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Let $M$ be a compact complex manifold, and $D\, \subset\, M$ a reduced normal crossing divisor on it, such that the logarithmic tangent bundle $TM(-\log D)$ is holomorphically trivial. Let ${\mathbb A}$ denote the maximal connected subgroup of the group of all holomorphic automorphisms of $M$ that preserve the divisor $D$. Take a holomorphic Cartan geometry $(E_H,\,\Theta)$ of type $(G,\, H)$ on $M$, where $H\, \subset\, G$ are complex Lie groups. We prove that $(E_H,\,\Theta)$ is isomorphic to $(\rho^* E_H,\,\rho^* \Theta)$ for every $\rho\, \in\, \mathbb A$ if and only if the principal $H$--bundle $E_H$ admits a logarithmic connection $\Delta$ singular on $D$ such that $\Theta$ is preserved by the connection $\Delta$., Comment: Final version to appear in Journal of Differential Geometry and its Applications

We give a conjectural formula for the characteristic number of rational cuspidal curves in the projective plane by extending the idea of Kontsevich's recursion formula (namely, pulling back the equality of two divisors in the four pointed moduli space). The key geometric input that is needed here is that in the closure of rational cuspidal curves, there are two component rational curves which are tangent to each other at the nodal point. While this fact is geometrically quite believable, we haven't as yet proved it; hence our formula is for the moment conjectural. The answers that we obtain agree with what has been computed earlier Ran, Pandharipande, Zinger and Ernstrom and Kennedy. We extend this technique (modulo another conjecture) to obtain the characteristic number of rational quartics with an E6 singularity., Comment: 19 pages, 8 figures. Comments are welcome

Given a holomorphic Lie algebroid on an m-pointed Riemann surface, we define parabolic Lie algebroid connections on any parabolic vector bundle equipped with parabolic structure over the marked points. An analogue of the Atiyah exact sequence for parabolic Lie algebroids is constructed. For any Lie algebroid whose underlying holomorphic vector bundle is stable, we give a complete characterization of all the parabolic vector bundles that admit a parabolic Lie algebroid connection., Comment: 23 pages

Let $X$ be a compact connected Riemann surface and $(V, \phi)$ a holomorphic Lie algebroid on $X$ such that the holomorphic vector bundle $V$ is stable. We give a necessary and sufficient condition on holomorphic vector bundles $E$ on $X$ to admit a Lie algebroid connection., Comment: Final version; to appear in Geometriae Dedicata

We prove that on the product of two elliptic curves a generic nonsingular turbulent foliation does not admit any transversely projective structure.

We prove that any holomorphic geometric structure of affine type on an Oeljeklaus- Toma manifold is locally homogeneous. For locally conformal K\"ahler Oeljeklaus-Toma manifolds we prove that all holomorphic geometric structures, and also all holomorphic Cartan geometries, on them are locally homogeneous., Comment: Final version to appear in Manuscripta Mathematica

We prove that any holomorphic vector bundle admitting a holomorphic connection, over a compact K\"ahler Calabi-Yau manifold, also admits a flat holomorphic connection. This addresses a particular case of a question asked by Atiyah and generalizes a result previously obtained in \cite{BD} for simply connected compact K\"ahler Calabi-Yau manifolds. We give some applications of it in the framework of Cartan geometries and foliated Cartan geometries on K\"ahler Calabi-Yau manifolds., Comment: Final version

