Z})$ similarity class $\mathfrak{N}=[N]\in \mathcal{N}^+/\sim$ and positive integer $r\in\mathbb{N}^*$, we have a finite set of deformation classes of type II Inoue surfaces. This set is parameterised by the quotient of $\mathbb{Z}^2/(I_2-N)\mathbb{Z}^2+r\mathbb{Z}^2$ by an action of the "positive centraliser" $Z^+_{{\rm GL}(2,\mathbb{Z})}(N)$ of $N$ in ${\rm GL}(2,\mathbb{Z})$. The set of biholomorphism types corresponding to a deformation class, endowed with its natural topology, can be identified with either $\mathbb{C}^*$ or $\mathbb{C}$. For any ${\rm GL}(2,\mathbb{Z})$-similarity class $\mathfrak{N}=[N]\in \mathcal{N}^-/\sim$ and positive integer $r\in\mathbb{N}^*$, we have a finite set of biholomorphism classes of type III Inoue surfaces. This set is parameterised by the quotient of $\mathbb{Z}^2/(I_2+N)\mathbb{Z}^2+r\mathbb{Z}^2$ by an action of $Z^+_{{\rm GL}(2,\mathbb{Z})}(N)$. In both cases the group $Z^+_{{\rm GL}(2,\mathbb{Z})}(N)$ is infinite cyclic (see section 5).

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arXiv2023

Holomorphic bundles framed along a real hypersurface and the Riemann-Hilbert problem

Let $X$ be a connected, compact complex manifold and $S\subset X$ a separating real hypersurface, so that $X$ decomposes as a union of compact complex manifolds with boundary $\bar X^\pm$. Let $\mathcal{M}$ be the moduli space of $S$-framed holomorphic bundles, i.e. of pairs $(E,\theta)$ of fixed topological type consisting of a holomorphic bundle $E$ on $X$ and a trivialization $\theta$ - belonging to a fixed H\"older regularity class $\mathcal{C}^{\kappa+1}$ - of its restriction to $S$. The restrictions to $\bar X^\pm$ of an $S$-framed holomorphic bundle $(E,\theta)$ are boundary framed formally holomorphic bundles $(E^\pm,\theta^\pm)$ which induce, via $\theta^\pm$, the same tangential Cauchy-Riemann operators on the trivial bundle on $S$, so one obtains a natural map from $\mathcal{M}$ into the fiber product $\mathcal{M}^-\times_\mathcal{C}\mathcal{M}^+$ over the space $\mathcal{C}$ of Cauchy-Riemann operators on the trivial bundle on $S$. Our main result states: this map is a homeomorphism for $\kappa\in (0,\infty]\setminus\mathbb{N}$. The proof is based on a gluing principle for formally holomorphic bundles along a real hypersurface. This principle can also be used to give a complex geometric interpretation of the space of solutions of a large class of Riemann-Hilbert type problems. The results generalize in two directions: first one can replace the decomposition associated with a separating hypersurface by the the manifold with boundary $\widehat X_S$ obtained by cutting $X$ along any oriented hypersurface $S$. Second one can consider principal $G$ bundles for an arbitrary complex Lie group $G$. We give explicit examples of moduli spaces of (boundary) framed holomorphic bundles and explicit formulae for the homeomorphisms provided by the general results.

