Tangent bundle of punctual Hilbert scheme and distinguishing products of varieties

1. Problem Statement

The paper addresses two interconnected problems in algebraic geometry over the field of complex numbers $\mathbb{C}$:

1. Decomposition of the Tangent Bundle: Determine the indecomposable components of the tangent bundle $T_{X[n]}$ of the punctual Hilbert scheme $X[n]$ (parametrizing length $n$ subschemes) for a smooth projective surface $X$. While the geometry of $X[n]$ is well-studied (e.g., it is smooth and projective of dimension $2n$), the specific structure of its tangent bundle as a direct sum of indecomposable vector bundles was not fully characterized for all surfaces.
2. Uniqueness of Product Decompositions: Investigate whether a product of varieties is uniquely determined by its factors. Specifically, given an isomorphism $\prod X_i \cong \prod Y_j$, can one conclude that the multisets of factors are identical (up to renumbering)? This is a variation of the Zariski cancellation problem. The author focuses on products of punctual Hilbert schemes $X[n]$ and symmetric powers $X^{(n)}$.

2. Methodology

The author employs a combination of bundle theory, group actions, and local geometric analysis.

• Symmetric Group Action ($S_n$): The core strategy involves lifting the decomposition of the tangent bundle of the Hilbert scheme $X[n]$ to the tangent bundle of the Cartesian product $X^n$. Since $X[n]$ is a resolution of the symmetric product $X^{(n)} = X^n / S_n$, the author analyzes $S_n$-invariant subbundles of $T_{X^n}$.
• Lifting and Descent:
  • Theorem 4.1 and 4.2 classify possible $S_n$-invariant direct sum decompositions of $T_{X^n}$.
  • Section 5 introduces the concept of extendability. A subbundle of $T_{X^n}$ is "extendable" if it arises as the pullback of a subbundle on $X[n]$ via the Hilbert-Chow morphism. The author proves that specific natural subbundles of $T_{X^n}$ (constructed from line bundles on $X$) are not extendable to $X[n]$. This is done via a local computation involving the total space of vector bundles and fiber dimension arguments near the singular locus of the symmetric product.
• Classification of Surfaces: The proof relies heavily on the classification of surfaces based on the splitting of their tangent bundles $T_X$. The author utilizes results from [6], [12], and [13] to categorize surfaces where $T_X$ splits (is a direct sum of line bundles) versus those where it is indecomposable.
• Induction and Singularity Analysis: For the product classification (Theorem C), the author uses induction on the dimension of the product space and analyzes the singular loci of symmetric powers. The normalization of the singular locus of $X^{(m)}$ is used to distinguish different symmetric powers.

3. Key Contributions and Results

A. Classification of the Tangent Bundle of $X[n]$ (Theorem A)

The paper provides a complete classification of the indecomposable components of $T_{X[n]}$ based on the geometry of the surface $X$. The author defines a class $\tilde{\mathcal{C}}$ of surfaces including:

• Surfaces of the form $(D \times Y)/G$ (where $D$ is elliptic, $Y$ is general type, $G$ finite).
• Bielliptic surfaces.
• $\mathbb{P}^1$-bundles over an elliptic curve with a split tangent bundle.

Theorem A states:

1. Abelian Surfaces: If $X$ is an abelian surface, $T_{X[n]} \cong \mathcal{O}_{X[n]}^{\oplus 2} \oplus E$, where $E$ is indecomposable.
2. Class $\tilde{\mathcal{C}}$: If $X \in \tilde{\mathcal{C}}$, $T_{X[n]} \cong \mathcal{O}_{X[n]} \oplus E$, where $E$ is indecomposable.
3. All Other Cases: If $X$ is not of the above types, $T_{X[n]}$ is indecomposable.

In cases (1) and (2), the bundle $E$ is identified as the kernel of the differential of a smooth morphism $\pi: X[n] \to F'$, where $F'$ is an abelian variety.

B. Distinguishing Products of Hilbert Schemes (Theorem B)

The author proves a conjecture regarding the uniqueness of products of punctual Hilbert schemes.
Theorem B: Let $X$ be a smooth projective surface. If $\prod_{i=1}^r X[a_i] \cong \prod_{j=1}^s X[b_j]$, then $r=s$ and the multisets $\{a_1, \dots, a_r\}$ and $\{b_1, \dots, b_s\}$ are identical.

• Method: The proof uses the decomposition of the tangent bundle derived in Theorem A. The number of indecomposable summands of a specific rank in $T_{\prod X[a_i]}$ corresponds directly to the multiplicity of the integer $k$ in the multiset $\{a_i\}$. Since the tangent bundle is an isomorphism invariant, the multisets must match.

C. Distinguishing Products of Symmetric Powers (Theorem C)

The result is extended to symmetric powers $X^{(n)}$ for smooth varieties of dimension $\ge 2$.
Theorem C: Let $X$ be a smooth variety of dimension $\ge 2$. If $\prod X(a_i) \cong \prod X(b_j)$, then the multisets of exponents are identical.

• Method: This proof relies on the structure of the singular locus $\text{Sing}(X^{(n)})$. The author analyzes the normalization of the irreducible components of the singular locus of the product space. By comparing the dimensions and the structure of these normalizations, the author derives a contradiction if the multisets differ.

4. Significance

• Geometric Invariants: The paper establishes the tangent bundle as a powerful invariant for distinguishing products of Hilbert schemes and symmetric powers, extending previous results that were limited to curves or specific surface types.
• Resolution of Conjectures: It confirms a recent conjecture (from [10]) regarding the classification of products of Hilbert schemes of points on surfaces.
• Structural Insight: The detailed analysis of when $T_{X[n]}$ splits versus when it remains indecomposable provides new geometric insight into the moduli spaces of subschemes on surfaces, linking the global geometry of $X[n]$ to the local splitting properties of $T_X$.
• Technical Innovation: The technique of proving non-extendability of specific subbundles from $X^n$ to $X[n]$ via local fiber dimension analysis is a novel contribution to the study of vector bundles on Hilbert schemes.

5. Conclusion

Supravat Sarkar successfully characterizes the tangent bundle of punctual Hilbert schemes for all smooth projective surfaces, distinguishing between abelian, bielliptic, and general types. This structural result is then leveraged to solve the isomorphism problem for products of these schemes and symmetric powers, demonstrating that the "length" parameters of the factors are uniquely recoverable from the product variety itself.