First we survey and explain the strategy of some recent results that construct holomorphic $\text{sl}(2, \mathbb C)$-differential systems over some Riemann surfaces $\Sigma_g$ of genus $g\geq 2$, satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian \cite{BDHH} or some cocompact Kleinian subgroup $$\Gamma \subset \text{SL}(2, \mathbb C)$$ as in \cite{BDHH2}. As a consequence, there exist holomorphic maps from $\Sigma_g$ to the quotient space $\text{SL}(2, \mathbb C)/ \Gamma$, where $\Gamma \subset \text{SL}(2, \mathbb C)$ is a cocompact lattice, that do not factor through any elliptic curve \cite{BDHH2}. This answers positively a question of Ghys in \cite{Gh}; the question was also raised by Huckleberry and Winkelmann in \cite{HW}. Then we prove that when $M$ is a Riemann surface, a Torelli type theorem holds for the affine group scheme over $\mathbb C$ obtained from the category of holomorphic connections on {\it \'etale trivial} holomorphic bundles. After that, we explain how to compute in a simple way the holonomy of a holomorphic connection on a free vector bundle. Finally, for a compact K\"ahler manifold $M$, we investigate the neutral Tannakian category given by the holomorphic connections on \'etale trivial holomorphic bundles over $M$. If $\varpi$ (respectively, $\Theta$) stands for the affine group scheme over $\mathbb C$ obtained from the category of connections (respectively, connections on free (trivial) vector bundles), then the natural inclusion produces a morphism $v:{\mathcal O}(\Theta)\longrightarrow {\mathcal O}(\varpi)$ of Hopf algebras. We present a description of the transpose of $v$ in terms of the iterated integrals., Comment: To appear in the Int. Jour. Math. volume in honor of Oscar Garcia-Prada

We prove that a compact Sasakian manifolds whose first and second basic Chern classes vanish is locally isomorphic to the real Heisenberg group equipped with the standard left invariant Sasakian structure up to deformation associated to a basic $1$-from., Comment: 7 pages. arXiv admin note: text overlap with arXiv:2211.16713

There are two canonical projective structures on any compact Riemann surface of genus at least two: one coming from the uniformization theorem, and the other from Hodge theory. They produce two (different) families of projective structures over the moduli space $M_g$ of compact Riemann surfaces. A recent work of Biswas, Favale, Pirola, and Torelli shows that families of projective structures over $M_g$ admit an equivalent characterization in terms of complex connections on the dual $\mathcal{L}$ of the determinant of the Hodge line bundle over $M_g$; the same work gave the connection on $\mathcal L$ corresponding to the projective structures coming from uniformization. Here we construct the connection on $\mathcal L$ corresponding to the family of Hodge theoretic projective structures. This connection is described in three different ways: firstly as the connection induced on $\mathcal L$ by the Chern connection of the $L^2$-metric on the Hodge bundle, secondly as an appropriate root of the Quillen metric induced by the (square of the) Theta line bundle on the universal family of abelian varieties, endowed with the natural Hermitian metric given by the polarization, and finally as Quillen connection gotten using the Arakelov metric on the universal curve, modified by Faltings' delta invariant., Comment: Revised version. To appear on Communications in Mathematical Physics

We have proven the existence of new higher-genus maxfaces with Enneper end. These maxfaces are not the companions of any existing minimal surfaces, and furthermore, the singularity set is located away from the ends. The nature of the singularities is systematically investigated., Comment: 27 pages, Final version. To appear in "The Journal of Geometric Analysis"

We study which representations $\rho$ of the fundamental group of a compact oriented surface $X$ admit Higgs data that depend holomorphically on the Riemann surface $\Sigma\,=\, (X,\, J)$ via non-abelian Hodge correspondence. For representations $\rho$ into $\mathrm{SL}(2,\mathbb C)$ we show that holomorphic dependency is equivalent to $\rho$ being unitary. For higher ranks this equivalence fails -- we show the existence of non-unitary and irreducible representations of the fundamental group into $\mathrm{SL}(n,\mathbb C)$ admitting Higgs data that are holomorphic in $\Sigma$, for $n$ large enough., Comment: 14 pages

Abstract. Let G be a complex reductive group and A be an Abelian variety of dimension d over $\mathbb{C}$. We determine the Poincar\'e polynomials and also the mixed Hodge polynomials of the moduli space $\mathcal{M}_{A}^{H}(G)$ of G-Higgs bundles over A. We show that these are normal varieties with symplectic singularities, when G is a classical semisimple group. For $G=GL_{n}(\mathbb{C})$, we also compute Poincar\'e polynomials of natural desingularizations of $\mathcal{M}_{A}^{H}(G)$ and of G-character varieties of free abelian groups, in some cases. In particular, explicit formulas are obtained when dim A=d=1, and also for rank 2 and 3 Higgs bundles, for arbitrary d>1., Comment: Dedicated to Peter E. Newstead

