[Paper Review] Fibration of log-general type space over quasi-abelian varieties

Chuanhao Wei|arXiv (Cornell University)|Sep 10, 2016

Algebraic Geometry and Number Theory2 references3 citations

TL;DR

This paper proves that there is no smooth fibration of a log-general type variety over a quasi-abelian variety, using Higgs bundle techniques and vanishing theorems. It establishes two weak versions of a conjecture on the non-existence of global log-one-forms on log-general type pairs, relying on Koszul complexes and Hodge module theory.

ABSTRACT

We show that there exists no smooth fibration of a smooth complex quasi-projective variety of log-general type over a quasi-abelian variety. The proof uses M. Popa and C. Schnell's construction of Higgs bundle.

Motivation & Objective

• To investigate the non-existence of smooth fibrations of log-general type varieties over quasi-abelian varieties.
• To prove weak versions of a conjecture stating that log-general type pairs admit no non-vanishing global log-one-forms.
• To extend Popa and Schnell’s vanishing result for holomorphic one-forms on general type varieties to the logarithmic setting.
• To use Hodge module theory and Higgs bundle constructions to analyze the zero locus of log-one-forms.
• To establish a contradiction via Serre duality and vanishing theorems when assuming a nowhere-vanishing log-one-form exists.

Proposed method

• Construct a Koszul complex using a global log-one-form θ, leading to a sequence of sheaves E^i twisted by O_X(L - D).
• Apply Akizuki-Kodaira-Nakano vanishing to show that certain cohomology groups H^{n-i+1}(Ω^i_X(log D)(L - D)) vanish.
• Use the surjectivity of connecting maps in the long exact sequence to deduce H^n(E^1(L - D)) = 0.
• Identify E^1(L - D) with ω_X(−E), and apply Serre duality to conclude H^0(O_X(E)) = 0, contradicting E being effective.
• Use base change and normalization to lift the fibration to a smooth compactification, applying [10, Theorem 9.4] to the induced morphism f': X' → P^{r,d}.
• Apply the bigness of a line bundle G and the non-vanishing of H^0(B^k) to derive a contradiction via the ampleness of ω_{P^{r,d}}(D_f).

Experimental results

Research questions

• RQ1 Does there exist a smooth fibration of a log-general type variety over a quasi-abelian variety?
• RQ2 Can a global log-one-form on a log-general type pair vanish identically?
• RQ3 Is the zero locus of any global log-one-form on a log-general type pair necessarily non-empty?
• RQ4 Does the non-vanishing of a log-one-form on a log-general type pair lead to a contradiction under Hodge-theoretic vanishing?
• RQ5 Can the conjecture on the non-existence of global log-one-forms be reduced to a contradiction via base change and Hodge module techniques?

Key findings

• There exists no smooth fibration of a smooth complex quasi-projective variety of log-general type over a quasi-abelian variety.
• For any log-smooth pair (X,D) with ω_X(D) containing an ample line bundle, every global log-one-form has a non-empty zero locus.
• If (X,D) is a projective log-smooth pair of log-general type and f: X∖D → T^{r,d} is a surjective projective morphism to a quasi-abelian variety, then f cannot be smooth.
• The pullback of a general log-one-form on P^{r,d} under a morphism f: X → P^{r,d} has a simple pole and does not vanish on the boundary divisor D.
• The contradiction in the proof arises from H^0(O_X(E)) = 0 when E is effective, violating the non-vanishing of sections.
• The application of [10, Theorem 9.4] remains valid even without connected fibers, provided the morphism is surjective and the sheaf conditions are met.