On the Second Cohomology of K?hler Groups

Carlson and Toledo conjectured that if an infinite group Γ is the fundamental group of a compact K?hler manifold, then virtually H²(G, \mathbb R) 1 0{H^{2}(\Gamma, {\mathbb R}) \ne 0} . We assume that Γ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ( \mathbbC{\mathbb{C}} -VHS) on the K?hler manifold. We prove the conjecture under some assumption on the \mathbbC{\mathbb{C}} -VHS. We also study some related geometric/topological properties of period domains associated to such a \mathbbC{\mathbb{C}} -VHS.     

Compact Surfaces with Constant Gaussian Curvature in Product Spaces

We prove that the only compact surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.     

The $${{\rm GL}(2,\mathbb{C})}$$ McKay correspondence

In this paper we show that for any affine complete rational surface singularity the quiver of the reconstruction algebra can be determined combinatorially from the dual graph of the minimal resolution. As a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Also, for any finite subgroup G of GL(2,\mathbbC){{\rm GL}(2,\mathbb{C})}, it means that the endomorphism ring of the special CM \mathbbC{\mathbb{C}} [[x, y]] ^G -modules can be used to build the dual graph of the minimal resolution of \mathbbC²/G{\mathbb{C}^{2}/G}, extending McKay’s observation (McKay, Proc Symp Pure Math, 37:183–186, 1980) for finite subgroups of SL(2,\mathbbC){{\rm SL}(2,\mathbb{C})} to all finite subgroups of GL(2,\mathbbC){{\rm GL}(2,\mathbb{C})}.     

A Berger type normal holonomy theorem for complex submanifolds

We prove a Berger type theorem for the normal holonomy F^{^}{\Phi^\perp} (i.e., the holonomy group of the normal connection) of a full complete complex submanifold M of the complex projective space \mathbbC Pⁿ{\mathbb{C} P^n}. Namely, if F^{^}{\Phi^\perp} does not act transitively, then M is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible Hermitian symmetric space of rank greater or equal to 3. Moreover, we show that for complete irreducible complex submanifolds of \mathbbCⁿ{\mathbb{C}^n} the normal holonomy is generic, i.e., it acts transitively on the unit sphere of the normal space. The methods in the proofs rely heavily on the singular data of appropriate holonomy tubes (after lifting the submanifold to the complex Euclidean space, in the \mathbbC Pⁿ{\mathbb{C} P^n} case) and basic facts of complex submanifolds.     

Algebraic characterization of the isometries of the hyperbolic 5-space

Let \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} be the group of invertible 2 × 2 matrices over the division algebra \mathbbH{\mathbb{H}} of quaternions. \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} acts on the hyperbolic 5-space as the group of orientation-preserving isometries. Using this action we give an algebraic characterization of the orientation-preserving isometries of the hyperbolic 5-space. Along the way we also determine the conjugacy classes and the conjugacy classes of centralizers or the z-classes in \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} .     

A generalisation of the Hopf construction and harmonic morphisms into {\mathbb {S}^2}

In this paper, we construct a new family of harmonic morphisms ${\varphi:V^5\to\mathbb{S}^2}In this paper, we construct a new family of harmonic morphisms j:V⁵?\mathbbS²{\varphi:V^5\to\mathbb{S}^2}, where V ⁵ is a 5-dimensional open manifold contained in an ellipsoidal hypersurface of \mathbbC⁴ = \mathbbR⁸{\mathbb{C}^4\,=\,\mathbb{R}^8}. These harmonic morphisms admit a continuous extension to the completion V^*5{{V^{\ast}}^5}, which turns out to be an explicit real algebraic variety. We work in the context of a generalization of the Hopf construction and equivariant theory.     

Biharmonic submanifolds of {\mathbb{C}P^n}

We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifold [`(M)]{\bar{M}} in \mathbbCPⁿ{\mathbb{C}P^n} and the bitension field of the inclusion of the corresponding Hopf-tube in \mathbbS²ⁿ⁺¹{\mathbb{S}^{2n+1}}. Using this relation we produce new families of proper-biharmonic submanifolds of \mathbbCPⁿ{\mathbb{C}P^n}. We study the geometry of biharmonic curves of \mathbbCPⁿ{\mathbb{C}P^n} and we characterize the proper-biharmonic curves in terms of their curvatures and complex torsions.     