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arXiv2023

The Newlander-Nirenberg Theorem for principal bundles

Let $G$ be an arbitrary (not necessarily isomorphic to a closed subgroup of $\mathrm{GL}(r,\mathbb{C})$) complex Lie group, $U$ a complex manifold and $p:P\to U$ a $\mathcal{C}^\infty$ principal $G$-bundle on $U$. We introduce and study the space $\mathcal{J}^\kappa_P$ of bundle almost complex structures of H{\"o}lder class $\mathcal{C}^\kappa$ on $P$. To any $J\in \mathcal{J}^\kappa_P$ we associate an $\mathrm{Ad}(P)$-valued form $\mathfrak{f}_J$ of type (0,2) on $U$ which should be interpreted as the obstruction to the integrability of $J$. For $\kappa\geq 1$ we have $\mathfrak{f}_J\in\mathcal{C}^{\kappa-1}(U,\bigwedge\hspace{-3.5pt}^{0,2}_{\,\,U}\otimes\mathrm{Ad}(P))$ whereas, for $\kappa\in[0,1)$, $\mathfrak{f}_J$ is a form with distribution coefficients. Let $J\in \mathcal{J}^\kappa_P$ with $\kappa\in (0,+\infty]\setminus\mathbb{N}$. We prove that $J$ admits locally $J$-pseudo-holomorphic sections of class $\mathcal{C}^{\kappa+1}$ if and only if $\mathfrak{f}_J=0$. If this is the case, $J$ defines a holomorphic reduction of the underlying $\mathcal{C}^{\kappa+1}$-bundle of $P$ in the sense of the theory of principal bundles on complex manifolds. The proof is based on classical regularity results for the $\bar\partial$-Neumann operator on compact, strictly pseudo-convex complex manifolds with boundary.The result will be used in forthcoming articles dedicated to moduli spaces of holomorphic bundles (on a compact complex manifold $X$) framed along a real hypersurface $S\subset X$.

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arXiv2022

Holomorphic bundles on complex manifolds with boundary

Let $\Omega$ be a complex manifold, and let $X\subset \Omega$ be an open submanifold whose closure $\bar X$ is a (not necessarily compact) submanifold with smooth boundary. Let $G$ be a complex Lie group, $\Pi$ be a differentiable principal $G$-bundle on $\Omega$ and $J$ a formally integrable bundle almost complex structure on the restriction $\bar P:= \Pi|_{\bar X}$. We prove that, if the boundary of $\bar X$ is strictly pseudoconvex, $J$ extends to a holomorphic structure on the restriction of $\Pi$ to a neighborhood of $\bar X$ in $\Omega$. This answers positively and generalizes a problem stated in the article "Boundary value problems for Yang-Mills fields" by S. Donaldson. We obtain a gauge theoretical interpretation of the quotient $\mathcal{C}^\infty(\partial \bar X,G)/\mathcal{O}^\infty(\bar X,G)$ associated with any compact Stein manifold with boundary $\bar X$ endowed with a Hermitian metric. For a fixed differentiable $G$-bundle $\bar P$ on a complex manifold $\bar X$ with non-pseudoconvex boundary, we study the set of formally integrable almost complex structures on $\bar P$ which admit formally holomorphic local trivializations at boundary points. We give an example where a "generic" formally integrable almost complex on $\bar P$ admits formally holomorphic local trivializations at no boundary point, whereas the set of formally integrable almost complex structures which admit formally holomorphic local trivializations at all boundary points is dense.

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arXiv2020

Infinitesimal homogeneity and bundles

Let $Q\to M$ be a principal $G$-bundle, and $B_0$ a connection on $Q$. We introduce an infinitesimal homogeneity condition for sections in an associated vector bundle $Q\times_GV$ with respect to $B_0$, and, inspired by the well known Ambrose-Singer theorem, we prove the existence of a connection which satisfies a system of parallelism conditions. We explain how this general theorem can be used to prove the known Ambrose-Singer type theorems by an appropriate choice of the initial system of data.We also obtain new applications, which cannot be obtained using the known formalisms, e.g. a classification theorem for locally homogeneous spinors. Finally we introduce natural local homogeneity and local symmetry conditions for triples $(g,P\stackrel{p}{\to} M,A)$ consisting of a Riemannian metric on $M$, a principal bundle on $M$, and a connection on $P$. Our main results concern locally homogeneous and locally symmetric triples, and they can be viewed as bundle versions of the Ambrose-Singer and Cartan theorem.

