We give inﬁnite series of groups Γ and of compact complex surfaces of general type S with fundamental group Γ such that 1) Any surface S with the same Euler number as S, and fundamental group Γ, is diﬀeomorphic to S. 2) The moduli space of S consists of exactly two connected components, exchanged by complex conjugation. Whence, i) On the one hand we give simple counterexamples to the DEF = DIFF question whether deformation type and diﬀeomorphism type coincide for algebraic surfaces. ii) On the other hand we get examples of moduli spaces without real points. iii)...
Let G be a connected reductive group. The late Ramanathan gave a notion of (semi)stable principal G-bundle on a Riemann surface and constructed a projective moduli space of such objects. We generalize Ramanathan’s notion and construction to higher dimension, allowing also objects which we call semistable principal G-sheaves, in order to obtain a projective moduli space: a principal G-sheaf on a projective variety X is a triple (P, E, ψ), where E is a torsion free sheaf on X, P is a principal G-bundle on the open set U where E is locally free and ψ is an isomorphism...
By means of analytic methods the quasi-projectivity of the moduli space of algebraically polarized varieties with a not necessarily reduced complex structure is proven including the case of nonuniruled polarized varieties. Contents Introduction Singular hermitian metrics Deformation theory of framed manifolds; V -structures Cyclic coverings Canonically polarized framed manifolds Singular Hermitian metrics for families of canonically polarized framed manifolds 7.
We show that every smooth toric variety (and many other algebraic spaces as well) can be realized as a moduli space for smooth, projective, polarized varieties. Some of these are not quasi-projective. This contradicts a recent paper (Quasi-projectivity of moduli spaces of polarized varieties, Ann. of Math. 159 (2004) 597–639.). A polarized variety is a pair (X, H) consisting of a smooth projective variety X and a linear equivalence class of ample divisors H on X. For simplicity, we look at the case when X is smooth, numerical and linear equivalence coincide for divisors on X, H is very...
Annals of Mathematics
By Curtis T. McMullen*
.Annals of Mathematics, 165 (2007), 397–456
Dynamics of SL2(R) over moduli space in genus two
By Curtis T. McMullen*
Abstract This paper classiﬁes orbit closures and invariant measures for the natural action of SL2 (R) on ΩM2 , the bundle of holomorphic 1-forms over the moduli space of Riemann surfaces of genus two. Contents 1. Introduction 2. Dynamics and Lie groups 3. Riemann surfaces and holomorphic 1-forms 4. Abelian varieties with real multiplication 5. Recognizing eigenforms 6. Algebraic sums of 1-forms 7.
D. Mumford conjectured in  that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes κi of dimension 2i. For the purpose of calculating rational cohomology, one may replace the stable moduli space of Riemann surfaces by BΓ∞ , where Γ∞ is the group of isotopy classes of automorphisms of a smooth oriented connected surface of “large” genus.
We prove Maruyama’s conjecture on the boundedness of slope semistable sheaves on a projective variety deﬁned over a noetherian ring. Our approach also gives a new proof of the boundedness for varieties deﬁned over a characteristic zero ﬁeld. This result implies that in mixed characteristic the moduli spaces of Gieseker semistable sheaves are projective schemes of ﬁnite type. The proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant.
The purpose of this paper is to carry out the abelianization program proposed by Atiyah  and Hitchin  for the geometric quantization of SU(2) Wess-Zumino-Witten model. Let C be a Riemann surface of genus g. Let Mg be the moduli space of semi-stable rank 2 holomorphic vector bundles on C with trivial determinant. For a positive integer k, let Γ(Mg , Lk ) be the space of holomorphic sections of the k-th tensor product of the determinant line bundle L on Mg . An element of Γ(Mg , Lk ) is called a rank 2 theta function...
Given a holomorphic vector bundle E over a compact K¨hler manifold X, a one deﬁnes twisted Gromov-Witten invariants of X to be intersection numbers in moduli spaces of stable maps f : Σ → X with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle H 0 (Σ, f ∗ E) H 1 (Σ, f ∗ E). Using the formalism of quantized quadratic Hamiltonians , we express the descendant potential for the twisted theory in terms of that for X. ...
For a transversal pair of closed Lagrangian submanifolds L, L of a symplectic manifold M such that π1 (L) = π1 (L ) = 0 = c1 |π2 (M ) = ω|π2 (M ) and for a generic almost complex structure J, we construct an invariant with a high homotopical content which consists in the pages of order ≥ 2 of a spectral sequence whose diﬀerentials provide an algebraic measure of the highdimensional moduli spaces of pseudo-holomorpic strips of ﬁnite energy that join L and L . When L and L are Hamiltonian isotopic, we show that the pages...
In this paper, we study the growth of sX(L), the number of simple closed
geodesics of length ≤ L on a complete hyperbolic surface X of finite area.
We also study the frequencies of different types of simple closed geodesics on
X and their relationship with the Weil-Petersson volumes of moduli spaces of
bordered Riemann surfaces.
We prove a closed formula for integrals of the cotangent line classes against the top Chern class of the Hodge bundle on the moduli space of stable pointed curves. These integrals are computed via relations obtained from virtual localization in Gromov-Witten theory. An analysis of several natural matrices indexed by partitions is required.
0. Introduction 0.1. Overview. Let Mg,n denote the moduli space of nonsingular genus g curves with n distinct marked points (over C).