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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.
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Special Bohr - Sommerfeld geometry, first formulated for simply connected symplectic manifolds (or for simple connected algebraic varieties), gives rise to some natural problems for the simplest example in non simply connected case. Namely for any algebraic curve one can define a correspondence between holomorphic differentials and certain finite graphs. Here we ask some natural questions appear with this correspondence. It is a partial answer to the question of A. Varchenko about possibility of applications of Special Bohr -Sommerfeld geometry in non simply connected case. The russian version has been translated.
An electrical potential U on a bordered Riemann surface X with conductivity function sigma>0 satisfies equation d(sigma d^cU)=0. The problem of effective reconstruction of sigma is studied. We extend to the case of Riemann surfaces the reconstruction scheme given, firstly, by R.Novikov (1988) for simply connected X. We apply for this new kernels for dbar on affine algebraic Riemann surfaces constructed in Henkin, arXiv:0804.3761
We extend the Boutet de Monvel Toeplitz index theorem to complex manifold with isolated singularities following the relative $K$-homology theory of Baum, Douglas, and Taylor for manifold with boundary. We apply this index theorem to study the Arveson-Douglas conjecture. Let $ball^m$ be the unit ball in $mathbb{C}^m$, and $I$ an ideal in the polynomial algebra $mathbb{C}[z_1, cdots, z_m]$. We prove that when the zero variety $Z_I$ is a complete intersection space with only isolated singularities and intersects with the unit sphere $mathbb{S}^{2m-1}$ transversely, the representations of $mathbb{C}[z_1, cdots, z_m]$ on the closure of $I$ in $L^2_a(ball^m)$ and also the corresponding quotient space $Q_I$ are essentially normal. Furthermore, we prove an index theorem for Toeplitz operators on $Q_I$ by showing that the representation of $mathbb{C}[z_1, cdots, z_m]$ on the quotient space $Q_I$ gives the fundamental class of the boundary $Z_Icap mathbb{S}^{2m-1}$. In the appendix, we prove with Kai Wang that if $fin L^2_a(ball^m)$ vanishes on $Z_Icap ball ^m$, then $f$ is contained inside the closure of the ideal $I$ in $L^2_a(ball^m)$.
Once first answers in any dimension to the Green-Griffiths and Kobayashi conjectures for generic algebraic hypersurfaces $mathbb{X}^{n-1} subset mathbb{P}^n(mathbb{C})$ have been reached, the principal goal is to decrease (to improve) the degree bounds, knowing that the `celestial horizon lies near $d geqslant 2n$. For Green-Griffiths algebraic degeneracy of entire holomorphic curves, we obtain: [ d ,geqslant, big(sqrt{n},{sf log},nbig)^n, ] and for Kobayashi-hyperbolicity (constancy of entire curves), we obtain: [ d ,geqslant, big(n,{sf log},nbig)^n. ] The latter improves $d geqslant n^{2n}$ obtained by Merker in arxiv.org/1807/11309/. Admitting a certain technical conjecture $I_0 geqslant widetilde{I}_0$, the method employed (Diverio-Merker-Rousseau, Berczi, Darondeau) conducts to constant power $n$, namely to: [ d ,geqslant, 2^{5n} qquad text{and, respectively, to:} qquad d ,geqslant, 4^{5n}. ] In Spring 2019, a forthcoming prepublication based on intensive computer explorations will present several subconjectures supporting the belief that $I_0 geqslant widetilde{I}_0$, a conjecture which will be established up to dimension $n = 50$.
A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quotient of a nilpotent Lie group equipped with a left-invariant hypercomplex structure. Such a manifold admits a whole 2-dimensional sphere $S^2$ of complex structures induced by quaternions. We prove that for any hypercomplex nilmanifold $M$ and a generic complex structure $Lin S^2$, the complex manifold $(M,L)$ has algebraic dimension 0. A stronger result is proven when the hypercomplex nilmanifold is abelian. Consider the Lie algebra of left-invariant vector fields of Hodge type (1,0) on the corresponding nilpotent Lie group with respect to some complex structure $Iin S^2$. A hypercomplex nilmanifold is called abelian when this Lie algebra is abelian. We prove that all complex subvarieties of $(M,L)$ for generic $Lin S^2$ on a hypercomplex abelian nilmanifold are also hypercomplex nilmanifolds.
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