An arithmetic Yau-Zaslow formula: $K_0(Var_k)$, $GW(k)$ and quadratic GW invariants

Recently two papers have appeared using constructions in $K_0(Var_k)$ combined with a $GW(k)$-valued motivic measure to construct interesting quadratic refinements of two important results in modern enumerative geometry: the Yau-Zaslow formula {YZ} for counting rational curves on K3 surfaces by Pajwani-Pâl {PP} and the local Donaldson-Thomas invariants for $Hilb^n_0(\mathbb{A}^3)$ by Espreafico-Walcher {EW}. The goal of the seminar this semester is to first go over some of the original work on $K_0(Var_k)$ and the Yau-Zaslow formula, by Batyrev {Batyrev}, Beauville {BeauvilleYZ}, Göttsche {GMotiveHilb}, {GBettiHilb}, Ellingsrud-Strømme {ES}, {ESBetti}, Kontsevich {Kontsevich}, Mukai {Mukai}, and others, and with this as background, go through the Pajwani-Pâl paper {PP}.

We will meet on Tuesdays, 16:15-18 Uhr, in WSC-N-U-3.04.

Previous Motives Seminars

Follow the Teaching link at the top of this page to find links to previous motives seminars.

Program

Lecture 0. October 10: Marc Levine.  An overview
Lecture 0 Video

We give an overview of Beauville's proof of the Yau-Zaslow formula for the count $e(g)$ of rational curves in a complete linear system on a K3 surface having general member a curve of genus $g$: \[ 1+\sum_{g\ge1}e(g)t^g =\frac{t}{\Delta(t)}=\prod_{n\ge1}(1-t^n)^{-24},\] as well as some indications of the quadratic refinement of Pajwani-Pâl.

Lecture 1. October 17:  Marc Levine.  $Hilb^n$ for surfaces and K3 surfaces.
Lecture 1 Video

The main points are:

i. For a smooth projective surface $S$, $S^{[n]}:=Hilb^n(S)$ is a smooth projective variety, with a birational morphism to the $n$th symmetric power of $S$, the Hilbert-Chow map $S^{[n]}\to S^{(n)}$. see [Theorem 2.4, Cor. 2.6]{Fogarty}.
ii. For $S$ a K3 surface or an abelian surface, $S^{[n]}$ is a holomorphic symplectic variety [Prop. 5]{Beauville83}. This relies on a result of Iarrobino [equation (5)]{Iar}.
iii. Mukai's theorem: [Corollary 0.2]{Mukai}. Let $S$ be a K3 surface, together with an ample divisor $H$. Let $M(r, l, s)$ be the coarse moduli space of $H$-stable sheaves $E$ on $S$ of rank $r>0$, sheaf Euler characteristic $\chi(E)=r+s$ and with 1st Chern class $c_1(E)=l\in Pic(S)$. Then $M(r,l,s)$ is a smooth quasi-projective symplectic variety of dimension $l^{(2)}-2rs+2$, and is projective if $gcd(r, s, (H\cdot l))=1$.

The last item needs only to be stated, if there is no time for details.

Lectures 2,3 cover Batyrev's theorem on the invariance Betti numbers of CY varieties under birational maps, and Kontsevich's extension to the Hodge numbers. You can use the notes of Mallory {Mallory} or Raibaut {Raibaut} (or any of the other numerous notes summarizing Kontsevich's work) as a guide. Note that Batyrev's theorem follows from Kontsevich's.

Lecture 2: October 24:  Lukas Bröring.  Statement of main results, and an introduction to $K_0(Var_k)$.
Lecture 2 Video

This is covered in [§1,2]{Mallory} or {Raibault} or ....

Lecture 3: October 31:  Andrei Konovalov. Arc schemes, proof of Kontsevich's theorem.
Lecture 3 Video

[§3,4]{Mallory} or {Raibaut} or .....

Lecture 4. November 7:  Clémentine Lemarie-Rieusset. Göttsche's formula for the Betti numbers of $S^{[n]}$.
Lecture 4 Video

Present this as a consequence of Göttsche's computation of $[S^{[n]}]\in K_0(Var_k)$ in terms of symmetric products $S^{(\alpha)}$. To go from this to the formula ($e(-)=$ Euler characteristic) \[ \sum_ge(S^{[g]})t^g=\prod_{n=1}^\infty(1-t^n)^{-e(S)} (=\frac{t}{\Delta(t)} \text{ for $S$ a K3 surface}) \] of [Theorem 0.1(2)]{GBettiHilb}, you can use the computation of the Euler characteristic/Poincaré polynomial for a symmetric product found in {MacDonald}. Particular points:

Discuss the cellular decomposition of $Hilb^n(\mathbb{A}^2,0)$, found in {ES} (see also {ESBetti}).
Present the main result (Theorem 1.1) of {GMotiveHilb} and use this to give Göttsche's formula for the Betti numbers of $S^{[n]}$ [Theorem 0.1, esp. part (2)]{GBettiHilb}.

