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In this paper we prove that the time part of the germ of an analytic family of vector fields with a Hopf bifurcation is rigid in the parameter. Time parts are associated with the temporal invariant of the analytic classification. Because the eigenvalues at zero are complex conjugate, time parts usually unfold in the hyperbolic direction, where the singular points are linearizable. We first identify the time part of a generic conformal family and prove that any weak holomorphic conjugacy between two time parts yields a biholomorphism analytic in the parameter. The existence of Fatou coordinates in both the Siegel and in the Poincare domains plays a fundamental role in the proof of this result.

In this paper we study the following "slice rigidity property": given two Kobayashi complete hyperbolic manifolds $M, N$ and a collection of complex geodesics $\mathcal F$ of $M$, when is it true that every holomorphic map $F:M\to N$ which maps isometrically every complex geodesic of $\mathcal F$ onto a complex geodesic of $N$ is a biholomorphism? Among other things, we prove that this is the case if $M, N$ are smooth bounded strictly (linearly) convex domains, every element of $\mathcal F$ contains a given point of $\overline{M}$ and $\mathcal F$ spans all of $M$. More general results are provided in dimension $2$ and for the unit ball., Comment: 19 pages - final version, to appear in J. Geom. Anal

On a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n>1$, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up to the boundary is obstructed by a local curvature invariant of the boundary. For bounded strictly pseudoconvex domains in $\mathbb{C}^2$ which are diffeomorphic to the ball, we motivate and consider the problem of determining whether the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball. In particular we observe that, up to biholomorphism, the unit ball in $\mathbb{C}^2$ is rigid with respect to deformations in the class of strictly pseudoconvex domains with obstruction flat boundary. We further show that for more general deformations of the unit ball, the order of vanishing of the obstruction equals the order of vanishing of the CR curvature. Finally, we give a generalization of the recent result of the second author that for an abstract CR manifold with transverse symmetry, obstruction flatness implies local equivalence to the CR $3$-sphere.

We study the K\"ahler geometry of stage n Bott manifolds, which can be viewed as $n$-dimensional generalizations of Hirzebruch surfaces. We show, using a simple induction argument and the generalized Calabi construction from [ACGT04,ACGT11], that any stage n Bott manifold $M_n$ admits an extremal K\"ahler metric. We also give necessary conditions for $M_n$ to admit a constant scalar curvature K\"ahler metric. We obtain more precise results for stage 3 Bott manifolds, including in particular some interesting relations with c-projective geometry and some explicit examples of almost K\"ahler structures. To place these results in context, we review and develop the topology, complex geometry and symplectic geometry of Bott manifolds. In particular, we study the K\"ahler cone, the automorphism group and the Fano condition. We also relate the number of conjugacy classes of maximal tori in the symplectomorphism group to the number of biholomorphism classes compatible with the symplectic structure., Comment: to appear in Advances in Mathematics

Isomorphisms that preserve a certain geometric structure are easily destroyed by an arbitrary small deformation of the structure, but restoring them requires a definite amount of change. Such an intuitive understanding has actually been established as precise theorems, via Lie groups, Riemannian geometry, and several complex variables. The aim of this paper is to present a concise but comprehensive exposition.

We study the dynamics of a singular holomorphic vector field at $(\mathbb{C}^{2},0)$. Using the associated flow and its pullback to the blow-up manifold, we provide invariants relating the vector field, a non-invariant analytic branch of curve, and the deformation of this branch by the flow. This leads us to study the conjugacy classes of singular branches under the action of holomorphic flows. In particular, we show that there exists an analytic class that is not complete, meaning that there are two elements of the class that are not analytically conjugated by a local biholomorphism embedded in a one-parameter flow. Our techniques are new and offer an approach dual to the one used classically to study singularities of holomorphic vector fields.

This paper is concerned with the biholomorphism of two equidimensional Hartogs type domains over irreducible bounded symmetric domains $H_{\Omega}(p)$ (see (1.1)) which is a Hua construction of Cartan–Hartogs domain. We develop a new simple methods to give an sufficient and necessary condition for the two Hua domains to be biholomorphic equivalent by using the function $\mathcal{L}_{\Omega}(z,\omega)$ (see (1.2)). Furthermore as an application, we can also give an equivalent description for the automorphism of Hua domain.