A criterion for the splitting of a principal bundle over a projective space

Let E_G be an algebraic principal G-bundle over \mathbbC\mathbbPⁿ ,\mathbb{C}\mathbb{P}^n , n  \mathbbC.\mathbb{C}. We prove that E_G admits a reduction of structure group to a one-parameter subgroup of G if and only if

Upper Bounds for $${\mathbb R}$$-Linear Resolvents

We discuss upper bounds for the resolvent of an \mathbbR{\mathbb{R}}-linear operator in \mathbbC^d{\mathbb{C}^d}.     

Deformations of Symplectic Cohomology and Exact Lagrangians in ALE Spaces

ALE spaces are the simply connected hyperkähler manifolds which at infinity look like ${\mathbb{C}^{2}/G}ALE spaces are the simply connected hyperk?hler manifolds which at infinity look like \mathbbC²/G{\mathbb{C}^{2}/G}, for any finite subgroup G ì SL₂(\mathbbC){G \subset SL_2(\mathbb{C})}. We prove that all exact Lagrangians inside ALE spaces must be spheres. The proof relies on showing the vanishing of a twisted version of symplectic cohomology.     

On the rationality of the moduli space of Lüroth quartics

We prove that the moduli space \mathfrakM_L{\mathfrak{M}_L} of Lüroth quartics in \mathbbP²{\mathbb{P}^2}, i.e. the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of PGL₃ (\mathbbC){\mathrm{PGL}_3 (\mathbb{C})} is rational, as is the related moduli space of Bateman seven-tuples of points in \mathbbP²{\mathbb{P}^2}.     

Complex cycles on algebraic models of smooth manifolds

Every compact smooth manifold M is diffeomorphic to the set X(\mathbbR){X(\mathbb{R})} of real points of a nonsingular projective real algebraic variety X, which is called an algebraic model of M. Each algebraic cycle of codimension k on the complex variety X_\mathbbC=X×_\mathbbR\mathbbC{X_{\mathbb{C}}=X\times_{\mathbb{R}}\mathbb{C}} determines a cohomology class in H^2k(X(\mathbbR);\mathbbD){H^{2k}(X(\mathbb{R});\mathbb{D})} , where \mathbbD{\mathbb{D}} denotes \mathbbZ{\mathbb{Z}} or \mathbbQ{\mathbb{Q}} . We investigate the behavior of such cohomology classes as X runs through the class of algebraic models of M.     

Complex Surfaces with Cat(0) Metrics

We study complex surfaces with locally CAT(0) polyhedral K?hler metrics and construct such metrics on \mathbbCP²{\mathbb{C}P^{2}} with various orbifold structures. In particular, in relation to questions of Gromov and Davis–Moussong we construct such metrics on a compact quotient of the two-dimensional unit complex ball. In the course of the proof of these results we give criteria for Sasakian 3-manifolds to be globally CAT(1). We show further that for certain Kummer coverings of \mathbbCP²{\mathbb{C}P^{2}} of sufficiently high degree their desingularizations are of type K(π, 1).     

All 4-Edge-Connected HHD-Free Graphs are {\mathbb{Z}_3}-Connected

An undirected graph G = (V, E) is called \mathbbZ₃{\mathbb{Z}_3}-connected if for all b: V ? \mathbbZ₃{b: V \rightarrow \mathbb{Z}_3} with ?_{v ? V}b(v)=0{\sum_{v \in V}b(v)=0}, an orientation D = (V, A) of G has a \mathbbZ₃{\mathbb{Z}_3}-valued nowhere-zero flow f: A? \mathbbZ₃-{0}{f: A\rightarrow \mathbb{Z}_3-\{0\}} such that ?_{e ? d⁺(v)}f(e)-?_{e ? d^-(v)}f(e)=b(v){\sum_{e \in \delta^+(v)}f(e)-\sum_{e \in \delta^-(v)}f(e)=b(v)} for all v ? V{v \in V}. We show that all 4-edge-connected HHD-free graphs are \mathbbZ₃{\mathbb{Z}_3}-connected. This extends the result due to Lai (Graphs Comb 16:165–176, 2000), which proves the \mathbbZ₃{\mathbb{Z}_3}-connectivity for 4-edge-connected chordal graphs.     