Let $X$ be a compact complex manifold such that its canonical bundle $K_X$ is numerically trivial. Assume additionally that $X$ is Moishezon or $X$ is Fujiki with dimension at most four. Using the MMP and classical results in foliation theory, we prove a Beauville-Bogomolov type decomposition theorem for $X$. We deduce that holomorphic geometric structures of affine type on $X$ are in fact locally homogeneous away from an analytic subset of complex codimension at least two, and that they cannot be rigid unless $X$ is an \'etale quotient of a compact complex torus. Moreover, we establish a characterization of torus quotients using the vanishing of the first two Chern classes which is valid for any compact complex $n$-folds of algebraic dimension at least $n-1$. Finally, we show that a compact complex manifold with trivial canonical bundle bearing a rigid geometric structure must have infinite fundamental group if either $X$ is Fujiki, $X$ is a threefold, or $X$ is of algebraic dimension at most one., Comment: 40 pages, v3: Final version; Math. Annalen (to appear)

In the paper "The Second cohomology of nilpotent orbits in classical Lie algebras, Kyoto J. Math. 60 (2020), no. 2, 717-799" by I. Biswas, P. Chatterjee, and C. Maity, computable explicit descriptions of the second and first cohomology groups of a general homogeneous space of a Lie group are given, extending an earlier result in "On the exactness of Kostant-Kirillov form and the second cohomology of nilpotent orbits, Internat. J. Math. 23 (2012), no. 8, 1250086" by I. Biswas and P. Chatterjee. They turned out to be crucial in our computations related to the cohomology of nilpotent orbits, as done in the above two papers. Here we describe the third and fourth cohomologies of a general homogeneous space along the line mentioned above. We also draw various consequences of our main results in numerous special settings. In particular, we obtain an interesting invariant by showing that for a large class of homogeneous spaces, the difference between the third and fourth Betti numbers coincides with the difference in the numbers of simple factors of the ambient group and the associated closed subgroup., Comment: 43 pages

For any smooth projective variety with holomorphic locally homogeneous structure modelled on a homogeneous algebraic variety, we determine all the subvarieties of it which develop to the model.

Through the action of an anti-holomorphic involution $\sigma$ (a real structure) on a Riemann surface $X$, we consider the induced actions on ${\rm SL}(r,\mathbb{C})$-opers and study the real slices fixed by such actions. By constructing this involution for different descriptions of the space of ${\rm SL}(r,\mathbb{C})$-opers, we are able to give a natural parametrization of the fixed point locus via differentials on the Riemann surface, which in turn allows us to study their geometric properties.

We obtain a recursive formula for the number of rational degree $d$ curves in $\mathbb{CP}^2$ that pass through $3d+1-m$ generic points and that have an $m$-fold singular point. The special case of counting curves with a triple point was solved earlier by other authors. We obtain the formula by considering a family version of Kontsevich's recursion formula, in contrast to the excess intersection theoretic approach of others. A large number of low degree cases have been worked out explicitly., Comment: 16 pages, 4 figures

Take an irreducible smooth projective curve $X$ defined over an algebraically closed field of characteristic zero, and fix finitely many distinct point $D\, =\, \{x_1,\, \cdots,\, x_n\}$ of it; for each point $x\, \in\, D$ fix a positive integer $N_x$. Take a nonconstant map $f\, :\, Y\, \longrightarrow \, X$ from an irreducible smooth projective curve. We construct a natural subbundle $\mathcal{F}\, \subset\, f_*{\mathcal O}_Y$ using $(D,\, \{N_x\}_{x\in D})$. Let $E_*$ be a stable parabolic vector bundle whose parabolic weights at each $x\, \in\, D$ are integral multiples of $\frac{1}{N_x}$. We prove that the pullback $f^*E_*$ is also parabolic stable, if ${\rm rank}(\mathcal{F})\,=\, 1$., Comment: Final version

