[Paper Review] A note on Grothendieck's (noncommutative) standard conjecture of type D

Gonçalo Tabuada|arXiv (Cornell University)|May 17, 2016

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper extends Grothendieck's noncommutative standard conjecture of type D from smooth projective schemes to smooth proper dg categories, proving its invariance under homological projective duality. As a result, the conjecture is verified for intersections of quadrics, intersections of bilinear divisors, and quadric fibrations.

ABSTRACT

Grothendieck conjectured in the sixties that the homological equivalence relation on algebraic cycles coincides with the numerical equivalence relation. In this note we extend this celebrated conjecture from smooth projective schemes to the broad setting of smooth proper dg categories. As an application, we prove that Grothendieck's original conjecture is invariant under homological projective duality. This leads to a proof of Grothendieck's conjecture in the case of intersections of quadrics and intersections of bilinear divisors. Along the way, we prove also the case of quadric fibrations.

Motivation & Objective

• To generalize Grothendieck's standard conjecture of type D from smooth projective schemes to the broader framework of smooth proper dg categories.
• To establish that the conjecture is preserved under homological projective duality, a key duality theory in algebraic geometry.
• To apply this invariance to prove the conjecture in new geometric settings, including intersections of quadrics and bilinear divisors.
• To extend the validity of the conjecture to quadric fibrations, providing new cases of its truth.

Proposed method

• Adopt the framework of smooth proper dg categories as a noncommutative generalization of smooth projective schemes.
• Utilize homological projective duality to relate the derived categories of dual varieties and transfer the conjecture across dual pairs.
• Apply the theory of noncommutative motives and noncommutative Chow motives to analyze cycle classes and equivalence relations.
• Use the invariance of the standard conjecture under derived equivalence and the structure of the derived category to verify the conjecture in specific geometric cases.
• Leverage known results on the derived categories of quadric fibrations and complete intersections to establish the conjecture in these settings.
• Employ the machinery of semiorthogonal decompositions and mutation functors to analyze the structure of the relevant dg categories.

Experimental results

Research questions

• RQ1 Does Grothendieck's noncommutative standard conjecture of type D extend from smooth projective schemes to smooth proper dg categories?
• RQ2 Is the standard conjecture invariant under homological projective duality in the noncommutative setting?
• RQ3 Can the conjecture be verified for intersections of quadrics using this extended framework?
• RQ4 Does the conjecture hold for intersections of bilinear divisors via the new invariance result?
• RQ5 Can the conjecture be established for quadric fibrations through the noncommutative extension?

Key findings

• The standard conjecture of type D is extended from smooth projective schemes to smooth proper dg categories, generalizing Grothendieck's original conjecture.
• The conjecture is invariant under homological projective duality, meaning it holds for a dual variety if it holds for the original.
• The conjecture is proven true for intersections of quadrics, a significant new case in algebraic geometry.
• The conjecture is verified for intersections of bilinear divisors, demonstrating the power of the extended framework.
• The conjecture is established for quadric fibrations, providing a new class of examples where the homological and numerical equivalence relations coincide.