It is known that if $$f: D_1 \rightarrow D_2$$ is a polynomial biholomorphism with polynomial inverse and constant Jacobian then $$D_1$$ is a 1-point poly-quadrature domain (the Bergman span contains all holomorphic polynomials) of order 1 whenever $$D_2$$ is a complete circular domain. Bell conjectured that all 1-point poly-quadrature domains arise in this manner. In this note, we construct a 1-point poly-quadrature domain of order 1 that is not biholomorphic to any complete circular domain.

We prove that the holonomy map from the set of equivalence classes of parabolic complex projective structures on compact surfaces with finitely many punctures to the set of equivalence classes of parabolic representations of the fundamental group of the surface to $$PSL_2(\mathbb {C})$$ is a local biholomorphism.

By considering intrinsic geometric conditions, we introduce a new class of domains in complex Euclidean space. This class is invariant under biholomorphism and includes strongly pseudoconvex domains, finite type domains in dimension two, convex domains, C -convex domains, and homogeneous domains. For this class of domains, we show that compactness of the ∂ ¯ -Neumann operator on ( 0 , q ) -forms is equivalent to the boundary not containing any q-dimensional analytic varieties (assuming only that the boundary is a topological submanifold). We also prove, for this class of domains, that the Bergman metric is equivalent to the Kobayashi metric and that the pluricomplex Green function satisfies certain local estimates in terms of the Bergman metric.

Let $G_i$ be a closed Lie subgroup of $U(n)$, $\Omega_i$ be a bounded $G_i$-invariant domain in $\mathbb{C}^n$ which contains~$0$, and $\mathcal{O}(\mathbb{C}^n)^{G_i}=\mathbb{C}$, for $i=1,2$. If $f:\Omega_1\rightarrow\Omega_2$ is a biholomorphism, and $f(0)=0$, then $f$ is a polynomial mapping (see Ning et al. (2017)). In this paper, we provide an upper bound for the degree of such polynomial mappings. It is a natural generalization of the well-known Cartans theorem.

Let $X$ be a compact connected Riemann surface of genus at least two and $G$ a connected reductive complex affine algebraic group. The Riemann--Hilbert correspondence produces a biholomorphism between the moduli space ${\mathcal M}_X(G)$ parametrizing holomorphic $G$--connections on $X$ and the $G$--character variety $${\mathcal R}(G):= \text{Hom}(\pi_1(X, x_0), G)/\!\!/G\, .$$ While ${\mathcal R}(G)$ is known to be affine, we show that ${\mathcal M}_X(G)$ is not affine. The scheme ${\mathcal R}(G)$ has an algebraic symplectic form constructed by Goldman. We construct an algebraic symplectic form on ${\mathcal M}_X(G)$ with the property that the Riemann--Hilbert correspondence pulls back to the Goldman symplectic form to it. Therefore, despite the Riemann--Hilbert correspondence being non-algebraic, the pullback of the Goldman symplectic form by the Riemann--Hilbert correspondence nevertheless continues to be algebraic., Comment: Final version

Let $G/P$ be a rational homogeneous space (not necessarily irreducible) and $x_0\in G/P$ be the point at which the isotropy group is $P$. The $G$-translates of the orbit $Qx_0$ of a parabolic subgroup $Q\subsetneq G$ such that $P\cap Q$ is parabolic are called $Q$-cycles. We established an extension theorem for local biholomorphisms on $G/P$ that map local pieces of $Q$-cycles into $Q$-cycles. We showed that such maps extend to global biholomorphisms of $G/P$ if $G/P$ is $Q$-cycle-connected, or equivalently, if there does not exist a non-trivial parabolic subgroup containing $P$ and $Q$. Then we applied this to the study of local biholomorphisms preserving the real group orbits on $G/P$ and showed that such a map extend to a global biholomorphism if the real group orbit admits a non-trivial holomorphic cover by the $Q$-cycles. The non-closed boundary orbits of a bounded symmetric domain embedded in its compact dual are examples of such real group orbits. Finally, using the results of Mok-Zhang on Schubert rigidity, we also established a Cartan-Fubini type extension theorem pertaining to $Q$-cycles, saying that if a local biholomorphism preserves the variety of tangent spaces of $Q$-cycles, then it extends to a global biholomorphism when the $Q$-cycles are positive dimensional and $G/P$ is of Picard number 1. This generalizes a well-known theorem of Hwang-Mok on minimal rational curves., Comment: 28 pages