We compute the infinitesimal deformations of quadruples of the form $$(X, S, E_*, D),$$ where $(X, S)$ is a compact Riemann surface with $n$ marked points, $E_*$ is a parabolic vector bundle on $X$ with parabolic structure over $S$, and $D$ is a parabolic connection on $E_*$. Using it we compute the infinitesimal deformations of $(X, S, D)$, where $D$ is a parabolic SL(r, C)-oper on $(X, S)$. It is shown that the monodromy map, from the moduli space of triples $(X, S, D)$, where $D$ is a parabolic SL(r, C)-oper on $(X, S)$, to the SL(r, C)-character variety of X - S, is an immersion.

In this continuation of \cite{BDS}, we investigate the deformations of holomorphic Cartan geometries where the underlying complex manifold is allowed to move. The space of infinitesimal deformations of a flat holomorphic Cartan geometry is computed. We show that the natural forgetful map, from the infinitesimal deformations of a flat holomorphic Cartan geometry to the infinitesimal deformations of the underlying flat principal bundle on the topological manifold, is an isomorphism.

We prove the existence of a pair $(\Sigma ,\, \Gamma)$, where $\Sigma$ is a compact Riemann surface with $\text{genus}(\Sigma)\, \geq\, 2$, and $\Gamma\, \subset\, {\mathrm SL}(2, \mathbb C)$ is a cocompact lattice, such that there is a generically injective holomorphic map $\Sigma \, \longrightarrow\, {\mathrm SL}(2, \mathbb C)/\Gamma$. This gives an affirmative answer to a question raised by Huckleberry and Winkelmann \cite{HW} and by Ghys \cite{Gh}., Comment: 33 pages; 3 figures; comments are very welcome

Let $M$ be a compact connected Fujiki manifold, $G$ a semisimple affine algebraic group over $\mathbb C$ with one simple factor and $P$ a fixed proper parabolic subgroup of $G$. For a holomorphic principal $G$--bundle $E_G$ over $M$, let ${\mathcal E}_P$ be the holomorphic principal $P$-bundle $E_G\rightarrow E_G/P$ given by the quotient map. We prove that the following three statements are equivalent: (1) ${\rm ad}(E_G)$ is numerically flat, (2) the holomorphic line bundle $\bigwedge^{\rm top} {\rm ad}({\mathcal E}_P)^*$ is nef, and (3) for every reduced irreducible compact complex analytic space $Z$ with a K\"ahler form $\omega$, holomorphic map $\gamma : Z \rightarrow M$, and holomorphic reduction of structure group $E_P \subset \gamma^*E_G$ to $P$, the inequality ${\rm degree}({\rm ad}(E_P)) \leq 0$ holds., Comment: Final version; to appear in "Differential Geometry and its Applications"

In this continuation of \cite{BK} we investigate the non-abelian Hodge correspondence on compact Sasakian manifolds with emphasis on the quasi-regular case. On quasi-regular Sasakian manifolds, we introduce the notions of quasi-regularity and regularity of basic vector bundles. These notions are useful in relating the vector bundles over a quasi-regular Sasakian manifold with the orbibundles over the orbifold defined by the orbits of the Reeb foliation of the Sasakian manifold. We note that the non-abelian Hodge correspondence on quasi-regular Sasakian manifolds gives a canonical correspondence between the semi-simple representations of the orbifold fundamental groups and the Higgs orbibundles on locally cyclic complex orbifolds admitting Hodge metrics. Under the quasi-regularity of Sasakian manifolds and vector bundles, we extend this correspondence to one between the flat bundles and the basic Higgs bundles. We also prove a Sasakian analogue of the characterization of numerically flat bundles given by Demailly, Peternell and Schneider., Comment: Final version; to appear in the "Annali della Scuola Normale Superiore di Pisa"

