Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

On a blowup formula for sheaf-theoretic virtualenumerative invariants on projective surfaces
Yuuji Tanaka, Kyoto University
Pacific Rim Complex and Symplectic Geometry Conference,
Kyoto, 2022
Based on joint work arXiv:2107.08155 with Nikolas Kuhn
and arXiv:2205.12953 with Nikolas Kuhn and Oliver Leigh.
5 August 2022

1 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

Plan of talk:
1. Donaldson invariatns on closed four-manifolds
2. Mochizuki’s virtual Donaldson invariants on projective surfaces
3. A blowup formula for sheaf-theoretic virtual enumerative
invariants on projective surfaces

2 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

The anti-self-dual instanton moduli spaces
For now, let X be a closed, oriented, simply-connected, smooth
four-manifold, and let P → X be a principal SO(3)-bundle with thefirst Pontryagin class p1 and the second Stiefel–Whitney class w2 .
Fix a Riemannian metric g on X , and consider the Hodge star
operator ∗g on Λ2X := (Λ2 T ∗ X ). This satisfies ∗2g = 1, so Λ2X−
decomposes as Λ2X = Λ+X ⊕ ΛX .
Definition: A connection A on P is said to be an anti-self-dual
instanton, if the curvature FA of A satisfies FA+ := π+ (FA ) = 0,
where π+ : Γ(gP ⊗ Λ2X ) → Γ(gP ⊗ Λ+X ) is the projection and gP is
the adjoint bundle of P.
We denote by AP the set of all connections on P and by
GP := Aut(P) the set of all gauge transformations on P, or thegauge group.
3 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

The gauge group GP acts on AP , and we consider theanti-self-dual instanton moduli space:
MX ,g (w2 , p1 ) := {A ∈ AP : FA+ = 0}/GP .
This is an oriented smooth manifold of expected dimensions
for a generic choice of Riemannian metrics of X , if bX+ > 0, wherebX+ is the number of positive eigenvalues of the intersection formon H 2 (X , Z).
Uhlenbeck compactification:−p1 /4

M X ,g (w2 , p1 ) :=⨿MX ,g (w2 , p1 + 4ℓ) × S ℓ X ,ℓ=0

where S ℓ X is the ℓ-th symmetric product of X . This is equippedwith a natural topology, and it is a compact Hausdorﬀ space.
4 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

Donaldson’s polynomial invariants
Assume bX+ > 0. Denote by 2d the expected dimension of
MX ,g (w2 , p1 ) and by Ad the symmetric algebra of degree dgenerated by H2 (X ) and H0 (X ) with α ∈ H2 (X ) degree 1 andp ∈ H0 (X ) degree 2.
One can define µ̄ : Hp (X , Z) → H 4−p (M X ,g (w2 , p1 ), Z) for
p = 0, 2 via a universal bundle on the moduli space. We thendefine Donaldson’s polynomial invariant qX ,d : Ad → Z of degree dby∫qX ,d (σ1 , . . . , σn , p m ) :=µ̄(σ1 ) ∪ · · · ∪ µ̄(σn ) ∪ µ̄(p)m ,[M X ,g (w2 ,p1 )]where σ1 , . . . , σn ∈ H2 (X ), p ∈ H0 (X ) and d = n + 2m. This isindependent of the choice of Riemannian metrics of X , if bX+ > 1.
5 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

If w2 ̸= 0, then there is no trivial bundle in lower strata of the
Uhlenbeck compactification, and the fundamental class[M X ,g (w2 , p1 )] is well-defined. However, if w2 = 0, then there is atrivial bundle in a lower stratum of the compactification. In thiscase, we introduce the notion of stable range (e.g. one requires−p1 is suﬃciently large) to have a well-defined fundamental class[M X ,g (w2 , p1 )].
Blowup formula by Friedman–Morgan: Consider the blowup
Xb → X at a point in X and denote by e its exceptional divisor.
Then Friedman and Morgan prove:
qXb ,d+1 (σ1 , . . . , σd , e, e, e, e) = −2qX ,d (σ1 , . . . , σd ).
This can be used to define the Donaldson invariants for unstable
range.
Moreover, blowup formulae played fantastic roles in figuring out
algebraic structures in the theory of the Donaldson invariants suchas:
6 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

Kronheimer–Mrowka (’95) proved the structure theorem for the
Donaldson invariants. Denote by qX : ⊕d Ad → Q the polynomial
invariants. They considered the following Donaldson series:∑ qX (Σd ) 1 ∑ qX (pΣd )
+
.
d!
2
d!
under the assumption of X simple type, and proved that thisformal series can be written by a finite collection of basic classes bydeveloping a theory of singular connections and a blowup formula.
Fintsushel–Stern (’96) gave an alternative proof of the above
structure theorem by improving the blowup formula. They further
obtained a universal form of blowup formula, surprisingly, it isexpressed in terms of modular forms.
Göttsche (’96) determined the wall-crossing term under theassumption that Kotschick–Morgan conjecture is true by aneﬀective use of the blowup formula, and it turned out to be written
in terms of modular forms.
7 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

