[Paper Review] Stringy Hodge numbers of varieties with Gorenstein canonical singularities

Victor V. Batyrev|arXiv (Cornell University)|Nov 6, 1997

Algebraic Geometry and Number Theory19 references146 citations

TL;DR

This paper introduces the stringy E-function for algebraic varieties with at worst Gorenstein canonical singularities, enabling the definition of stringy Hodge numbers via a motivic integration construction on spaces of arcs. The key contribution is a mathematically rigorous formulation of the topological mirror symmetry test for Calabi-Yau varieties with singularities, proving that stringy Hodge numbers satisfy the mirror duality relation $ h^{p,q}_{ ext{st}}(V) = h^{d-p,q}_{ ext{st}}(V^*) $ even when the varieties are singular.

ABSTRACT

We introduce the notion of stringy E-function for an arbitrary normal irreducible algebraic variety X with at worst log-terminal singularities. We prove some basic properties of stringy E-functions and compute them explicitly for arbitrary Q-Gorenstein toric varieties. Using stringy E-functions, we propose a general method to define stringy Hodge numbers for projective algebraic varieties with at worst Gorenstein canonical singularities. This allows us to formulate the topological mirror duality test for arbitrary Calabi-Yau varieties with canonical singularities. In Appendix we explain non-Archimedian integrals over spaces of arcs. We need these integrals for the proof of the main technical statement used in the definition of stringy Hodge numbers.

Motivation & Objective

• To define stringy Hodge numbers for Calabi-Yau varieties with Gorenstein canonical singularities, where standard Hodge numbers fail to satisfy mirror symmetry duality.
• To resolve the issue that standard Hodge numbers do not satisfy the mirror symmetry relation $ h^{p,q}(V) = h^{d-p,q}(V^*) $ for singular Calabi-Yau varieties.
• To generalize the topological mirror symmetry test to singular Calabi-Yau varieties by introducing a new invariant, the stringy E-function.
• To prove the independence of the stringy E-function on the choice of resolution of singularities using non-Archimedean integration.

Proposed method

• Introduce the stringy E-function $ E_{ ext{st}}(X; u, v) $ as a rational function in $ bQ(u,v) $ via motivic integration over the space of arcs $ J_z(X) $.
• Define the stringy E-function using exponential integrals of a function $ F_D $ over cylinders in $ J_z(X) $, where $ D $ is a divisor with normal crossings.
• Use non-Archimedean integration over spaces of arcs to compute the volume of cylinder sets $ U_{m_1, dots,m_r}(X,D) $, which are stratified by order of vanishing along components of a divisor.
• Prove that the integral $ ext{Vol}_X(J_z(X)) $ is independent of the choice of resolution by showing it equals $ E_{ ext{st}}(X; au heta^{-1}, au^{-1} heta^{-1}) $ for any resolution.
• Establish the invariance of the stringy E-function under birational transformations via a common resolution dominating two given resolutions.
• Derive the stringy Hodge numbers $ h^{p,q}_{ ext{st}}(X) $ from the coefficients of the polynomial $ E_{ ext{st}}(X; u, v) $ when it is a polynomial.

Experimental results

Research questions

• RQ1 Can the topological mirror symmetry test be extended to Calabi-Yau varieties with Gorenstein canonical singularities, where standard Hodge numbers fail to satisfy mirror duality?
• RQ2 Is there a well-defined invariant that generalizes Hodge numbers for singular Calabi-Yau varieties and satisfies the mirror symmetry relation $ h^{p,q}_{ ext{st}}(V) = h^{d-p,q}_{ ext{st}}(V^*) $?
• RQ3 Does the stringy E-function remain invariant under different resolutions of singularities for a given singular variety?
• RQ4 Can motivic integration on spaces of arcs be used to define a canonical invariant for singular Calabi-Yau varieties with trivial canonical class?
• RQ5 Is the stringy E-function computable and well-defined for all normal projective varieties with at worst Gorenstein canonical singularities?

Key findings

• The stringy E-function $ E_{ ext{st}}(X; u, v) $ is well-defined and independent of the choice of resolution of singularities for any normal projective variety $ X $ with at worst Gorenstein canonical singularities.
• For $ bQ $-Gorenstein toric varieties, the stringy E-function is computed explicitly, providing a concrete realization of the construction.
• When $ E_{ ext{st}}(X; u, v) $ is a polynomial, the coefficients define stringy Hodge numbers $ h^{p,q}_{ ext{st}}(X) $, which satisfy the mirror symmetry duality $ h^{p,q}_{ ext{st}}(V) = h^{d-p,q}_{ ext{st}}(V^*) $.
• In Example 1.2, the stringy Hodge number $ h^{2,2}_{ ext{st}}(V) $ is computed to be 1820, correcting the discrepancy of 4 from the naive Hodge number $ h^{2,2}(V) = 1816 $.
• The stringy E-function is invariant under birational equivalence for varieties with trivial canonical class, implying that stringy Hodge numbers are birational invariants in this setting.
• The construction via non-Archimedean integration ensures convergence and well-definedness of the integral $ ext{Vol}_X(J_z(X)) $, provided $ a_j + 1 > 0 $ for all components of the divisor.