Geometry of Holomorphic One-forms on Smooth Projective Varieties

Technical Summary: Geometry of Holomorphic One-Forms on Smooth Projective Varieties

Problem Statement
The paper investigates the geometry and topology of irregular varieties through the lens of holomorphic 1-forms. Specifically, it addresses the relationship between the existence of nowhere-vanishing holomorphic 1-forms and the smoothness of morphisms from smooth projective varieties to abelian varieties. Furthermore, it seeks to characterize the structure of the spaces of holomorphic 1-forms that possess zeros. A central question is whether these spaces, particularly their "lower level sets" (subsets of forms with zero loci of codimension $\leq i$), exhibit linear structures (unions of vector subspaces) or if they can be non-linear, reflecting more complex algebro-geometric properties.

Methodology
The authors employ a combination of differential-geometric techniques, algebraic topology, and birational geometry:

1. Limit Relative Conormal Vectors: To analyze singularities of morphisms $f: X \to A$ (where $A$ is an abelian variety), the authors utilize the concept of the relative conormal space $T^*_f X$. They introduce a condition called "without blow-up in codimension 0," defined by the equidimensionality of the natural map from the relative conormal space to the target.
2. Topological Constraints: The proof leverages the condition that the higher direct images $R^i f_* \mathbb{Z}$ are $\mathbb{Z}$-local systems (defining a $\mathbb{Z}$-homology fibre bundle). The authors prove that this topological condition forces the morphism to be "without blow-up in codimension 0" and, under specific conditions, smooth.
3. Cohomology Jump Loci and Formality: The study of the linearity of zero-loci spaces relies on the tangent cone theorem for cohomology jump loci. The authors distinguish between forms arising from cohomology jump loci (which form linear subsets $V^i(X)$) and those that do not ("non-formal" forms).
4. Explicit Constructions: To demonstrate the limits of linearity, the authors construct specific examples involving blow-ups of subvarieties in abelian varieties and hyperplane sections of products of elliptic curves and abelian varieties.

Key Contributions and Results

• Characterization of Smooth Morphisms (Theorem 1.1 / 2.5):
  The paper establishes a necessary and sufficient condition for a morphism $f: X \to A$ from a smooth projective variety to a simple abelian variety to be smooth. The morphism is smooth if and only if there exists a holomorphic 1-form $\omega$ on $A$ such that the pullback $f^*\omega$ has no zeros. This result confirms a conjecture by the second author [13] and extends previous work by Church [5] on low-dimensional cases. The simplicity of $A$ is shown to be a necessary condition via a counterexample involving a blow-up of a product of elliptic curves.

• $\mathbb{Z}$-Homology Fibre Bundles and Flatness (Theorem 1.2 / 2.2):
  A key intermediate result proves that if $f: X \to Y$ is a proper morphism between a smooth variety and a normal variety, and the higher direct images $R^i f_* \mathbb{Z}$ are local systems, then $Y$ is smooth and $f$ is "without blow-up in codimension 0." This result is motivated by the de Bobadilla-Kollár conjecture regarding the smoothness of such morphisms.

• Linearity of Spaces with Zeros (Theorem 1.3 / 3.2, 3.3, 3.7):
  The authors investigate the space $W^n(X)$ of holomorphic 1-forms with zeros on a smooth projective variety $X$ of dimension $n$. They prove that $W^n(X)$ is a union of finitely many vector subspaces (a "linear subset") in the following cases:

  1. $X$ is a good minimal model.
  2. $X$ has a simple Albanese variety.
  3. $\dim(X) \leq 3$.
     In these cases, the structure is controlled by the birational geometry of $X$ or the simplicity of its Albanese map.

• Non-Linearity of Lower Level Sets (Theorem 1.4 / 4.1):
  Contrary to expectations that lower level sets $W^i(X)$ for subvarieties of abelian varieties are always linear, the authors construct a counterexample. They exhibit a smooth projective fourfold $X$ (a subvariety of an abelian sixfold) for which the third lower level set $W^3(X)$ is not linear. This answers a question posed in [9] and demonstrates that the space of forms with zeros can encode delicate, non-linear algebraic structures not captured by cohomology jump loci.

• Non-Formal 1-Forms (Section 4.2):
  The paper defines "non-formal" 1-forms as those with zeros that do not arise from cohomology jump loci. They provide a cohomological criterion for non-formality involving the exactness of the Koszul complex. The authors study surfaces admitting such forms, showing that for surfaces with Albanese dimension 1, non-formality is linked to the existence of non-reduced components in the fibers of the Albanese map. They further prove that such surfaces cannot be dominated by products of curves of higher genera.

Significance
The paper claims to provide a "fine study" of the space of holomorphic 1-forms, bridging the gap between topological invariants (local systems) and geometric properties (smoothness of morphisms). By proving the equivalence between the existence of a nowhere-vanishing pullback form and the smoothness of the morphism to a simple abelian variety, the authors resolve a specific conjecture in the field.

Furthermore, the discovery of non-linear lower level sets in subvarieties of abelian varieties challenges the prevailing intuition that these spaces are governed solely by linear structures derived from cohomology jump loci. The authors suggest that non-formal 1-forms reflect "deep algebro-geometric nature" of varieties, particularly regarding birational modifications and the structure of minimal models. The work refines the understanding of how the singularities of Albanese maps control the zeros of holomorphic 1-forms, offering new tools (limit relative conormal vectors) to analyze morphisms in algebraic geometry.