Lecture 5. November 14:  Andrés Jaramillo-Puentes. Jacobians, generalized Jacobians and compactified Jacobians (see e.g. {FGvS} for some of the details).
Lecture 5 Video

Main Points:

i. Review the classical theory of the Jacobian $J(C)$ of a smooth projective curve $C$ over an algebraically closed field, especially the identity $J(C)=Pic^0(C)\cong Pic^d(C)$ and the description of $J(C)$ as a complex torus, for $C$ a curve over $\mathbb{C}$.
ii. For $C$ an integral curve over an algebraically closed field $k$ with normalization $p:\tilde{C}\to C$, give the description of $Pic^0(C)$ as an extension \[ 1\to G\to Pic^0(C)\to J(\tilde{C})\to 0 \] where $G$ is a product of a unipotent group scheme with a torus $\mathbb{G}_m^n$, by using \[Pic^0(C)(k)=H^1(C, \mathcal{O}_C^\times) \] and the exact sequence \[ 1\to \mathcal{O}_C^\times\to p_*\mathcal{O}_{\tilde{C}}^\times\to p_*\mathcal{O}_{\tilde{C}}^\times/\mathcal{O}_C^\times\to 1 \] Factor $p$ as $p_1:\tilde{C}\to C^{sn}$, $p_2:C^{sn}\to C$, where $p_2:C^{sn}\to C$ is the semi-normalization of $C$, and show that $p_{1*}\mathcal{O}_{\tilde{C}}^\times/\mathcal{O}_{C^{sn}}^\times\cong (k^\times)^n$ for some $n$ and that $p_{2*}\mathcal{O}_{C^{sn}}^\times/\mathcal{O}_{C}^\times$ is a uni-potent subgroup of $GL_m(k)$ for some $m$.
iii. Describe the main existence theorem of Altman-Kleiman {AK} for the relative Picard scheme [Theorem 6.3, 6.4]{AK} for a flat proper family with integral fibers, and the compactified relative Picard scheme [Theorem 8.5]{AK} for a flat projective family of integral curves, as well as Rego's theorem [Theorem A]{Rego}, that the compactified generalized Jacobian of an integral curve $C$ with planar singularities is irreducible.
iv. Discuss the example of $C$ a projective curve over an algebraically closed field with only ordinary double points and describe the complement of $Pic^0(C)$ in the compactified version $\overline{Pic}^0(C)$, see e.g. the description of D'Souza's results in the introduction to {Rego}.
v. Mention Mukai's example [Example 0.5]{Mukai} about the relative compactified Jacobian $\bar{J}^d_{\mathcal{C}/|D|}$ for a complete linear system $|D|$ of curves on a K3 surface (assuming all curves in $|D|$ are integral), namely, that $\bar{J}^d_{\mathcal{C}/|D|}$ is a smooth projective holomorphic symplectic variety.

Lecture 6. November 21:  Svetlana Makarova.  Beauville's proof of the Yau-Zaslov formula
Lecture 6 Video

Present Beauville's paper {BeauvilleYZ} proving the Yau-Zaslov formula for counting rational curves on a K3 surface.

Lecture 7. November 28:  Chirantan Chowdhury.  The theorem of de Cataldo-Migliorini on the decomposition of $S^{[n]}$ in the category of rational Chow motive over $k$.
Lecture 7 Video

Main points:

Present the results of {dCM}. Note that the decomposition of $S^{[n]}$ in $K_0(Mot_{CH}(k)_\mathbb{Q})$ follows from Göttsche's computation of $[S^{[n]}]\in K_0(Var_k)$ if $k$ is algebraically closed, using the homomorphism $K_0(Var_k)\to K_0(Mot_{CH}(k)_\mathbb{Q})$, but we will need a description of $S^{[n]}$ for non-algebraically closed $k$. The paper {dCM} is written with $k$ algebraically closed, but it is claimed that this assumption is not necessary, so try to avoid this assumption if possible.

