We introduce an effective method to solve the $\bar\partial$-harmonic forms on the Kodaira-Thurston manifold endowed with an almost complex structure and an Hermitian metric. Using the Weil-Brezin transform, we reduce the elliptic PDE system to countably many linear ODE systems. By solving a fundamental problem on linear ODE systems, the problem of finding $\bar\partial$-harmonic forms is equivalent to a generalised Gauss circle problem.
We demonstrate two remarkable applications. First, the dimension of the almost complex $\bar\partial$-Hodge numbers on the Kodaira-Thurston manifold could be arbitrarily large. Second, Hodge numbers vary with different choices of Hermitian metrics. This answers a question of Kodaira and Spencer in Hirzebruch's 1954 problem list.
We define analogue of theta-functions on the Kodaira--Thurston manifold which is a compact 4-dimensional symplectic manifold and use them to construct canonical symplectic embedding of the Kodaira--Thurston manifold into the complex projective space (analogue of the Lefshetz theorem).
We prove that the Calabi-Yau equation can be solved on the Kodaira-Thurston manifold for all given $T^2$-invariant volume forms. This provides support for Donaldson's conjecture that Yau's theorem has an extension to symplectic four-manifolds with compatible but non-integrable almost complex structures.
The Kodaira-Thurston manifold is a quotient of a nilpotent Lie group by a cocompact lattice. We compute the family Gromov-Witten invariants which count pseudoholomorphic tori in the Kodaira-Thurston manifold. For a fixed symplectic form the Gromov-Witten invariant is trivial so we consider the twistor family of left-invariant symplectic forms which are orthogonal for some fixed metric on the Lie algebra. This family defines a loop in the space of symplectic forms. This is the first example of a genus one family Gromov-Witten computation for a non-Kähler manifold.
We review some previous results about the Calabi-Yau equation on the Kodaira-Thurston manifold equipped with an invariant almost-Kaehler structure and assuming the volume form invariant by the action of a torus. In particular, we observe that under some restrictions the problem is reduced to a Monge-Ampère equation by using the ansatz $\tilde ω=Ω-dJdu+da$, where $u$ is a $T^2$-invariant function and $a$ is a $1$-form depending on $u$. Furthermore, we extend our analysis to non-invariant almost-complex structures by considering some basic cases and we finally take into account a generalization …
The Kodaira--Thurston M manifold is a compact, 4-dimensional nilmanifold which is symplectic and complex but not Kaehler. We describe a construction of theta-functions associated to M which parallels the classical theory of theta-functions associated to the torus (from the point of view of representation theory and geometry), and yields pseudoperiodic complex-valued functions on R^4. There exists a three-step nilpotent Lie group G which acts transitively on the Kodaira--Thurston manifold M in a Hamiltonian fashion. The theta-functions discussed in this paper are intimately related to the repre…
For any positive natural number $r\in\N^+$ we construct new explicit proper $r$-harmonic functions on the celebrated $3$-dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $\H^2\times\rn$ and $\s^2\times\rn$.
In this expository article, we outline the theory of harmonic differential forms and its consequences. We provide self-contained proofs of the following important results in differential geometry: (1) Hodge theorem, which states that for a compact complex manifold, the de Rham cohomology group is isomorphic to the group of harmonic forms, (2) Hodge decomposition theorem, which states that for a Kähler manifold, the de Rham cohomology group decomposes into the Dolbeault cohomology groups, and (3) The Kodaira Embedding theorem, which gives a criterion of when a compact complex manifold is in fac…
Let $M^3$ be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, $F_{best}$, of a \emph{harmonic} map $f: M^3 \to S^1$ with Morse-type singularities delivers the Thurston norm $χ_-([F_{best}])$ of its homology class $[F_{best}] \in H_2(M^3; \Z)$. In particular, for a map $f$ with connected fibers and any well-positioned oriented surface $Σ\subset M$ in the homology class of a fiber, we show that the Thurston number $χ_-(Σ)$ satisfies an inequality $$χ_-(Σ) \geq χ_-(F_{best}) - ρ^\circ(Σ, f)\cdot Var_{χ_-}(f).$$ Here the variation $Var_{χ_-}(f)$ i…
Grafting is a surgery on Riemann surfaces introduced by Thurston which connects hyperbolic geometry and the theory of projective structures on surfaces. We will discuss the space of projective structures in terms of the Thurston's geometric parametrization given by grafting. From this approach we will prove that on any compact Riemann surface with genus greater than $1$ there exist infinitely many projective structures with Fuchsian holonomy representations. In course of the proof it will turn out that grafting is closely related to harmonic maps between surfaces.