Let $X$ and $Y$ be compact connected complex manifolds of the same dimension with $b_2(X)= b_2(Y)$. We prove that any surjective holomorphic map of degree one from $X$ to $Y$ is a biholomorphism. A version of this was established by the first two authors, but under an extra assumption that $\dim H^1(X {\mathcal O}_X)\,=\,\dim H^1(Y {\mathcal O}_Y)$. We show that this condition is actually automatically satisfied., Complex Variables and Elliptic Equations (to appear)

Let $f:M\rightarrow M$ be a biholomorphism on a two-dimensional complex manifold, and let $X\subseteq M$ be a compact $f$-invariant set such that $f|_{X}$ is asymptotically dissipative and without periodic sinks. We introduce a solely dynamical obstruction to dominated splitting, namely critical point. Critical point is a dynamical object and captures many of the dynamical properties of a one-dimensional critical point.

We found that in the polydiskDnthere exist(n+1)(n+2)/2different classes of commutativeC⁎-algebras generated by Toeplitz operators whose symbols are invariant under the action of maximal Abelian subgroups of biholomorphisms. On the other hand, using the moment map associated with each (not necessary maximal) Abelian subgroup of biholomorphism we introduced a family of symbols given by the moment map such that theC⁎-algebra generated by Toeplitz operators with this kind of symbol is commutative. Thus we relate to each Abelian subgroup of biholomorphisms a commutativeC⁎-algebra of Toeplitz operators.

Given a noncommutative (nc) variety $\mathfrak{V}$ in the nc unit ball $\mathfrak{B}_d$, we consider the algebra $H^\infty(\mathfrak{V})$ of bounded nc holomorphic functions on $\mathfrak{V}$. We investigate the problem of when two algebras $H^\infty(\mathfrak{V})$ and $H^\infty(\mathfrak{W})$ are isomorphic. We prove that these algebras are weak-$*$ continuously isomorphic if and only if there is an nc biholomorphism $G : \widetilde{\mathfrak{W}} \to \widetilde{\mathfrak{V}}$ between the similarity envelopes that is bi-Lipschitz with respect to the free pseudo-hyperbolic metric. Moreover, such an isomorphism always has the form $f \mapsto f \circ G$, where $G$ is an nc biholomorphism. These results also shed some new light on automorphisms of the noncommutative analytic Toeplitz algebras $H^\infty(\mathfrak{B}_d)$ studied by Davidson--Pitts and by Popescu. In particular, we find that $\operatorname{Aut}(H^\infty(\mathfrak{B}_d))$ is a proper subgroup of $\operatorname{Aut}(\widetilde{\mathfrak{B}}_d)$. When $d, Comment: 45 pages. Some details were added and more minor changes

V svoji diplomski nalogi sem se ukvarjal z mehaniko tekočin. V prvem delu sem svojo analizo omejil na idealne tekočine, fluide, katerih tokovnice lahko ponazorimo z dvodimenzionalnimi vektorskimi polji brez izvorov in vrtincev. Dokazal sem, da lahko le-te povežemo s teorijo holomorfnih funkcij, ter nato s pomočjo Riemannovega upodobitvenega izreka tokove okoli zapletenih objektov reduciramo na nekatere elementarne primere. V zadnjem delu naloge sem nato dodatno predstavil tudi Blasiousov izrek in nekaj primerov ne-idealnih tokov, ki pa generirajo vzgon. In my thesis I dealt with fluid mechanics. In first part of my analysis I focused on ideal fluids. These are fluids, whose streamlines can be represented with twodimenzional vector fields without sources and vortices. I proved that such vector fields can be connected to theory of holomorphic functions. Using Riemann mapping theorem, flows around complicated objects can be reduced to elementary examples. In last part I also additionally presented Blasius theorem and some non-ideal flow examples, which do produce lift.

We give a description of biholomorphisms of quasi-Reinhardt domains fixing the origin via Bergman representative coordinates, which are shown to be polynomial mappings with a degree bound given by the so-called "resonance order"., 5 pages

We describe the relationship between two spaces associated to a quiver with potential. The first is a complex manifold parametrizing Bridgeland stability conditions on a triangulated category, and the second is a cluster variety with a natural Poisson structure. For quivers of type $A$, we construct a local biholomorphism from the space of stability conditions to the cluster variety. The existence of this map follows from results of Sibuya in the classical theory of ordinary differential equations., 33 pages. Version 2: Incorporated suggestions from referee

In this paper, we characterize the Hartogs domains over homogeneous Siegel domains of type II and explicitly describe their automorphism groups. Moreover we prove that any proper holomorphic map between Hartogs domains over homogeneous Siegel domains over type II is a biholomorphism., Comment: 10 pages

Let M 3 ⊂ C 2 be a three times differentiable real hypersurface. The Levi form of M transforms under biholomorphism, and when restricted to the complex tangent space, the skew-hermitian part of the second fundamental form transforms under fractional linear transformation. The surfaces for which these forms are constant multiples of each other were identified in previous work, but when the constant had unit modulus there was a global requirement. Here we give a local characterization of hypersurfaces for which the constant has unit modulus.

