Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points

Guseĭn-Zade, S. M.; Luengo, I.; Melle-Hernández, A.

doi:10.1307/mmj/1156345599

Cited by 49 publications

(70 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then by the usual Künneth isomorphism (which respects the underlying mixed Hodge structures of Deligne), h (c) becomes a ring homomorphism. So, the generating series (5) for h (c) just tells us the well-known fact (compare [13,20,24]) that h (c) is a morphism of pre-lambda rings. And the corresponding morphism of pre-lambda rings e c factorizes over the motivic Grothendieck ring:…”

Section: Examples and Homomorphisms Of Pre-lambda Ringsmentioning

confidence: 99%

“…If X is a smooth projective curve, symmetric products are of fundamental importance in the study of the Jacobian variety of X and other aspects of its geometry, e.g., see [31]. If X is a smooth algebraic surface, X (n) is used to understand the topology of the nth Hilbert scheme X [n] , parametrizing closed zero-dimensional subschemes of length n of X , e.g., see [13,21,23], and also [13,24] for higher-dimensional generalizations. For the purpose of this note, we shall assume that X is a (possibly singular) complex quasi-projective variety, and therefore, its symmetric products are algebraic varieties as well.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Twisted genera of symmetric products

Maxim

Schürmann

2011

Sel. Math. New Ser.

View full text Add to dashboard Cite

We give a new proof of formulae for the generating series of (Hodge) genera of symmetric products X (n) with coefficients, which hold for complex quasiprojective varieties X with any kind of singularities and which include many of the classical results in the literature as special cases. Important specializations of our results include generating series for extensions of Hodge numbers and Hirzebruch's χ y -genus to the singular setting and, in particular, generating series for intersection cohomology Hodge numbers and Goresky-MacPherson intersection cohomology signatures of symmetric products of complex projective varieties. Our proof applies to more general situations and is based on equivariant Künneth formulae and pre-lambda structures on the coefficient theory of a point,K 0 (A( pt)), with A( pt) a Karoubian Q-linear tensor category. Moreover, Atiyah's approach to power operations in K -theory also applies in this context toK 0 (A( pt)), giving a nice description of the important related Adams operations. This last approach also allows us to introduce very interesting coefficients on the symmetric products X (n) .

show abstract

Section: Examples and Homomorphisms Of Pre-lambda Ringsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Twisted genera of symmetric products

Maxim

Schürmann

2011

Sel. Math. New Ser.

View full text Add to dashboard Cite

show abstract

“…Let

a

be the

A

‐action on

Y \times A

by translation in the second factor, and let

a^{false[n false]}

be the induced action on

{Hilb}^{n} false(Y \times A false)

. By the stratification of

{Hilb}^{n} false(Y \times A false)

(compare ) we have

0true \sum_{n = 0}^{\infty} [{Hilb}^{n} false(Y \times A false), a^{[n]}] q^{n} = {(0true \sum_{m = 0}^{\infty} false[H_{m} false] q^{m})}^{false[Y \times A, a false]} .

…”

Section: Equivariant Grothendieck Ringsmentioning

confidence: 99%

“…For example, for a smooth projective variety X of dimension d we have the identity in the Grothendieck ring of varieties , (1) where Hilb n (X) is the Hilbert scheme of points on X, and Hilb n (C d ) 0 is the punctual Hilbert scheme in the affine space C d [16]. In case X = Y × A where A is a simple abelian variety acting on X by translation in the second factor, a straightforward argument shows that (1) lifts to the A-equivariant Grothendieck ring.…”

Section: Equivariant Hall Algebrasmentioning

confidence: 99%

“…This construction arises naturally in applications. For example, for a smooth projective variety

X

of dimension

d

we have the identity in the Grothendieck ring of varieties

0true \sum_{n = 0}^{\infty} [{prefixHilb}^{n} false(X false)] q^{n} = {(0true \sum_{n = 0}^{\infty} false[{Hilb}^{n} {false({double-struckC}^{d} false)}_{0} false] q^{n})}^{false[X false]},

where

{prefixHilb}^{n} false(X false)

is the Hilbert scheme of points on

X

, and

{prefixHilb}^{n} {(C^{d})}_{0}

is the punctual Hilbert scheme in the affine space

{double-struckC}^{d}

. In case

X = Y \times A

where

A

is a simple abelian variety acting on

X

by translation in the second factor, a straightforward argument shows that lifts to the

A

‐equivariant Grothendieck ring.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Reduced Donaldson-Thomas invariants and the ring of dual numbers

Oberdieck

Shen

2018

Proc. London Math. Soc.

View full text Add to dashboard Cite

Let A be an abelian variety. We introduce A‐equivariant Grothendieck rings and A‐equivariant motivic Hall algebras, and endow them with natural integration maps to the ring of dual numbers. The construction allows a systematic treatment of reduced Donaldson–Thomas (DT) invariants by Hall algebra techniques. We calculate reduced DT invariants for K3×E and abelian threefolds for several imprimitive curve classes. This verifies (in special cases) multiple cover formulas conjectured by Oberdieck–Pandharipande and Bryan–Oberdieck–Pandharipande–Yin.

show abstract

The Hilbert Scheme of Points

Ricolfi

2022

An Invitation to Modern Enumerative Geometry

View full text Add to dashboard Cite

Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points

Cited by 49 publications

References 9 publications

Twisted genera of symmetric products

Twisted genera of symmetric products

Reduced Donaldson-Thomas invariants and the ring of dual numbers

The Hilbert Scheme of Points

Contact Info

Product

Resources

About