Given a simple, simply connected, complex algebraic group G, a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic G-bundles over any family of smooth projective curves with marked points was constructed by the authors in an earlier paper. Here, it is shown that the identification between the bundle of nonabelian theta functions and the bundle of WZNW conformal blocks is flat with respect to this connection and the one constructed by Tsuchiya-Ueno-Yamada. As an application, we give a geometric construction of the Knizhnik-Zamolodchikov connection on the trivial bundle over the configuration space of points in the projective line whose typical fiber is the space of invariants of tensor product of representations., Comment: 29 pp, Exposition improved and separated to appendix, proof of a theorem has been moved to arXiv:2103.03792, comments are welcome!

Let ${\mathcal B}_g(r)$ be the moduli space of triples of the form $(X,\, K^{1/2}_X,\, F)$, where $X$ is a compact connected Riemann surface of genus $g$, with $g\, \geq\, 2$, $K^{1/2}_X$ is a theta characteristic on $X$, and $F$ is a stable vector bundle on $X$ of rank $r$ and degree zero. We construct a $T^*{\mathcal B}_g(r)$--torsor ${\mathcal H}_g(r)$ over ${\mathcal B}_g(r)$. This generalizes on the one hand the torsor over the moduli space of stable vector bundles of rank $r$, on a fixed Riemann surface $Y$, given by the moduli space of holomorphic connections on the stable vector bundles of rank $r$ on $Y$, and on the other hand the torsor over the moduli space of Riemann surfaces given by the moduli space of Riemann surfaces with a projective structure. It is shown that ${\mathcal H}_g(r)$ has a holomorphic symplectic structure compatible with the $T^*{\mathcal B}_g(r)$--torsor structure. We also describe ${\mathcal H}_g(r)$ in terms of the second order matrix valued differential operators. It is shown that ${\mathcal H}_g(r)$ is identified with the $T^*{\mathcal B}_g(r)$--torsor given by the sheaf of holomorphic connections on the theta line bundle over ${\mathcal B}_g(r)$.

This is a survey paper dealing with holomorphic G-structures and holomorphic Cartan geometries on compact complex manifolds. Our emphasis is on the foliated case: holomorphic foliations with transverse (branched or generalized) holomorphic Cartan geometries.

The Quillen connection on ${\mathcal L} \rightarrow {\mathcal M}_g$, where ${\mathcal L}^*$ is the Hodge line bundle over the moduli stack of smooth complex projective curves curves ${\mathcal M}_g$, $g \geq 5$, is uniquely determined by the condition that its curvature is the Weil--Petersson form on ${\mathcal M}_g$. The bundle of holomorphic connections on ${\mathcal L}$ has a unique holomorphic isomorphism with the bundle on ${\mathcal M}_g$ given by the moduli stack of projective structures. This isomorphism takes the $C^\infty$ section of the first bundle given by the Quillen connection on ${\mathcal L}$ to the $C^\infty$ section of the second bundle given by the uniformization theorem. Therefore, any one of these two sections determines the other uniquely.

Over a smooth and proper complex scheme, the differential Galois group of an integrable connection may be obtained as the closure of the transcendental monodromy representation. In this paper, we employ a completely algebraic variation of this idea by restricting attention to connections on trivial vector bundles and replacing the fundamental group by a certain Lie algebra constructed from the regular forms. In more detail, we show that the differential Galois group is a certain ``closure'' of the aforementioned Lie algebra. This is then applied to construct connections on curves with prescribed differential Galois group., Comment: A reader pointed out an error; we have removed the results concerning applications to the connections on the projective line. To appear in C. R. Math. Acad. Sci. Paris

For every integer $g \,\geq\, 2$ we show the existence of a compact Riemann surface $\Sigma$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${\mathcal O}^{\oplus 2}_{\Sigma}$ admits holomorphic connections with $\text{SL}(2,{\mathbb R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. The construction carries over to all very stable and compatible real holomorphic structures for the topologically trivial rank two bundle over $\Sigma$ and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well., Comment: 38 pages, 2 figures