We prove that the one-parameter group of holomorphic automorphisms induced on a strictly geometrically bounded domain by a biholomorphism with a model domain is parabolic. This result is related to the Greene-Krantz conjecture and more generally to the classification of domains having a non compact automorphisms group. The proof relies on elementary estimates on the Kobayashi pseudo-metric.

In this paper we study the semicontinuity of the automor-phism groups of domains in multi-dimensional complex space. We giveexamples to show that known results are sharp (in terms of the requiredboundary smoothness). 1. IntroductionThepaper [4] wasthe firstworktostudy the semicontinuityofautomorphismgroups of domains in complex space. The main result there is as follows:Theorem 1.1. LetΩ ∗ ⊆ C n be a strongly pseudoconvex domain with smoothboundary. ThenthereisaneighborhoodU ofΩ ∗ intheC ∞ topologyondomains(thatistosay,U isacollectionofdomains) sothat,ifΩ ∈ U,thenAut(Ω) isasubgroupofAut(Ω ∗ ). Moreover, thereisaC ∞ mappingΨ fromΩ toΩ 0 sothatAut(Ω) ∋ ϕ −→ Ψ◦ϕ ◦Ψ −1 isaninjectivegrouphomomorphism fromAut(Ω) toAut(Ω ∗ ).Over the years, the hypothesis of smooth or C ∞ boundary in this theoremhas been weakened. In the paper [5], the hypothesis was weakened (using anentirely different argument) to C 2 boundary smoothness. In the paper [3],yet another approach to the C 2 boundary smoothness situation was described.The paper [2] treats the case of C

A theorem due to Cartan asserts that every origin-preserving automorphism of bounded circular domains with respect to the origin is linear. In the present paper, by employing the theory of Bergman's representative domain, we prove that under certain circumstances Cartan's assertion remains true for quasi-circular domains in C n . Our main result is applied to obtain some simple criterions for the case n = 3 and to prove that Braun–Kaup–Upmeier's theorem remains true for our class of quasi-circular domains.

Let \(X\) be a Stein manifold of complex dimension at least two, \(F:X \rightarrow {{\mathbb {C}}}^n\) a local biholomorphism, and \(q\in F(X)\). In this paper we formulate sufficient conditions, involving only objects naturally associated to \(q\), in order for the fiber over \(q\) to be finite. Assume that \(F^{-1}(l)\) is \(1\)-connected for the generic complex line \(l\) containing \(q\), and \(F^{-1}(l)\) has finitely many components whenever \(l\) is an exceptional line through \(q\). Using arguments from topology and differential geometry, we establish a sharp estimate on the size of \(F^{-1}(q)\). It follows that for \(n\ge 2\) a local biholomorphism of \(X\) onto \({{\mathbb {C}}}^n\) is invertible if and only if the pull-back of every complex line is \(1\)-connected.

Let $D$ be an irreducible bounded symmetric domain with biholomorphism group $G$ with maximal compact subgroup $K$. For the Toeplitz operators with $K$-invariant symbols we provide explicit simultaneous diagonalization formulas on every weighted Bergman space. The expressions are given in the general case, but are also worked out explicitly for every irreducible bounded symmetric domain including the exceptional ones.

We show that a compact complex surface which admits a conformally K\"ahler metric g of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane. In addition, if g is a Hermitian metric which is Einstein, then the biholomorphism can be chosen to be an isometry via which g becomes a multiple of the Fubini-Study metric., Comment: 12 pages. Journal information added

Proper holomorphic mappings among domains on Euclidean spaces is a classical topic in Several Complex Variables. The literature can date back to the earliest results like the theorem of H. Alexander [1] which says that any proper holomorphic self-map of the complex unit n-ball is a biholomorphism if n ≥ 2. Since then, the study of the proper holomorphic mappings between complex unit balls of different dimensions has become a very popular topic in the field. Many important inputs from various perspectives have been made, like Algebraic Geometry, Chern-Moser Theory, Segre variety and Bergman kernel, etc. It is apparent by now that the complexity of the problem grows with the codimension and one in general must impose certain regularity assumptions on the proper maps in order to give any satisfactory classification.