Lectures 8-13. Present the paper of Pajwani-Pâl. {PP}

Lecture 8. December 5:  Marc Levine. Statement of main results and an introduction to Euler characteristics [§1,2]{PP}
Lecture 8 Video
Lecture 8 Notes

Lecture 9. December 19:  Marc Levine. Equivariant motivic Euler characteristics [§3]{PP}
Lecture 9 Video

Lecture 10. January 9:  Jan Hennig.  Transfer maps [§4]{PP}
Lecture 10 Video

Lecture 11. January 16:  Pietro Gigli.  Fubini theorems [§5]{PP}
Lecture 11 Video

You can skip §5.3, just mention that there are examples of $f:X\to U$, $U\subset \mathbb{P}^1$ open, $f$ smooth and projective with fibers elliptic curves, but with $\chi(X/k)\neq0$, so the ``Saito factors'' discussed in this section make a non-zero contribution in general.

Lecture 12. January 23:  Herman Rohrbach.  The motivic Euler characteristic of compactifed Jacobians [§6]{PP}
Lecture 12 Video

Lecture 13. January 30:  Clémentine Lemarie-Rieusset.  The motivic Euler characteristic of CY varieties, the motivic Göttsche formula, and the arithmetic Yau-Zaslow formula [§7-9]{PP}

References

{AK} Altman, Allen B.; Kleiman, Steven L., Compactifying the Picard scheme. Adv. in Math. 35 (1980), no. 1, 50--112.

{Batyrev} V.V. Batyrev, Birational Calabi-Yau $n$-folds have equal Betti numbers. London Math. Soc. Lecture Note Ser., 264 Cambridge University Press, Cambridge, 1999, 1--11.

{BeauvilleYZ} Beauville, Arnaud, Counting rational curves on K3 surfaces. Duke Math. J. 97 (1999), no. 1, 99--108.

{Beauville83} Beauville, Arnaud, Vari\'et\'es k\"ahl\'eriennes compactes avec $c_1=0$. Ast\'erisque(1985), no. 126, 181--192.

{dCM} de Cataldo, Mark Andrea A.; Migliorini, Luca, The Chow groups and the motive of the Hilbert scheme of points on a surface J. Algebra 251 (2002), no. 2, 824--848.

{ES} Ellingsrud, Geir; Strømme, Stein Arild, On a cell decomposition of the Hilbert scheme of points in the plane Invent. Math. 91 (1988), no. 2, 365--370.

{ESBetti} Ellingsrud, Geir; Strømme, Stein Arild, On the homology of the Hilbert scheme of points in the plane Invent. Math. 87 (1987), no. 2, 343--352.

{EW} Felipe Espreafico, Johannes Walcher, On Motivic and Arithmetic Refinements of Donaldson-Thomas invariants. arXiv:2307.03655.

{FGvS} Fantechi, B.; Göttsche, L.; van Straten, D., Euler number of the compactified Jacobian and multiplicity of rational curves. J. Algebraic Geom. 8 (1999), no. 1, 115--133.

{Fogarty} Fogarty, John, Algebraic families on an algebraic surface. Amer. J. Math.90(1968), 511--521.

{GMotiveHilb} Göttsche, Lothar, On the motive of the Hilbert scheme of points on a surface Math. Res. Lett. 8 (2001), no. 5-6, 613--627.

{GBettiHilb} Göttsche, Lothar, The Betti numbers of the Hilbert scheme of points on a smooth projective surface. Math. Ann. 286 (1990), no. 1-3, 193--207.

{Iar} Iarrobino, A., Punctual Hilbert schemes. Bull. Amer. Math. Soc. 78 (1972), 819-823.

{Kontsevich} M. Kontsevich, Motivic Integration, Lecture Orsay, 1995.

{MacDonald} Macdonald, I. G., The Poincaré polynomial of a symmetric product. Proc. Cambridge Philos. Soc. 58 (1962), 563--568.

{Mallory} D. Mallory, Motivic Integration. {http://www-personal.umich.edu/\~malloryd/main.pdf

{Mukai} Mukai, Shigeru, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface. Inventiones mathematicae volume 77, pages 101--116 (1984).

{PP} Jesse Pajwani, Ambrus Pâl. An arithmetic Yau-Zaslow formula. arXiv:2210.15788.

{Raibaut} M. Raibaut, Notes on motivic integration http://raibautm.perso.math.cnrs.fr/site/an_introduction_to_motivic_integration_theory.pdf

{Rego} Rego, C. J., The compactified Jacobian. Ann. Sci. École Norm. Sup. (4)13(1980), no.2, 211--223.

{Saito} T. Saito, Jacobi sum Hecke characters, de Rham discriminant, and the determinant of $l$-adic cohomologies. J. Algebraic Geom. 3 (1994), 411--434.

{YZ} Yau, Shing-Tung; Zaslow, Eric, BPS states, string duality, and nodal curves on K3 Nuclear Phys. B 471 (1996), no. 3, 503--512.