We prove that on a compact almost Hermitian 4-manifold the space of $\bar\partial$-harmonic $(1,1)$-forms always has dimension $h_{\bar\partial}^{1,1} = b_- +1$ or $b_-$, whilst the space of Bott-Chern harmonic $(1,1)$-forms always has dimension $h_{BC}^{1,1} = b_- +1$. We also perform calculations of $h^{2,1}_{BC}$ and $h^{1,2}_{BC}$ on the Kodaira-Thurston manifold, thereby providing a full account of when $h^{p,q}_{BC}$ is or is not invariant of the choice of almost Hermitian metric. Finally, we introduce a decomposition of the space of $L^2$ functions on all torus bundles over $S^1$, which…
For a homotopically energy-minimizing map $u: N^3\to S^1$ on a compact, oriented $3$-manifold $N$ with boundary, we establish an identity relating the average Euler characteristic of the level sets $u^{-1}\{θ\}$ to the scalar curvature of $N$ and the mean curvature of the boundary $\partial N$. As an application, we obtain some natural geometric estimates for the Thurston norm on $3$-manifolds with boundary, generalizing results of Kronheimer-Mrowka and the second named author from the closed setting. By combining these techniques with results from minimal surface theory, we obtain moreover a …
In 1995, Dan Guan constructed examples of non-Kahler, simply-connected holomorphically symplectic manifolds. An alternative construction, using the Hilbert scheme of Kodaira-Thurston surface, was given by F. Bogomolov. We investigate topology and deformation theory of Bogomolov-Guan manifolds and show that it is similar to that of hyperkahler manifolds. We prove the local Torelli theorem, showing that holomorphically symplectic deformations of BG-manifolds are unobstructed, and the corresponding period map is locally a diffeomorphism. Using the local Torelli theorem, we prove the Fujiki formul…
In recent years a lot of attention has been paid to topological spaces which are a bit more general than smooth manifolds - orbifolds. Orbifolds are intuitively speaking manifolds with some singularities. The formal definition is also modelled on that of manifolds, an orbifold is a topological space which locally is homeomorphic to the orbit space of a finite group acting on $R^n$. Orbifolds were defined by Satake, as V-manifolds, then studied by W. Thurston, who introduced the term "orbifold". Due to their importance in physics, and in particular in the string theory, orbifolds have been draw…
For a harmonic map $u:M^3\to S^1$ on a closed, oriented $3$--manifold, we establish the identity $$2π\int_{θ\in S^1}χ(Σ_θ)\geq \frac{1}{2}\int_{θ\in S^1}\int_{Σ_θ}(|du|^{-2}|Hess(u)|^2+R_M)$$ relating the scalar curvature $R_M$ of $M$ to the average Euler characteristic of the level sets $Σ_θ=u^{-1}\{θ\}$. As our primary application, we extend the Kronheimer--Mrowka characterization of the Thurston norm on $H_2(M;\mathbb{Z})$ in terms of $\|R_M^-\|_{L^2}$ and the harmonic norm to any closed $3$--manifold containing no nonseparating spheres. Additional corollaries include the Bray--Brendle--Nev…
This paper gives a quantitative version of Thurston's hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of non-hyperbolic Dehn fillings on a cusped hyperbolic 3-manifold, and estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling. The proofs involve the construction of a family of hyperbolic cone-manifold structures, using infinitesimal harmonic deformations and analysis of geometric limits.
We study the relationship between two norms on the first cohomology of a hyperbolic 3-manifold: the purely topological Thurston norm and the more geometric harmonic norm. Refining recent results of Bergeron, Şengün, and Venkatesh as well as older work of Kronheimer and Mrowka, we show that these norms are roughly proportional with explicit constants depending only on the volume and injectivity radius of the hyperbolic 3-manifold itself. Moreover, we give families of examples showing that some (but not all) qualitative aspects of our estimates are sharp. Finally, we exhibit closed hyperbolic 3-…
In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3-manifolds with ``tubular boundary''. In particular, this applies to complements of tubes of radius at least $R_0 = \arctanh(1/\sqrt{3}) \approx 0.65848$ around the singular set of hyperbolic cone manifolds, removing the previous restrictions on cone angles. We then apply this to obtain a new quantitative version of Thurston's hyperbolic Dehn surgery theorem, showing that all generalized Dehn surgery coefficients outside a disc of ``uniform'' size yield hyperbolic structures. Here the size of a…
Given a complex manifold $X$ and a smooth positive function $η$ thereon, we perturb the standard differential operator $d=\partial + \bar\partial$ acting on differential forms to a first-order differential operator $D_η$ whose principal part is $η\partial + \bar\partial$. The role of the zero-th order part is to force the integrability property $D_η^2=0$ that leads to a cohomology isomorphic to the de Rham cohomology of $X$, while the components of types $(0,\,1)$ and $(1,\,0)$ of $D_η$ induce cohomologies isomorphic to the Dolbeault and conjugate-Dolbeault cohomologies. We compute Bochner-Kod…