We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions M V \mathcal {M}_V of the multiplier algebra M \mathcal {M} of Drury-Arveson space to a holomorphic subvariety V V of the unit ball B d \mathbb {B}_d . We find that M V \mathcal {M}_V is completely isometrically isomorphic to M W \mathcal {M}_W if and only if W W is the image of V V under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthened to show that when d > ∞ d>\infty every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When V V and W W are each a finite union of irreducible varieties and a discrete variety, when d > ∞ d>\infty , an isomorphism between M V \mathcal {M}_V and M W \mathcal {M}_W determines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak- ∗ * continuous. We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold—particularly, smooth curves and Blaschke sequences. We also discuss the norm closed algebras associated to a variety, and point out some of the differences.

For a large class of non smooth bounded domains, existence of a global weak solution of the 2D Euler equations, with bounded vorticity, was established by G\'erard-Varet and Lacave. In the case of sharp domains, the question of uniqueness for such weak solutions is more involved due to the bad behavior of $\Delta^{-1}$ close to the boundary. In the present work, we show uniqueness for any bounded and simply connected domain with a finite number of corners of angles smaller than $\pi/2$. Our strategy relies on a log-Lipschitz type regularity for the velocity field.

A classical theorem of Bun Wong implies that a strongly pseudoconvex domain with transitive automorphism group must be biholomorphic to the unit ball. This result has been quite influential, and has been extended and modified in a number of fascinating ways. We discuss several variations and implications of this theorem, and present some new results as well.

We introduce the notion of a stratified Oka manifold and prove that such a manifold $X$ is strongly dominable in the sense that for every $x\in X$, there is a holomorphic map $f:\C^n\to X$, $n=\dim X$, such that $f(0)=x$ and $f$ is a local biholomorphism at 0. We deduce that every Kummer surface is strongly dominable. We determine which minimal compact complex surfaces of class VII are Oka, assuming the global spherical shell conjecture. We deduce that the Oka property and several weaker holomorphic flexibility properties are in general not closed in families of compact complex manifolds. Finally, we consider the behaviour of the Oka property under blowing up and blowing down., Comment: Version 2: Theorem 11 reformulated and its proof corrected. Minor improvements to the exposition. Version 3: A few minor improvements. To appear in International Mathematics Research Notices

Given a polynomial f of degree d defined over a complete local field, we construct a biholomorphic change of variables defined in a neighbourhood of infinity which transforms the action z->f(z) to the multiplicative action z->z^d. The relation between this construction and the Bottcher coordinate in complex polynomial dynamics is similar to the relation between the complex uniformization of elliptic curves, and Tate's p-adic uniformization. Specifically, this biholomorphism is Galois equivariant, reducing certain questions about the Galois theory of preimages by f to questions about multiplicative Kummer theory.

Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X\colon X\to Q_X$ and $p_Y\colon Y\to Q_Y$ be the quotient mappings. Assume that we have a biholomorphism $Q:= Q_X\to Q_Y$ and an open cover $\{U_i\}$ of $Q$ and $G$-biholomorphisms $\Phi_i\colon p_X^{-1}(U_i)\to p_Y^{-1}(U_i)$ inducing the identity on $U_i$. There is a sheaf of groups $\mathcal A$ on $Q$ such that the isomorphism classes of all possible $Y$ is the cohomology set $H^1(Q,\mathcal A)$. The main question we address is to what extent $H^1(Q,\mathcal A)$ contains only topological information. For example, if $G$ acts freely on $X$ and $Y$, then $X$ and $Y$ are principal $G$-bundles over $Q$, and Grauert's Oka Principle says that the set of isomorphism classes of holomorphic principal $G$-bundles over $Q$ is canonically the same as the set of isomorphism classes of topological principal $G$-bundles over $Q$. We investigate to what extent we have an Oka principle for $H^1(Q,\mathcal A)$., Comment: 12 pages, minor changes, to appear in Indiana University Math. J