For a simple, simply connected, complex group G, we prove the existence of a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic G-bundles over families of smooth projective curves with marked points., Comment: Final version: To appear in Crelle's Journal. Major Changes based on comments of anonymous referees: Section on Ginzburg algebra is split from this to form a separate paper, Theorem 6.3 from arXiv2110.00430v1 (also will be updated soon) is now moved to this paper with additional details

We study the holomorphic symplectic geometry of (the smooth locus of) the space of holomorphic sections of a twistor space with rotating circle action. The twistor space has a meromorphic connection constructed by Hitchin. We give an interpretation of Hitchin's meromorphic connection in the context of the Atiyah--Ward transform of the corresponding hyperholomorphic line bundle. It is shown that the residue of the meromorphic connection serves as a moment map for the induced circle action, and its critical points are studied. Particular emphasis is given to the example of Deligne--Hitchin moduli spaces., Comment: 36 pages

We investigate the symplectic geometric and differential geometric aspects of the moduli space of connections on a compact Riemann surface $X$. Fix a theta characteristic $K^{1/2}_X$ on $X$; it defines a theta divisor on the moduli space ${\mathcal M}$ of stable vector bundles on $X$ of rank $r$ degree zero. Given a vector bundle $E \in {\mathcal M}$ lying outside the theta divisor, we construct a natural holomorphic connection on $E$ that depends holomorphically on $E$. Using this holomorphic connection, we construct a canonical holomorphic isomorphism between the following two: \begin{enumerate} \item the moduli space $\mathcal C$ of pairs $(E, D)$, where $E\in {\mathcal M}$ and $D$ is a holomorphic connection on $E$, and \item the space ${\rm Conn}(\Theta)$ given by the sheaf of holomorphic connections on the line bundle on $\mathcal M$ associated to the theta divisor. \end{enumerate} The above isomorphism between $\mathcal C$ and ${\rm Conn}(\Theta)$ is symplectic structure preserving, and it moves holomorphically as $X$ runs over a holomorphic family of Riemann surfaces., Comment: Final version

We prove that any one-relator group $G$ is the fundamental group of a compact Sasakian manifold if and only if $G$ is either finite cyclic or isomorphic to the fundamental group of a compact Riemann surface of genus g > 0 with at most one orbifold point of order $n \geq 1$. We also classify all groups of deficiency at least two that are also the fundamental group of some compact Sasakian manifold.

Given a compact Riemann surface $\Sigma$ of genus $g_\Sigma\, \geq\, 2$, and an effective divisor $D\, =\, \sum_i n_i x_i$ on $\Sigma$ with $\text{degree}(D)\, <\, 2(g_\Sigma -1)$, there is a unique cone metric on $\Sigma$ of constant negative curvature $-4$ such that the cone angle at each $x_i$ is $2\pi n_i$ (see McOwen and Troyanov [McO,Tr]). We describe the Higgs bundle corresponding to this uniformization associated to the above conical metric. We also give a family of Higgs bundles on $\Sigma$ parametrized by a nonempty open subset of $H^0(\Sigma,\,K_\Sigma^{\otimes 2}\otimes{\mathcal O}_\Sigma(-2D))$ that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin's results in [Hi1], for the case $D\,=\, 0$., Comment: minor changings; to appear in International Journal of Mathematics

Let $\mathbb{X}=[X_1\rightrightarrows X_0]$ be a Lie groupoid equipped with a connection, given by a smooth distribution $\mathcal{H} \subset T X_1$ transversal to the fibers of the source map. Under the assumption that the distribution $\mathcal{H}$is integrable, we define a version of de Rham cohomology for the pair $(\mathbb{X}, \mathcal{H})$, and we study connections on principal $G$-bundles over $(\mathbb{X}, \mathcal{H})$ in terms of the associated Atiyah sequence of vector bundles. We also discuss associated constructions for differentiable stacks. Finally, we develop the corresponding Chern-Weil theory and describe characteristic classes of principal G-bundles over a pair $(\mathbb{X}, \mathcal{H})$., Comment: To appear in Reviews in Mathematical Physics

