Moduli Spaces of Maslov Complex Germs

Maslov complex germs (complex vector bundles, satisfying certain additional conditions, over isotropic submanifolds of the phase space) are one of the central objects in the theory of semiclassical quantization. To these bundles one assigns spectral series (quasimodes) of partial differential operators. We describe the moduli spaces of Maslov complex germs over a point and a closed trajectory and find the moduli of complex germs generated by a given symplectic connection over an invariant torus.     

Moduli Space of Fedosov Structures

We consider the space of germs of Fedosov structures at a point, under the action of origin-preserving diffeomorphisms. We calculate dimensions of moduli spaces of k-jets of generic structures and construct the Poincaré series of the moduli space. It is shown to be a rational function.Mathematics Subject Classifications (2000): primary: 53A55; secondary: 53B15, 53D15, 58J60.     

Kähler geometry of Douady spaces

We consider the generalized Petersson–Weil metric on the moduli space of compact submanifolds of a Kähler manifold or a projective variety. It is extended as a positive current to the space of points corresponding to reduced fibers, and estimates are shown. For moduli of projective varieties the Petersson–Weil form is the curvature of a certain determinant line bundle equipped with a Quillen metric. We investigate its extension to the compactified moduli space.     

On Simple ℤ2-Invariant and Corner Function Germs

Mathematical Notes - V. I. Arnold has classified simple (i.e., having no moduli for the classification) singularities (function germs), and also simple boundary singularities: function germs...     

Holomorphic germs on certain locally convex spaces

Summary We study the bounded sets in the space of holomorphic germs defined on compact subsets of non-metrizable locally convex spaces. We relate this problem to the problem of existence of uniform Cauchy estimates for the bounded subsets. We show that the space of holomorphic germs defined on a compact subset of a reflexive dual Fréchet space is regular if the bounded subsets of the space of holomorphic germs defined at the origin have uniform Cauchy estimates.     

Fredholm theory for pseudoholomorphic curves with brake symmetry

We study the pseudoholomorphic curves with brake symmetry in symplectization of a closed contact manifold. We introduce the pseudo-holomorphic curves with brake symmetry and the corresponding moduli space. Then we get the virtual dimension of the moduli space.     

The $${\textit{SU}}(2)$$-character variety of the closed surface of genus 2

We study the symplectic geometry of the \({\text {SU}}(2)\)-representation variety of the compact oriented surface of genus 2. We use the Goldman flows to identify subsets of the moduli space with corresponding subsets of \({\mathbb {P}}^3(\mathbb {C})\). We also define and study two antisymplectic involutions on the moduli space and their fixed point sets.     

On Simple $${\mathbb Z}_3$$ -Invariant Function Germs

Functional Analysis and Its Applications - V. I. Arnold classified simple (i.e., having no moduli for classification) singularities (function germs) and also simple boundary singularities, that is,...     

Cohomology of the moduli of parabolic vector bundles

The purpose of this paper is to compute the Betti numbers of the moduli space ofparabolic vector bundles on a curve (see Seshadri [7], [8] and Mehta & Seshadri [4]), in the case where every semi-stable parabolic bundle is necessarily stable. We do this by generalizing the method of Atiyah and Bott [1] in the case of moduli of ordinary vector bundles. Recall that (see Seshadri [7]) the underlying topological space of the moduli of parabolic vector bundles is the space of equivalence classes of certain unitary representations of a discrete subgroup Γ which is a lattice in PSL (2,R). (The lattice Γ need not necessarily be co-compact). While the structure of the proof is essentially the same as that of Atiyah and Bott, there are some difficulties of a technical nature in the parabolic case. For instance the Harder-Narasimhan stratification has to be further refined in order to get the connected strata. These connected strata turn out to have different codimensions even when they are part of the same Harder-Narasimhan strata. If in addition to ‘stable = semistable’ the rank and degree are coprime, then the moduli space turns out to be torsion-free in its cohomology. The arrangement of the paper is as follows. In § 1 we prove the necessary basic results about algebraic families of parabolic bundles. These are generalizations of the corresponding results proved by Shatz [9]. Following this, in § 2 we generalize the analytical part of the argument of Atiyah and Bott (§ 14 of [1]). Finally in § 3 we show how to obtain an inductive formula for the Betti numbers of the moduli space. We illustrate our method by computing explicitly the Betti numbers in the special case of rank = 2, and one parabolic point.     

Moduli of reflexive sheaves on smooth projective 3-folds

We compute the expected dimension of the moduli space of torsion-free rank 2 sheaves at a point corresponding to a stable reflexive sheaf, and give conditions for the existence of a perfect tangent-obstruction complex on a class of smooth projective threefolds; this class includes Fano and Calabi-Yau threefolds. We also explore both local and global relationships between moduli spaces of reflexive rank 2 sheaves and the Hilbert scheme of curves.     

Hyperelliptic Surfaces and their Moduli

The hyperelliptic portion of the moduli space of compact Riemann surfaces of genus g2 is decomposed into a lattice of nondisjoint subvarieties corresponding precisely with the lattice of maximal g-hyperelliptic group actions (classified up to topological equivalence). The resulting stratification of the hyperelliptic moduli space exhibits regularities which depend on the parity of g and can be detected at the level of groups of order 8.     

On the monodromy of the moduli space of Calabi–Yau threefolds coming from eight planes in {\mathbb{P}^3}

It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi–Yau threefolds coming from eight planes in ${\mathbb{P}^3}$ does not have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is naturally isomorphic to wedge product of a variation of Hodge structures of weight one. It turns out the hyperelliptic locus does not extend to a Shimura subvariety of type III (Siegel space) within the moduli space. Besides general Hodge theory, representation theory and computational commutative algebra, one of the proofs depends on a new result on the tensor product decomposition of complex polarized variations of Hodge structures.     

Weil-Petersson Areas of The Moduli Spaces Of Tori

We study the Teichmüller spaces of torus with one branch point of order v and of torus with a totally geodesic boundary curve of length m, respectively. Applying the obtained results for the corresponding moduli spaces we find that the Weil-Petersson area of the moduli space of torus with one conical point of order v is (π²/6)(1 - l/v²) and that of the moduli space of torus with a totally geodesic boundary curve of length m is π²/6 + m²/24.     

Moduli of theta-characteristics via Nikulin surfaces

We study moduli spaces of K3 surfaces endowed with a Nikulin involution and their image in the moduli space R _g of Prym curves of genus g. We observe a striking analogy with Mukai’s well-known work on ordinary K3 surfaces. Many of Mukai’s results have a very precise Prym-Nikulin analogue, for instance a general Prym curve from R _g is a section of a Nikulin surface if and only if g ≤ 7 and g ≠ 6. Furthermore, R ₇ has the structure of a fibre space over the corresponding moduli space of polarized Nikulin surfaces. We then use these results to study the geometry of the moduli space of even spin curves, with special emphasis on the transition case of which is a 21-dimensional Calabi-Yau variety.     

Geometric aspects of the moduli space of riemann surfaces

We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kähler metrics were introduced on the moduli space and Teichmüller space. Their curvature properties and asymptotic behavior were studied in details. These natural metrics served as bridges to connect all the known canonical metrics, especially the Kähler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincaré type growth. Furthermore, the Kähler-Einstein metric has strongly bounded geometry. This also implied that the logarithm cotangent bundle of the moduli space is stable in the sense of Mumford.