When is the diagonal contractible?

Technical Summary: When is the Diagonal Contractible?

Problem Statement
This paper investigates the geometric conditions under which the diagonal $\Delta_X \subset X \times X$ of a smooth projective complex variety $X$ can be contracted via a birational morphism $\pi: X \times X \to Y$ to a subvariety of smaller dimension. Specifically, the authors address two primary questions:

1. Point Contraction: When does there exist a birational projective morphism such that $\pi(\Delta_X)$ is a single point?
2. General Contraction: When does there exist a contraction where $\dim \pi(\Delta_X) = m < \dim X$?

The authors note that while the case $\dim X = 1$ (curves) is classical, the higher-dimensional case depends on subtle positivity properties of $X$. The paper aims to resolve this by linking the contractibility of the diagonal to the properties of the Albanese morphism.

Methodology
The core methodology relies on the interplay between the geometry of $X$, its Albanese variety $A = \text{Alb}(X)$, and the induced difference map. The authors utilize the following structural tools:

• The Difference Map: They define the map $\lambda: X \times X \to A$ by $\lambda(x_1, x_2) = \text{alb}(x_1) - \text{alb}(x_2)$, where $\text{alb}: X \to A$ is the Albanese morphism.
• Divisor Theory and Intersection Numbers: The proofs utilize the theory of big and semi-ample divisors. By assuming a contraction exists, the authors construct a divisor $D$ on $X \times X$ that is big and nef but vanishes on the diagonal (or its image).
• Künneth Decomposition and Picard Groups: A crucial technical step involves decomposing divisors on $X \times X$ using the Künneth formula. Lemma 3.2 establishes that any divisor in $\text{Pic}_{\mathbb{Q}}(X \times X)$ can be written as a sum of pullbacks from $X$ and a pullback from the Albanese variety $A \times A$. This allows the authors to relate the positivity of divisors on $X \times X$ to the geometry of the Albanese image.
• Symmetrization: To handle the symmetry of the product $X \times X$, the authors often symmetrize divisors (using the involution $\sigma(x_1, x_2) = (x_2, x_1)$) to derive contradictions or necessary conditions regarding the dimension of the image of $\lambda$.

Key Contributions and Results

1. Characterization of Point Contraction (Theorem 1.4)
The paper establishes that the diagonal $\Delta_X$ is contractible to a point if and only if a set of equivalent conditions holds, primarily involving the Albanese morphism. Let $n = \dim X$ and $q = h^1(X, \mathcal{O}_X) = \dim \text{Alb}(X)$. The following are equivalent:

• (A) $\Delta_X$ is contractible to a point.
• (B) The image of the difference map $\lambda$ has dimension $2n$ (i.e., $\lambda$ is generically finite onto its image).
• (C) The Albanese variety satisfies $\dim \text{Alb}(X) \ge 2n$, and for every abelian subvariety $B \subset \text{Alb}(X)$ and general point $a$, $\dim B \ge 2 \dim((B+a) \cap \text{alb}(X))$.
• (D) Two general points of $X$ impose independent conditions on the holomorphic cotangent bundle $\Omega^1_X$. Equivalently, the evaluation map $H^0(\Omega^1_X) \to \Omega^1_{X,x} \oplus \Omega^1_{X,y}$ is surjective for general $x, y$.

The authors clarify that condition (C) is strictly stronger than mere "non-degeneracy" of the Albanese image; it requires a specific "geometric non-degeneracy" relative to abelian subvarieties.

2. Contraction to Higher Dimensions (Theorem 1.6)
For contractions to a variety of dimension $m < n$, the paper provides a necessary and sufficient condition: there must exist a dominant morphism $\pi: X \to W$ (where $\dim W = m$) such that for a general fiber $X_p$, the restriction of the difference map $\lambda$ to $X_p \times X_p$ is generically finite onto its image.

3. Isomorphism Outside the Diagonal (Theorem 1.8)
The authors address the stronger variant where the contraction is an isomorphism outside the diagonal. They prove this occurs if and only if the Albanese morphism is finite and birational over its image, and the difference map restricted to the Albanese image is finite outside the diagonal. This implies that the intersection of the Albanese image with its translates must be zero-dimensional.

4. Non-Contractibility in Higher Products (Theorem 1.9)
The paper proves that for any smooth projective variety $X$ with $\dim X > 0$ and $n \ge 3$, it is impossible to simultaneously contract all pairwise diagonals $\Delta_{ij} \subset X^n$ to points via a single birational morphism.

Examples and Classification

• Curves: The diagonal of a curve $C$ is contractible if and only if the genus $g \ge 2$.
• Surfaces: The diagonal of a surface $S$ is contractible to a point only if $S$ is of general type (Corollary 2.7). The paper classifies surfaces with $\kappa(S) \le 1$ satisfying the numerical conditions, showing they are quotients of products of curves but fail the geometric non-degeneracy required for actual contractibility.
• Products of Curves: The diagonal of a product of curves $C_1 \times \dots \times C_n$ (with $g_i \ge 2$) is contractible to a point. More generally, it can be contracted to a variety of dimension $t$ for any $0 \le t \le n-1$.
• Subvarieties of Abelian Varieties: Geometrically non-degenerate subvarieties $X \subset A$ with $\dim A \ge 2 \dim X$ satisfy the contractibility conditions.

Significance and Claims
The authors claim that the question of diagonal contractibility, while seemingly complex, is fundamentally equivalent to properties of the Albanese morphism. They correct a previous assumption in their own work (and potentially the literature) that the dimension of the Albanese image alone ($q \ge 2n$) is sufficient; they demonstrate that the interaction with abelian subvarieties (Condition C) is necessary when the Albanese variety is not simple.

The paper emphasizes that the image $Y$ of such a contraction is typically singular along the image of the diagonal, citing general results on the non-smoothness of such contractions. The work connects the positivity of the cycle $[\Delta_X]$ and the bundle $\Omega^1_X$ to the existence of these birational maps, showing that while ampleness of $\Omega^1_X$ is neither necessary nor sufficient, global generation properties at pairs of points are central.

The authors conclude by noting that the difference map to the Jacobian (or Albanese) is essentially the unique morphism contracting the diagonal in specific cases (e.g., when the contraction is an isomorphism outside the diagonal), though they leave open the question of whether this uniqueness holds generally for all generically finite contractions.