Characterizing holomorphic fiber spaces whose every holomorphic self-map preserves a fiber

Let $p:Y\to X$ be a holomorphic fiber space, i.e. a proper holomorphic surjective submersion with connected fibers over a connected complex manifold $X$ . Bharali and Biswas (2014) proved that if a holomorphic self-map $F:Y\to Y$ maps a single fiber of $p$ into another fiber, then $F$ is a map of fiber spaces: it descends to a holomorphic self-map $f:X\to X$ with $p\circ F=f\circ p$ . They explicitly posed, and flagged as difficult, the converse rigidity question: determine conditions on the triple $(Y,X,p)$ — geometric, topological, or cohomological — that force every holomorphic self-map of the total space $Y$ to preserve at least one fiber (to map some fiber into a fiber), equivalently to descend to the base. The characterization problem remains open: the only progress is strengthened hypotheses for the conditional descent theorem and a partial rigidity result for product fibrations involving a base factor of genus at least 2.

Background

A holomorphic fiber space is a triple $(Y,X,p)$ where $p:Y\to X$ is a proper holomorphic surjective submersion with connected fibers and connected base; each fiber $Y_x=p^{-1}(\{x\})$ is then a compact complex manifold, and by Ehresmann's theorem the triple is a $C^\infty$ -smooth fiber bundle, so all fibers are mutually diffeomorphic. In several complex variables, a rigidity result for holomorphic maps is a statement that a general class of holomorphic maps is structurally simple because of the geometry of the domain or target (in the sense of Remmert–Stein, Kaup, and Kobayashi). Bharali and Biswas studied two rigidity phenomena for such fiber spaces. First, a degree-one criterion: writing $\deg(f)$ for the topological degree (the integer satisfying $f^*(1_X)=\deg(f)\cdot 1_Y$ on top-degree cohomology), they proved that a degree-one holomorphic map $f:Y\to X$ between compact connected complex manifolds whose underlying real manifolds are diffeomorphic is a biholomorphism provided $\dim H^1(X,\mathcal{O}_X)=\dim H^1(Y,\mathcal{O}_Y)$ ; here $H^1(\,\cdot\,,\mathcal{O})$ is the sheaf cohomology group whose dimension equals that of the space of infinitesimal deformations of any holomorphic line bundle, and the hypothesis says these deformation spaces of $X$ and $Y$ agree. Second, a descent theorem for self-maps of the total space: if a holomorphic $F:Y\to Y$ maps some fiber $Y_a$ into another fiber $Y_b$ (the paper's meaning of "preserving a fiber" — mapping into, not onto), then the set of base points $x$ such that $F(Y_x)$ lies in a single fiber is nonempty, open, and closed, hence all of $X$ ; consequently $F$ descends, i.e. there is a holomorphic $f:X\to X$ with $p\circ F=f\circ p$ , and if additionally $F|_{Y_a}$ has degree one and $\dim H^1(Y_x,\mathcal{O}_{Y_x})$ is independent of $x$ , then $F$ is fiberwise biholomorphic. The authors observe that this theorem is conditional on a hypothesis about $F$ itself, and state that a genuine rigidity result would require determining conditions on $(Y,X,p)$ alone that cause any holomorphic self-map to preserve at least one fiber — a requirement they describe as difficult. Note that the property is not automatic: a constant self-map always preserves a fiber (its image is a single point), but on a product $X\times X$ fibered over the first projection, the factor-swap map sends each fiber $\{x\}\times X$ to the cross-section $X\times\{x\}$ , which lies in no fiber, so the property fails for trivial fibrations with repeated factors. In the positive direction, the same paper proves that any non-constant holomorphic map from a product of two compact connected complex manifolds into a compact Riemann surface of genus at least 2 depends on at most one factor (via the Kobayashi–Ochiai finiteness theorem, which asserts that dominant meromorphic maps onto a fixed compact complex space of general type — such as a Riemann surface of genus at least 2 — from a fixed compact complex space form a finite set); as a corollary, for a product fibration whose base has a genus-at-least-2 Riemann surface factor, every non-constant holomorphic self-map has a constrained two-variable form. Related work on proper holomorphic maps between products of hyperbolic Riemann surfaces (Janardhanan) treats the analogous product setting. Later work strengthened the inputs of the descent theorem but left the characterization itself untouched: Bharali–Biswas–Schumacher (2017) showed the $\dim H^1(\,\cdot\,,\mathcal{O})$ hypothesis in the degree-one criterion is automatic once the second Betti numbers agree (for surjective maps), and Meng (2018) similarly weakened the cohomological hypothesis in the fiberwise-biholomorphism part to equality of second Betti numbers — a condition automatic here, since the fibers of a proper submersion are mutually diffeomorphic — and separately gave degree-one-map criteria in terms of the Picard number (the rank of the Néron–Severi group of divisor classes).

Problem Statement

Let $p:Y\to X$ be a proper holomorphic surjective submersion with connected fibers between complex manifolds, with $X$ connected (a holomorphic fiber space). Say that a holomorphic self-map $F:Y\to Y$ preserves a fiber if there exist $a,b\in X$ with $F(Y_a)\subseteq Y_b$ , where $Y_x=p^{-1}(\{x\})$ ; by Theorem 1.3(a) of Bharali–Biswas (2014) this is equivalent to $F$ being a map of fiber spaces, i.e. the existence of a holomorphic $f:X\to X$ with $p\circ F=f\circ p$ . Determine necessary and sufficient conditions — geometric, topological, or cohomological — on the triple $(Y,X,p)$ under which every holomorphic self-map $F:Y\to Y$ preserves at least one fiber. No hypothesis of algebraicity, Kählerness, or hyperbolicity beyond the fiber-space definition above may be silently assumed.

References

Gautam Bharali and Indranil Biswas, Rigidity of holomorphic maps between fiber spaces, International Journal of Mathematics 25 (2014), no. 1, 1450006, DOI 10.1142/S0129167X14500062; arXiv:1309.3948. https://doi.org/10.1142/S0129167X14500062
Lingxu Meng, Holomorphic Gromov's partial order, The Journal of Geometric Analysis 28 (2018), no. 3, 2539–2549, DOI 10.1007/s12220-017-9917-x; arXiv:1610.07150. https://doi.org/10.1007/s12220-017-9917-x
Gautam Bharali, Indranil Biswas, and Georg Schumacher, A criterion for a degree-one holomorphic map to be a biholomorphism, Complex Variables and Elliptic Equations 62 (2017), no. 7, 914–918, DOI 10.1080/17476933.2016.1252342; arXiv:1610.06286. https://doi.org/10.1080/17476933.2016.1252342
Gautam Bharali, Indranil Biswas, and Mahan Mj, The Fujiki class and positive degree maps, Complex Manifolds 2 (2015), no. 1, DOI 10.1515/coma-2015-0002; arXiv:1312.5655. https://doi.org/10.1515/coma-2015-0002
Jaikrishnan Janardhanan, Proper holomorphic mappings between hyperbolic product manifolds, International Journal of Mathematics 25 (2014), no. 4, 1450039, DOI 10.1142/S0129167X14500396; arXiv:1105.2648. https://doi.org/10.1142/S0129167X14500396