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arXiv2019

Graded tilting for gauged Landau-Ginzburg models and geometric applications

In this paper we develop a graded tilting theory for gauged Landau-Ginzburg models of regular sections in vector bundles over projective varieties. Our main theoretical result describes - under certain conditions - the bounded derived category of the zero locus $Z(s)$ of such a section $s$ as a graded singularity category of a non-commutative quotient algebra $\Lambda/\langle s\rangle$: $D^b(\mathrm{coh} Z(s))\simeq D^{\mathrm{gr}}_{\mathrm{sg}}(\Lambda/\langle s\rangle)$. Our geometric applications all come from homogeneous gauged linear sigma models. In this case $\Lambda$ is a non-commutative resolution of the invariant ring which defines the $\mathbb{C}^*$-equivariant affine GIT quotient of the model. We obtain purely algebraic descriptions of the derived categories of the following families of varieties: - Complete intersections. - Isotropic symplectic and orthogonal Grassmannians. - Beauville-Donagi IHS 4-folds.

math.AG↗

arXiv2018

Locally homogeneous connections on principal bundles over hyperbolic Riemann surfaces

Let $g$ be locally homogeneous (LH) Riemannian metric on a differentiable compact manifold $M$, and $K$ be a compact Lie group endowed with an $\mathrm {ad}$-invariant inner product on its Lie algebra $\mathfrak{k}$. A connection $A$ on a principal $K$-bundle $p:P\to M$ on $M$ is locally homogeneous if for any two points $x_1$, $x_2\in M$ there exists an isometry $\varphi:U_1\to U_2$ between open neighborhoods $U_i\ni x_i$ which sends $x_1$ to $x_2$ and admits a $\varphi$-covering bundle isomorphism preserving the connection $A$. This condition is invariant under the action of the automorphism group (gauge group) of the bundle, so the classification problem for LH connections leads to an interesting moduli problem: for fixed objects $(M,g,K)$ as above describe geometrically the moduli space of all LH connections on principal $K$-bundles on $M$ (up to bundle isomorphisms). Note that if $A$ is LH, then the associated connection metric $g_A$ on $P$ is locally homogenous, so it defines a geometric structure (in the sense of Thurston) on the total space of the bundle. Therefore this moduli problem is related to the classification of LH (geometric) Riemannian manifolds which admit a Riemannian submersion onto the given manifold $M$. Omitting the details, our moduli problem concerns the classification of geometric fibre bundles over a given geometric base. We develop a general method for describing moduli spaces of LH connections on a given base. Using our method we give explicit descriptions of these moduli spaces when the base manifold is a hyperbolic Riemann surface $(M,g)$ and $K\in\{S^1,PU(2)\}$. The case $K=S^1$ leads to a new construction of the moduli spaces of Yang-Mills $S^1$-connections on hyperbolic Riemann surfaces, and the case $K=PU(2)$ leads to a one-parameter family of compact, 5-dimensional geometric manifolds, which we study in detail.

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arXiv2018

Smooth deformations of singular contractions of class VII surfaces

We consider normal compact surfaces $Y$ obtained from a minimal class VII surface $X$ by contraction of a cycle $C$ of $r$ rational curves with $C^2<0$. Our main result states that, if the obtained cusp is smoothable, then $Y$ is globally smoothable. The proof is based on a vanishing theorem for $H^2(\Theta_Y)$. If $r<b_2(X)$ any smooth small deformation of $Y$ is rational, and if $r=b_2(X)$ (i.e. when $X$ is a half-Inoue surface) any smooth small deformation of $Y$ is an Enriques surface. The condition "the cusp is smoothable" in our main theorem can be checked in terms of the intersection numbers of the cycle, using the Looijenga conjecture (which has recently become a theorem). Therefore this is a "decidable" condition. We prove that this condition is always satisfied if $r<b_2(X)\leq 11$. Therefore the singular surface $Y$ obtained by contracting a cycle $C$ of $r$ rational curves in a minimal class VII surface $X$ with $r<b_2(X)\leq 11$ is always smoothable by rational surfaces. The statement holds even for unknown class VII surfaces.