Lifting of convex functions on Carnot groups and lack of convexity for a gauge function

Let ${\mathbb{G}}Let \mathbbG{\mathbb{G}} be a Carnot group of step r and m generators and homogeneous dimension Q. Let \mathbbF_m,r{\mathbb{F}_{m,r}} denote the free Lie group of step r and m generators. Let also p:\mathbbF_m,r?\mathbbG{\pi:\mathbb{F}_{m,r}\to\mathbb{G}} be a lifting map. We show that any horizontally convex function u on \mathbbG{\mathbb{G}} lifts to a horizontally convex function u°p{u\circ \pi} on \mathbbF_m,r{\mathbb{F}_{m,r}} (with respect to a suitable horizontal frame on \mathbbF_m,r{\mathbb{F}_{m,r}}). One of the main aims of the paper is to exhibit an example of a sub-Laplacian L=?_j=1^m X_j²{\mathcal{L}=\sum_{j=1}^m X_j^2} on a Carnot group of step two such that the relevant L{\mathcal{L}}-gauge function d (i.e., d ^2-Q is the fundamental solution for L{\mathcal{L}}) is not h-convex with respect to the horizontal frame {X ₁, . . . , X _m }. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).     

Some remarks on the Kähler geometry of the Taub-NUT metrics

In this article, we investigate the balanced condition and the existence of an Engliš expansion for the Taub-NUT metrics on \mathbbC²{\mathbb{C}^2} . Our first result shows that a Taub-NUT metric on \mathbbC²{\mathbb{C}^2} is never balanced unless it is the flat metric. The second one shows that an Engliš expansion of the Rawnsley’s function associated to a Taub-NUT metric always exists, while the coefficient a ₃ of the expansion vanishes if and only if the Taub-NUT metric is indeed the flat one.     

Singular del Pezzo surfaces that are equivariant compactifications

We determine which singular del Pezzo surfaces are equivariant compactifications of \mathbbG_\texta² \mathbb{G}_{\text{a}}^2 , to assist with proofs of Manin’s conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an equivariant compactification of \mathbbG_\texta {\mathbb{G}_{\text{a}}} ⋊ \mathbbG_\textm {\mathbb{G}_{\text{m}}} . Bibliography: 32 titles.     

Relative universality and universality obtained by adding constants

A variety ${\mathbb{V}}${\mathbb{V}} is var-relatively universal if it contains a subvariety \mathbbW{\mathbb{W}} such that the class of all homomorphisms that do not factorize through any algebra in \mathbbW{\mathbb{W}} is algebraically universal. And \mathbbV{\mathbb{V}} has an algebraically universal α-expansion a\mathbbV{\alpha\mathbb{V}} if adding α nullary operations to all algebras in \mathbbV{\mathbb{V}} gives rise to a class a\mathbbV{\alpha\mathbb{V}} of algebras that is algebraically universal. The first two authors have conjectured that any varrelative universal variety \mathbbV{\mathbb{V}} has an algebraically universal α-expansion a\mathbbV{\alpha\mathbb{V}} . This note contains a more general result that proves this conjecture.     

An ODE’s Version of the Formula of Baker, Campbell, Dynkin and Hausdorff and the Construction of Lie Groups with Prescribed Lie Algebra

If ${\mathcal{L} = {\sum_{j=1}^m} {X_j^2} + X_0}If L = ?_j=1^m X_j² + X₀{\mathcal{L} = {\sum_{j=1}^m} {X_j^2} + X_0} is a H?rmander partial differential operator in \mathbbR^N{\mathbb{R}^N}, we give sufficient conditions on the vector fields X _j ’s for the existence of a Lie group structure \mathbbG = (\mathbbR^N, *){\mathbb{G} = (\mathbb{R}^N, *)} (and we exhibit its construction), not necessarily nilpotent nor homogeneous, such that L{\mathcal{L}} is left invariant on \mathbbG{\mathbb{G}}. The main tool is a formula of Baker-Campbell-Dynkin-Hausdorff type for the ODE’s naturally related to the system of vector fields {X ₀, . . . , X _m }. We provide a direct proof of this formula in the ODE’s context (which seems to be missing in literature), without invoking any result of Lie group theory, nor the abstract algebraic machinery usually involved in formulas of Baker-Campbell-Dynkin-Hausdorff type. Examples of operators to which our results apply are also furnished.     

Groups whose non-linear irreducible characters are rational valued

A finite group G all of whose nonlinear irreducible characters are rational is called a \mathbbQ₁{\mathbb{Q}_1}-group. In this paper, we obtain some results concerning the structure of \mathbbQ₁{\mathbb{Q}_1}-groups.