We construct and study general connections on Lie groupoids and differentiable stacks as well as on principal bundles over them using Atiyah sequences associated to transversal tangential distributions., Comment: To appear in Journal of Noncommutative Geometry

We introduce a class of $G$-invariant connections on a homogeneous principal bundle $Q$ over a hermitian symmetric space $M=G/K$. The parameter space carries the structure of normal variety and has a canonical anti-holomorphic involution. The fixed points of the anti-holomorphic involution are precisely the integrable invariant complex structures on $Q.$ This normal variety is closely related to quiver varieties and, more generally, to varieties of commuting matrix tuples modulo simultaneous conjugation., Comment: Final version; to appear in Proc. Ind. Acad. Sci

Let $M$ be the moduli space of complex Lagrangian submanifolds of a hyperK\"ahler manifold $X$, and let $\varpi : \widehat{\mathcal{A}} \rightarrow M$ be the relative Albanese over $M$. We prove that $\widehat{\mathcal{A}}$ has a natural holomorphic symplectic structure. The projection $\varpi$ defines a completely integrable structure on the symplectic manifold $\widehat{\mathcal{A}}$. In particular, the fibers of $\varpi$ are complex Lagrangians with respect to the symplectic form on $\widehat{\mathcal{A}}$. We also prove analogous results for the relative Picard over $M$., Comment: Final version

Given a compact Kaehler manifold X, it is shown that pairs of the form (E, D), where E is a trivial holomorphic vector bundle on X, and D is an integrable holomorphic connection on $E$, produce a neutral Tannakian category. The corresponding pro-algebraic affine group scheme is studied. In particular, it is shown that this pro-algebraic affine group scheme for a compact Riemann surface determines uniquely the isomorphism class of the Riemann surface., Comment: Final version; to appear in Canadian Journal of Mathematics

We prove that a holomorphic projective connection on a complex projective threefold is either flat, or it is a translation invariant holomorphic projective connection on an abelian threefold. In the second case, a generic translation invariant holomorphic affine connection on the abelian variety is not projectively flat. We also prove that a simply connected compact complex threefold with trivial canonical line bundle does not admit any holomorphic projective connection., Comment: This is the final version to appear in Math. Zeitschrift

Given a smooth complex projective variety $M$ and a smooth closed curve $X \subset M$ such that the homomorphism of fundamental groups $\pi_1(X) \rightarrow \pi_1(M)$ is surjective, we study the restriction map of Higgs bundles, namely from Higgs bundles on $M$ to those on $X$. In particular, we investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on $M$ and $X$. We also consider the set-up where a finite group is acting on $M$ via holomorphic automorphisms or anti-holomorphic involutions, and the curve $X$ is preserved by the action. Branes are studied in this context., Comment: Final version; to appear in the "Research in the Mathematical Sciences"

A vector bundle $E$ over a projective variety $M$ is called finite if it satisfies a nontrivial polynomial equation with nonnegative integral coefficients. Introducing finite bundles, Nori proved that $E$ is finite if and only if the pullback of $E$ to some finite \'etale covering of $M$ is trivializable \cite{No1}. The definition of finite bundles extends naturally to holomorphic vector bundles over compact complex manifolds. We prove that a holomorphic vector bundle over a compact complex manifold $M$ is finite if and only if the pullback of $E$ to some finite \'etale covering of $M$ is holomorphically trivializable. Therefore, $E$ is finite if and only if it admits a flat holomorphic connection with finite monodromy., Comment: Final version

We study holomorphic GL(2) and SL(2) geometries on compact complex manifolds. We show that a compact K\"ahler manifold of complex even dimension higher than two admitting a holomorphic GL(2)-geometry is covered by a compact complex torus. We classify compact K\"ahler-Einstein manifolds and Fano manifolds bearing holomorphic GL(2)-geometries. Among the compact K\"ahler-Einstein manifolds we prove that the only examples bearing holomorphic GL(2)-geometry are those covered by compact complex tori, the three dimensional quadric and those covered by the three dimensional Lie ball (the non compact dual of the quadric)., Comment: This is the final version to be published in Manuscripta Mathematica