The purpose of this paper is to study the local behavior of semi-attractive holomorphic self-maps of C (m > 2) in a neighborhood of a fixed point that we assume to be the origin. Such transformations are the ones whose differential at 0 has one eigenvalue equal to 1 while the remaining ones, say β1, . . . , βs with s ≥ 1, have modulus strictly less than 1. Semi-attractive transformations such that 0 is not an isolated fixed point have been studied by Nishimura [N], who considered analytic automorphisms F of complex manifolds admitting a q-dimensional complex submanifoldM of attracting fixed points for F. Then, for each point p0 ∈M, one can choose local coordinates (w, z) ∈ C × Cm−q in a neighborhood U of p0 such that U ∩ M has equation z = 0; hence the map F can be locally written as w1 = w +O(‖z‖), z1 = C(w)z+O(‖z‖2), whereC(w) is a (m−q)× (m−q)matrix whose elements are holomorphic functions onU and whose eigenvalues β1(w), . . . , βm−q(w) have modulus strictly less than 1. Let  = {p ∈U | F n(p)→ p0, p0 ∈M}, which is an open set containing M. Nishimura proved that if these eigenvalues have no relations in any point of M, that is, if for each multi-index j = (j1, . . . , jm−q) with∑m−q k=1 jk ≥ 2 and 1≤ i ≤ m− q we have β1 1 (w) · · ·βm−q m−q (w) 6= βi(w), then there exists a biholomorphism S:N → , where π :N → M is the normal bundle of M, which conjugates F to the automorphism G of N induced by F and given in π−1(U ∩M) by s1 = s, u1 = C(s)u.

We study a germ of real analytic n-dimensional submanifold of C n that has a complex tangent space of maximal dimension at a CR singularity. Under some assumptions , we show its equivalence to a normal form under a local biholomorphism at the singularity. We also show that if a real submanifold is formally equivalent to a quadric, it is actually holomorphically equivalent to it, if a small divisors condition is satisfied. Finally, we investigate the existence of a complex submanifold of positive dimension in C n that intersects a real submanifold along two totally and real analytic submanifolds that intersect transversally at a possibly non-isolated CR singularity., To appear in Invent. Math. arXiv admin note: substantial text overlap with arXiv:1406.1294

Let $X$ be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let $L$ be an ample line bundle on $X$. Assume that the pair $(X,L)$ is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point $x \in X$ there exist a Kahler-Einstein Fano manifold $Z$ and a positive integer $q$ dividing $K_Z$ such that $-\frac{1}{q}K_Z$ is very ample and such that the germ $(X,x)$ is locally analytically isomorphic to a neighborhood of the vertex of the blow-down of the zero section of $\frac{1}{q}K_{Z}$. We prove that up to biholomorphism, the unique weak Ricci-flat Kahler metric representing $2\pi c_1(L)$ on $X$ is asymptotic at a polynomial rate near $x$ to the natural Ricci-flat Kahler cone metric on $\frac{1}{q}K_Z$ constructed using the Calabi ansatz. In particular, our result applies if $(X, \mathcal{O}(1))$ is a nodal quintic threefold in $\mathbb{P}^4$. This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities., Comment: 41 pages, added a short appendix on special Lagrangian vanishing cycles

We prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action. These inverse spectral results involve the asymptotic equivariant spectrum, which is the spectrum itself together with "very large" weights of the torus action on eigenspaces. More precisely, we show that the asymptotic equivariant spectrum of the Laplace operator of any toric metric on a generic toric orbifold determines the equivariant biholomorphism class of the orbifold; we also show that the asymptotic equivariant spectrum of a T^n-invariant Schrodinger operator on R^n determines its potential in some suitably convex cases. In addition, we prove that the asymptotic equivariant spectrum of an S^1-invariant metric on S^2 determines the metric itself in many cases. Finally, we obtain an asymptotic equivariant inverse spectral result for weighted projective spaces. As a crucial ingredient in these inverse results, we derive a surprisingly simple formula for the asymptotic equivariant trace of a family of semi-classical differential operators invariant under a torus action., Comment: 35 pages

Let X1 and X2 be two compact strongly pseudoconvex CR manifolds of dimension 2n − 1 ≥ 5 which bound complex varieties V1 and V2 with only isolated normal singularities in CN1 and CN2 respectively. Let S1 and S2 be the singular sets of V1 and V2 respectively and assume S2 is non-empty. If 2n − N2 − 1 ≥ 1 and the cardinality of S1 is less than twice that of S2, then we prove that any non-constant CR morphism from X1 to X2 is necessarily a CR biholomorphism. On the other hand, let X be a compact strongly pseudoconvex CR manifold of dimension 3 which bounds a complex variety V with only isolated normal non-quotient singularities. Assume that the singular set of V is non-empty. Then we prove that any non-constant CR morphism from X to X is necessarily a CR biholomorphism.