Zeroes of the Ricci tensor of the base of Yau's torus-bundle structure theorem for compact complex submanifolds of complex tori

Yau's Theorem 5 (Inventiones mathematicae 25 (1974), Chapter III) states that every compact complex submanifold $M$ of a flat complex torus is a holomorphic torus bundle over a compact complex submanifold $N$ of a complex torus whose induced Kähler metric has negative-definite Ricci tensor on an open dense set, and his Remark 2 asks to understand the set of zeroes of the Ricci tensor of $N$, noting that for curves the number of zeroes is determined by the genus. For $\dim N \geq 2$ the question is open: the zero set — equivalently the locus where the second fundamental form of $N$ in its flat torus vanishes identically, or where the differential of the Gauss map of $N$ vanishes — has no characterization in the literature. The problem asks whether this set can be empty, which dimensions its components can have, what the cohomology class of its components is, and which features are specific to bases of the fibration.

Background

This problem concerns the local differential geometry of compact complex submanifolds of flat complex tori, as organized by Yau's 1974 splitting theorem.

A complex torus is a quotient $T^n = \mathbb{C}^n/\Lambda$ of $\mathbb{C}^n$ by a lattice of rank $2n$; the standard Hermitian metric of $\mathbb{C}^n$ descends to a flat Kähler metric on $T^n$ (a Kähler metric is a Hermitian metric whose associated $(1,1)$-form is closed). Complex submanifolds of Kähler manifolds are Kähler, and their curvature is governed by the second fundamental form: for a complex submanifold $N$ of a flat Hermitian ambient space, $\alpha$ is the symmetric bilinear form with values in the normal bundle $N_{N/T}$ measuring the failure of $N$ to be totally geodesic. By the Gauss equation, the Ricci tensor of the induced metric (the trace of the curvature tensor) satisfies $\mathrm{Ric}(X,X) = -2\sum_{i=1}^{m}|\alpha(V_i,X)|^2 \leq 0$ whenever $\{V_i, JV_i\}$ is a unitary frame of the tangent space, with $J$ the complex structure; Yau states this identity in Lemma 3 of Chapter III and observes that $\mathrm{Ric}(X,X) = 0$ exactly for $X$ in the kernel of $\alpha$.

Yau's Theorem 5 (Chapter III of the cited paper) is a splitting theorem: if $M$ is a compact complex submanifold of a complex torus $T^n$, then $M$ is a holomorphic fiber bundle over a compact complex submanifold $N$ of a complex torus with fiber a compact complex subtorus (a torus bundle), and the induced Kähler metric on the base $N$ has negative-definite Ricci tensor on an open dense subset. The proof splits the universal cover $\pi^{-1}(M) \subset \mathbb{C}^n$ as $\mathbb{C}^k \times \bar{N}$ along the foliation tangent to the kernel of $\alpha$ (the relative nullity foliation) and descends by the action of the covering group; a companion corollary shows the base is of general type, that is, its plurigenera grow maximally ($\limsup_{q \to \infty} q^{-\dim N} \dim H^0(N, K_N^{\otimes q}) > 0$, where $K_N$ is the canonical bundle, the determinant of the holomorphic cotangent bundle). Nagano and Smyth independently and simultaneously (1975) proved the real analog: a compact minimal submanifold of a flat real torus is a principal torus bundle over a base whose Ricci tensor is negative definite on an open dense set.

The relevant prior results about the Gauss map are as follows. For an $m$-dimensional complex submanifold $N$ of a flat complex torus $T^a$, the holomorphic Gauss map sends a point $p \in N$ to the translate to the origin of the tangent space $T_pN$, a point of the Grassmannian $\mathrm{Gr}(m,a)$ of $m$-planes in $\mathbb{C}^a$; in the flat trivialization its differential at $p$ is the second fundamental form $T_pN \to \mathrm{Hom}(T_pN, N_p)$. Consequently the zero set of the Ricci tensor of $N$ — the set of points where the Ricci tensor vanishes identically — is exactly the locus where $\alpha$ vanishes identically, that is, the locus where the differential of the Gauss map is zero (the stationary locus of the Gauss map); it is a proper analytic subvariety of $N$ whenever the Ricci tensor is somewhere negative definite, by holomorphicity of $\alpha$ and the identity above. The literature controls other aspects of the Gauss map: Griffiths and Harris showed that the generic fiber of the Gauss map of a subvariety of a complex torus is the maximal subtorus leaving the subvariety invariant; Ran (1984), proving a conjecture of Ueno, showed that the Gauss map of a smooth subvariety of a complex torus that is not invariant under any subtorus is finite, and that Gauss maps are, up to Stein factorization, locally trivial fibrations with homogeneous fibers; Smyth and Sommese (1984) bounded the degree of the Gauss map of a submanifold of an abelian variety with ample normal bundle by $|e(N)|/\mathrm{codim}(N)$, where $e$ is the topological Euler characteristic, with sharp examples given by curves of genus $g$ in complex 3-tori whose Gauss map has degree exactly $g-1$; Zak (1988) extended his tangency theorems to subvarieties of complex tori. These results concern generic finiteness, degree, and tangencies of the Gauss map; none characterizes its stationary locus.

In the algebraic category the analog of the splitting is understood: an abelian variety is a complex torus that admits a holomorphic embedding into projective space, and for subvarieties of abelian varieties Ueno's fibration theorem and Kawamata's Theorem 13 (1981) give, after a finite étale cover (an unramified finite covering), a product decomposition into an abelian variety and a variety of general type — the algebro-geometric counterpart of Yau's Remark 1, which asserts as a conditional (not as a question) that $M$ is actually a product $T^k \times N$ whenever every leaf subtorus of the fibration admits a complementary subtorus with $T^n = T^k \times T^{n-k}$. In non-algebraic tori such complementary subtori need not exist, since extensions of complex tori are parametrized by a nontrivial $\mathrm{Ext}^1$ group; this is the status of the product-splitting statement, related to but distinct from the question posed below.