Starting from the data of a nonsingular complex projective toric variety, we define an associated notion of toric co-Higgs bundle. We provide a Lie-theoretic classification of these objects by studying the interaction between Klyachko's fan filtration and the fiber of the co-Higgs bundle at a closed point in the open orbit of the torus action. This can be interpreted, under certain conditions, as the construction of a coarse moduli scheme of toric co-Higgs bundles of any rank and with any total equivariant Chern class., Comment: 11 pages, to appear in Illinois J. Math

We construct an irreducible holomorphic connection with SL(2,R)-monodromy on the trivial holomorphic vector bundle of rank two over a compact Riemann surface. This answers a question of Calsamiglia, Deroin, Heu and Loray in \cite{CDHL}., Comment: Final version; to appear in Journal de Math. Pures et Appl

Let $G$ be a connected reductive affine algebraic group defined over $\mathbb C$ and $\mathfrak g$ its Lie algebra. We study the monodromy map from the space of $\mathfrak g$-differential systems on a compact connected Riemann surface $\Sigma$ of genus $g \,\geq\, 2$ to the character variety of $G$-representations of the fundamental group of $\Sigma$. If the complex dimension of $G$ is at least three, we show that the monodromy map is an immersion at the generic point., Comment: Final version; to appear in Pub. RIMS

Given any compact Riemann surface $C$, there is a canonical meromorphic 2--form $\widehat\eta$ on $C\times C$, with pole of order two on the diagonal $\Delta\, \subset\, C\times C$, constructed in \cite{cfg}. This meromorphic 2--form $\widehat\eta$ produces a canonical projective structure on $C$. On the other hand the uniformization theorem provides another canonical projective structure on any compact Riemann surface $C$. We prove that these two projective structures differ in general. This is done by comparing the $(0,1)$--component of the differential of the corresponding sections of the moduli space of projective structures over the moduli space of curves. The $(0,1)$--component of the differential of the section corresponding to the projective structure given by the uniformization theorem was computed by Zograf and Takhtadzhyan in \cite{ZT} as the Weil--Petersson K\"ahler form $\omega_{wp}$ on the moduli space of curves. We prove that the $(0,1)$--component of the differential of the section of the moduli space of projective structures corresponding to $\widehat{\eta}$ is the pullback of a nonzero constant scalar multiple of the Siegel form, on the moduli space of principally polarized abelian varieties, by the Torelli map., Comment: Final version; to appear in Jour. Math. Pures. Appl

Opers were introduced by Beilinson-Drinfeld [arXiv:math.AG/0501398]. In [J. Math. Pures Appl. 82 (2003), 1-42] a higher rank analog was considered, where the successive quotients of the oper filtration are allowed to have higher rank. We dedicate this paper to introducing and studying generalized $B$-opers (where "$B$" stands for "bilinear"), obtained by endowing the underlying vector bundle with a bilinear form which is compatible with both the filtration and the connection. In particular, we study the structure of these $B$-opers, by considering the relationship of these structures with jet bundles and with geometric structures on a Riemann surface.

We study the holomorphic vector bundles E over the twistor space Tw(M) of a compact simply connected hyperk\"ahler manifold $M$. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic sections of the holomorphic twistor projection \pi :Tw(M)\rightarrow CP^1. It is shown that if E admits a holomorphic connection, then E is holomorphically trivial and the holomorphic connection on E is trivial as well. For any irreducible vector bundle E on Tw(M) of prime rank, we prove that its restriction to the generic fibre of \pi is stable. On the other hand, for a K3 surface M, we construct examples of irreducible vector bundles of any composite rank on Tw(M) whose restriction to every fibre of \pi is non-stable. We have obtained a new method of constructing irreducible vector bundles on hyperk\"ahler twistor spaces; this method is employed in constructing these examples., Comment: Final version; Math. Zeit. (to appear)