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arXiv2017

On the Donaldson-Uhlenbeck compactification of instanton moduli spaces on class VII surfaces

We study the following question: Let $(X,g)$ be a compact Gauduchon surface, $(E,h)$ be a differentiable rank $r$ vector bundle on $X$, ${\mathcal{D}}$ be a fixed holomorphic structure on $D:=\det(E)$ and $a$ be the Chern connection of the pair $(\mathcal{D},\det(h))$. Does the complex space structure on ${\mathcal{M}}_a^{\mathrm{ASD}}(E)^*$ induced by the Kobayashi-Hitchin correspondence extend to a complex space structure on the Donaldson-Uhlenbeck compactification $\overline{\mathcal{M}}_a^\mathrm{ASD}(E)$? Our results answer this question in detail for the moduli spaces of $\mathrm{SU}(2)$-instantons with $c_2=1$ on general (possibly unknown) class VII surfaces.

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arXiv2016

A continuity theorem for families of sheaves on complex surfaces

We prove that any flat family $(\mathcal{ F}_u)_{u\in U}$ of rank 2 torsion-free sheaves on a Gauduchon surface defines a continuous map on the semi-stable locus $U^{\mathrm {ss}}:=\{u\in U \ |\ \mathcal{ F}_u\hbox{ is slope semi-stable}\}$ with values in the Donaldson-Uhlenbeck compactification of the corresponding instanton moduli space. In the general (possibly non-K\"ahlerian) case, the Donaldson-Uhlenbeck compactification is not a complex space, and the set $U^{\mathrm {ss}}$ can be a complicated subset of the base space $U$ that is neither open or closed in the classical topology, nor locally closed in the Zariski topology. This result provides an efficient tool for the explicit description of Donaldson-Uhlenbeck compactifications on arbitrary Gauduchon surfaces.

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arXiv2015

Analytic cycles in flip passages and in instanton moduli spaces over non-K\"ahlerian surfaces

Let $\mathcal{M}^{\mathrm{st}}$ ($\mathcal{M}^{\mathrm{pst}}$) be a moduli space of stable (polystable) bundles with fixed determinant on a complex surface with $b_1=1$, $p_g=0$, and let $Z\subset \mathcal{M}^{\mathrm{st}}$ be a pure $k$-dimensional analytic set. We prove a general formula for the homological boundary $\delta[Z]^{BM}\in H_{2k-1}^{BM}(\partial\hat{\mathcal M}^{\mathrm{pst}},\mathbb{Z})$ of the Borel-Moore fundamental class of $Z$ in the boundary of the blow up moduli space $\hat {\mathcal M}^{\mathrm{pst}}$. The proof is based on the holomorphic model theorem (proved in a previous article), which identifies a neighborhood of a boundary component of $\hat {\mathcal M}^{\mathrm{pst}}$ with a neighborhood of the boundary of a "blow up flip passage". We then focus on a particular instanton moduli space which intervenes in our program for proving the existence of curves on class VII surfaces. Using our result, combined with general properties of the Donaldson cohomology classes, we prove incidence relations between the Zariski closures (in the considered moduli space) of certain families of extensions. These incidence relations are crucial for understanding the geometry of the moduli space, and cannot be obtained using classical complex geometric deformation theory.

math.DG↗

arXiv2014

Instanton moduli spaces on non-K\"ahlerian surfaces. Holomorphic models around the reduction loci

