[Paper Review] Vanishing theorems and the multigraded regularity of nonsingular subvarieties

Victor Lozovanu, Gregory G. Smith|arXiv (Cornell University)|Aug 2, 2012

Algebraic Geometry and Number Theory16 references3 citations

TL;DR

This paper establishes a new vanishing theorem for higher cohomology groups of twisted ideal sheaves on nonsingular subvarieties, leveraging big and nef line bundles associated with scheme-theoretic equations. The key contribution is linear bounds on multigraded Castelnuovo-Mumford regularity and new criteria for projective normality of adjoint line bundle embeddings, generalizing prior results without requiring a total order on nef divisors.

ABSTRACT

Given scheme-theoretic equations for a nonsingular subvariety, we prove that the higher cohomology groups for suitable twists of the corresponding ideal sheaf vanish. From this result, we obtain linear bounds on the multigraded Castelnuovo-Mumford regularity of a nonsingular subvariety, and new criteria for the embeddings by adjoint line bundles to be projectively normal. A special case of our work recovers the vanishing theorem of Bertram, Ein, and Lazarsfeld.

Motivation & Objective

• To establish a vanishing theorem for higher cohomology groups of twisted ideal sheaves on nonsingular subvarieties defined by nef divisors.
• To derive linear bounds on the multigraded Castelnuovo-Mumford regularity of such subvarieties.
• To provide new criteria for the surjectivity of multiplication maps between global sections of adjoint line bundles.
• To generalize the Bertram-Ein-Lazarsfeld vanishing theorem by removing the need for a total order on nef divisors.
• To extend the framework to ambient spaces with higher-dimensional nef cones, such as toric varieties and Mori dream spaces.

Proposed method

• The proof uses a geometric construction via the projective bundle associated with a quotient of line bundles, exploiting the structure of the nef cone.
• It applies the Kawamata-Viehweg vanishing theorem in a higher-codimensional setting through asymptotic multiplier ideal sheaves.
• The method relies on showing that the asymptotic multiplier ideal sheaf of a big and nef line bundle is trivial at every point via local regular systems of parameters.
• It uses the decomposition of the line bundle $ B = \pi^*(L \otimes \mathcal{O}_X(mD_{s_1} + \sum D_{s_j})) \otimes \mathcal{O}_{X'}(mF_{s_1} + \sum F_{s_j}) $ to analyze base-point-free linear series.
• It applies Theorem 11.2.12(ii) from [PAGII] to conclude vanishing of higher cohomology when the asymptotic multiplier ideal is trivial.
• The argument involves constructing effective divisors $ G $ with normal crossing support at each point to ensure $ \mathscr{J}(G) = \mathscr{O}_{X'} $, implying trivial multiplier ideal.

Experimental results

Research questions

• RQ1 Under what conditions do higher cohomology groups of twisted ideal sheaves on nonsingular subvarieties vanish?
• RQ2 How can multigraded Castelnuovo-Mumford regularity be bounded linearly in terms of defining equations?
• RQ3 What conditions ensure the surjectivity of multiplication maps between global sections of adjoint line bundles?
• RQ4 Can the vanishing theorem of Bertram, Ein, and Lazarsfeld be generalized without assuming a total order on the nef divisors?
• RQ5 To what extent can this framework be applied to ambient spaces with multidimensional nef cones, such as toric varieties or Mori dream spaces?

Key findings

• The higher cohomology groups $ H^i(X, \mathscr{I}_Y^{m+1} \otimes K_X \otimes L) $ vanish for all $ i > 0 $ when $ L \otimes \mathcal{O}_X(-(m+1)D_{s_1} - \cdots - D_{s_e}) $ is big and nef for all $ e $-element subsets $ \{s_1, \dots, s_e\} \subseteq \{1, \dots, r\} $.
• The multigraded Castelnuovo-Mumford regularity of a nonsingular subvariety $ Y \subseteq X $ grows at most linearly in the degrees of its defining equations.
• Corollary 1.2 provides a linear bound on the regularity of $ \mathscr{I}_Y $ with respect to $ P_1, \dots, P_\ell $, assuming $ L \otimes \mathcal{O}_X(-D_{s_1} - \cdots - D_{s_e} - \sum u_j P_j) $ is big and nef for all such subsets and $ \sum u_j = \dim(Y)+1 $.
• Corollary 1.3 gives a new criterion for surjectivity of multiplication maps between global sections of adjoint line bundles $ K_X \otimes L_1 $ and $ K_X \otimes L_2 $, under the condition that $ L_j \otimes A_1^{-1} \otimes \cdots \otimes A_d^{-1} $ is big and nef for $ j=1,2 $.
• The paper proves a new Griffiths-type vanishing theorem for symmetric powers of vector bundles, showing $ H^i(X, \operatorname{Sym}^m(E) \otimes B) = 0 $ for all $ i > 0 $, $ m \geq 0 $, and big, nef $ B $, when $ E $ is a quotient of $ \bigoplus \mathcal{O}_X(D_j) $.
• In the case $ X = \mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_\ell} $, the tangent bundle is a quotient of $ \bigoplus \mathcal{O}_X(D_k) $, so $ H^i(X, \operatorname{Sym}^m(T_X) \otimes B) = 0 $ for all $ i > 0 $, $ m \geq 0 $, and big, nef $ B $.