Problem Statement

Let $T^n = \mathbb{C}^n/\Lambda$ be a complex torus with its flat Kähler metric and let $M \subset T^n$ be a compact complex submanifold. By Yau's Theorem 5 (Inventiones mathematicae 25 (1974), Chapter III), $M$ is a holomorphic torus bundle over a compact complex submanifold $N$ of a complex torus (the base obtained by splitting the universal cover $\pi^{-1}(M) \subset \mathbb{C}^n$ along the relative-nullity foliation of the second fundamental form), and the induced Kähler metric on $N$ has negative-definite Ricci tensor on an open dense set. By the Gauss equation in the flat ambient metric, $\mathrm{Ric}(X,X) = -2\sum_i |\alpha(V_i,X)|^2$, so the zero set of the Ricci tensor of $N$, $Z = \{p \in N : \mathrm{Ric}_p \equiv 0\}$, is precisely the locus where the second fundamental form $\alpha$ vanishes identically — equivalently, the locus where the differential of the Gauss map of $N$ into the Grassmannian vanishes. For $\dim N = 1$ the number of zeroes is determined by the genus of the curve (Yau, Remark 2, loc. cit.). The problem, open for $\dim N \geq 2$, is to characterize $Z$:

(a) Can $Z$ be empty when $\dim N \geq 2$? (For $\dim N = 1$ it cannot be empty: the genus-determined count is positive for every curve that can arise as such a base. For $\dim N \geq 2$, writing $m = \dim N$ and $a$ for the dimension of the ambient torus, the second fundamental form is a holomorphic section of the bundle $S^2T^*N \otimes N_{N/T}$ (with $N_{N/T}$ the normal bundle of $N$ in its ambient torus), whose rank $\frac{m(m+1)}{2}(a-m)$ typically exceeds $m$; a generic section of such a bundle has empty vanishing locus, so no general mechanism is known that forces $Z$ to be nonempty, and none is known to force emptiness either.)

(b) When $Z$ is nonempty, which dimensions (equivalently, codimensions) can its irreducible components have in $N$?

(c) When $Z$ is nonempty, what is the cohomology class of a nonempty component of $Z$ of codimension $r$ (with $r$ the codimension actually attained by a nonempty component, as in (b)) in terms of the Chern classes of $N$ and of the normal bundle $N_{N/T}$, and how does it specialize on products of curves in products of tori and on smooth complete intersections in abelian varieties? Here $r$ is not the rank $\operatorname{rank}E = \frac{m(m+1)}{2}(a-m)$ of $E = S^2T^*N \otimes N_{N/T}$: the vanishing locus of a generic section of $E$ has that codimension, which exceeds $m = \dim N$ whenever $\dim N \geq 2$, so such a generic vanishing locus is empty and $r < \operatorname{rank}E$.

(d) Which features of the answer are specific to bases of Yau's fibration (for example, that $N$ is of general type and has negative-definite Ricci tensor on an open dense set), as opposed to holding for arbitrary compact complex submanifolds of flat complex tori?

References

1. S.-T. Yau, On the curvature of compact Hermitian manifolds, Inventiones mathematicae 25 (1974), 213–239. DOI 10.1007/BF01389728, https://doi.org/10.1007/BF01389728
2. T. Nagano and B. Smyth, Minimal varieties and harmonic maps in tori, Commentarii Mathematici Helvetici 50 (1975), 249–265. DOI 10.1007/BF02565749, https://doi.org/10.1007/BF02565749
3. A. J. Sommese, Submanifolds of abelian varieties to Rebecca, Mathematische Annalen 233 (1978), 229–256. DOI 10.1007/BF01405353, https://doi.org/10.1007/BF01405353
4. P. Griffiths and J. Harris, Algebraic geometry and local differential geometry, Annales scientifiques de l'École Normale Supérieure (4) 12 (1979), 355–452. DOI 10.24033/asens.1370, https://doi.org/10.24033/asens.1370
5. Y. Kawamata, Characterization of abelian varieties, Compositio Mathematica 43 (1981), 253–276. https://www.numdam.org/item/CM_1981__43_2_253_0/
6. Z. Ran, The structure of Gauss-like maps, Compositio Mathematica 52 (1984), 171–177. https://www.numdam.org/item/CM_1984__52_2_171_0/
7. B. Smyth and A. J. Sommese, On the degree of the Gauss mapping of a submanifold of an abelian variety, Commentarii Mathematici Helvetici 59 (1984), 341–346. DOI 10.1007/BF02566355, https://doi.org/10.1007/BF02566355
8. F. L. Zak, Theorems on tangents to subvarieties of complex tori, Mathematical Notes of the Academy of Sciences of the USSR 43 (1988), 219–225 (translation of Matematicheskie Zametki 43:3, 382–392). DOI 10.1007/BF01138845, https://doi.org/10.1007/BF01138845
9. K. Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics 439, Springer, Berlin, 1975. DOI 10.1007/BFb0070570, https://doi.org/10.1007/BFb0070570
10. T. Nagano and B. Smyth, Periodic minimal surfaces and Weyl groups, Acta Mathematica 145 (1980), 1–27. DOI 10.1007/BF02414183, https://doi.org/10.1007/BF02414183