Let $\mathcal{M}$ be a moduli space of polystable rank 2-bundles bundles with fixed determinant (a moduli space of $\mathrm{PU}(2)$-instantons) on a Gauduchon surface with $p_g=0$ and $b_1=1$. We study the holomorphic structure of $\mathcal{M}$ around a circle $\mathcal{T}$ of regular reductions. Our model space is a "blowup flip passage", which is a manifold with boundary whose boundary is a projective fibration, and whose interior comes with a natural complex structure. We prove that a neighborhood of the boundary of the blowup $\hat{\mathcal{M}}_{\mathcal{T}}$ of $\mathcal{M}$ at $\mathcal{T}$ can be smoothly identified with a neighborhood of the boundary of a "flip passage" $\hat Q$, the identification being holomorphic on $\mathrm{int}(\hat Q)$.

math.DG↗

arXiv2014

The $g$-areas and the commutator length

The commutator length of a Hamiltonian diffeomorphism $f\in \mathrm{Ham}(M, ω)$ of a closed symplectic manifold $(M,ω)$ is by definition the minimal $k$ such that $f$ can be written as a product of $k$ commutators in $\mathrm{Ham}(M, ω)$. We introduce a new invariant for Hamiltonian diffeomorphisms, called the $k_+$-area, which measures the "distance", in a certain sense, to the subspace $\mathcal{C}_k$ of all products of $k$ commutators. Therefore this invariant can be seen as the obstruction to writing a given Hamiltonian diffeomorphism as a product of $k$ commutators. We also consider an infinitesimal version of the commutator problem: what is the obstruction to writing a Hamiltonian vector field as a linear combination of $k$ Lie brackets of Hamiltonian vector fields? A natural problem related to this question is to describe explicitly, for every fixed $k$, the set of linear combinations of $k$ such Lie brackets. The problem can be obviously reformulated in terms of Hamiltonians and Poisson brackets. For a given Morse function $f$ on a symplectic Riemann surface $M$ (verifying a weak genericity condition) we describe the linear space of commutators of the form $\{f,g\}$, with $g\in\mathcal{C}^\infty(M,\mathbb{R})$.

math.SG↗

arXiv2014

A wall crossing formula for degrees of real central projections

The main result is a wall crossing formula for central projections defined on submanifolds of a real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a $\Z$-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the real subspace problem.

math.AG↗

arXiv2014

A variation formula for the determinant line bundle. Compact subspaces of moduli spaces of stable bundles over class VII surfaces

This article deals with two topics: the first, which has a general character, is a variation formula for the the determinant line bundle in non-Kählerian geometry. This formula, which is a consequence of the non-Kählerian version of the Grothendieck-Riemann Roch theorem proved recently by Bismut, gives the variation of the determinant line bundle corresponding to a perturbation of a Fourier-Mukai kernel ${\cal E}$ on a product $B\times X$ by a unitary flat line bundle on the fiber $X$. When this fiber is a complex surface and ${\cal E}$ is a holomorphic 2-bundle, the result can be interpreted as a Donaldson invariant. The second topic concerns a geometric application of our variation formula, namely we will study compact complex subspaces of the moduli spaces of stable bundles considered in our program for proving existence of curves on minimal class VII surfaces. Such a moduli space comes with a distinguished point $a=[{\cal A}]$ corresponding to the canonical extension ${\cal A}$ of $X$. The compact subspaces $Y\subset {\cal M}^\st$ containing this distinguished point play an important role in our program. We will prove a non-existence result: there exists no compact complex subspace of positive dimension $Y\subset {\cal M}^\st$ containing $a$ with an open neighborhood $a\in Y_a\subset Y$ such that $Y_a\setminus\{a\}$ consists only of non-filtrable bundles. In other words, within any compact complex subspace of positive dimension $Y\subset {\cal M}^\st$ containing $a$, the point $a$ can be approached by filtrable bundles. Specializing to the case $b_2=2$ we obtain a new way to complete the proof of a theorem in a previous article: any minimal class VII surface with $b_2=2$ has a cycle of curves. Applications to class VII surfaces with higher $b_2$ will be be discussed in a forthcoming article.

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