Mochizuki’s virtual Donaldson invariants
The Hitchin–Kobayashi correspondence tells the instanton
moduli space is the same as the moduli space of semistablesheaves on a complex projective surface. (cf. Algebraic Donaldson
invariants.) This is good, as we have a nice compactification of thelatter moduli space thanks to Gieseker and Maruyama.
Then there is one other way of defining Donaldson type invariant
from the moduli space of semistable sheaves in Algebraic Geometryby using virtual techniques developed by Li–Tian andBehrend–Fantechi. (cf. Li–Tian and Fukaya–Ono in Symplectic
Geometry).
Perfect obstruction theory. Denote by M the moduli space ofsemistable sheaves on a projective surface X . We consider the casethat semistability implies stability. Locally, M is the zero set of theKuranishi map κ : Ext10 (E , E ) → Ext20 (E , E ) around [E ] ∈ M.
8 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

A moral here is to consider the following local model attached to
it:
VW


s

M := Z (s) ⊂ X ,where X is an ambient space, V is a bundle on it, and the moduli
space is locally the zero set Z (s) of a section s of V.
This local model in fact yields a two-term locally-free resolution of
the Kuranishi map, and by dualising the above geometricdescription one may package it into the following diagram:V ∨ |Mds ∨ |M/ ΩX |M∨ s |M

2
IM/X /IM/X

id/ ΩX |M .

Then Behrend and Fantechi consider a globalisation of this.
9 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

They define a perfect obstruction theory on a Deligne–Mumford
stack M as a complex E • ∈ D b (M), locally quasi-isomorphic to acomplex of locally-free sheaves supported in degree [−1, 0] with a≥1morphism ϕ : E • → L≥1
M , where LM is [−1, 0]-truncation of thecotangent complex, so that h0 (ϕ) is isomorphism and h−1 (ϕ) issurjective. From these data, they construct a virtual fundamental
class in the Chow group of M of the right degree and prove that it
is a deformation invariant.
Then, one can use this class to define virtual Donaldson invariants
on a projective surface X .
∫DX ,c1 (σ1 . . . σn ) :=µ(σ1 ) · · · · · µ(σn ),[M]virwhere σ1 , . . . , σn ∈ H ∗ (X ), and µ : H i (X ) → H i (M), when themoduli space M does not contain strictly semistable sheaves.
10 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

Takuro Mochizuki performed this in full generality (even withparabolic structures). More precisely, Mochizuki• formulates the invariants where the moduli space may havestrictly semistable sheaves;• proves a weak wall-crossing formula for his invariants; and• expresses them in terms of Seiberg–Witten invariants.
These lead to the determination of the wall-crossing terms and a
resolution of Witten’s conjecture D = SW on a projective surface
both by Göttsche–Nakajima–Yoshioka both with analysis(blowup formulae) on the Nekrasov partition function.
One may think about other insertions. Fantechi–Göttsche andCiocan-Fontanine–Kapranov, for instance, define the virtual
Euler characteristic of M by∫vir ),
e vir (M) := [M]vir cvd (TMvir is the virtual tagent sheaf coming form the perfectwhere TMobstruction theory on M.
11 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

More generally, they define the virtual χy -genus of M by∫vir −1virχvir−y (M) := [M]vir (1 − y ch(T ) ) · td(T )) ∈ Q[y ].
These virtual Euler characteristic and virtual χy -genus of themoduli space of semistable sheaves can be thought of as theinstanton part of the Vafa-Witten invariant or a refinement of it.
Göttsche–Kool forms conjectures that the generating series ofthem and also other virtual enumerative invariants such as virtual
Segre and Verlinde numbers of the moduli spaces could be written
in terms of modular forms and Seiberg–Witten invariants.
They also raised conjectures on blowup formulae for them. The one
for the virtual Euler characteristics of the moduli spaces resembles
Li–Qin’s one for the virtual Hodge polynomials, whose origin is thework by Vafa–Witten, where the interesting appearance of modular
forms is explained by means of Conformal Field theory, or Vertex
Algebras. (cf. Nakajima’s pioneering works for these subjects)
12 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

A blowup formula for virtual enumerative invariants
We establish a Nakajima–Yoshioka style blowup formula for virtualenumerative invariants on a projective surface, which include thevirtual Euler characteristics, virtual χy -genera, virtual Segre andVerlinde numbers of the moduli spaces of semistable sheaves, andthe Donaldson–Mochizuki invariants, by constructing perfectobstruction theories on the moduli spaces of m-stable sheaves
which interpolate the moduli space of semistable sheaves on acomplex projective surface and that on its blowup at a point, via
enhanced master spaces.
Nakajima–Yoshioka’s m-stable sheaves. Let p : X̂ → X be theblowup at a point of a projective surface X . Fix a cohomology
class c := r + c1 + ch2 of X and consider cb = p ∗ c + ke of Xb ,where e := ch(OC (−1)) with C the exceptional divisor.
13 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

We would like to compare the (virtual) fundamental classes comingfrom the moduli spaces MX (c) and MXb (bc ) of semistable sheaves
on X and X̂ respectively.
To do so, we consider the moduli spaces Mcbm of m-stable sheaves
on Xb introduced by Nakajima–Yoshioka on the blowup, whichinterpolate the moduli spaces of semistable sheaves on the blowup
and on the original surface in the following manner:
Mcb0

Mcb1

···

Mcbm−1Mcbmp∗

MX (c)c ),MXb (bwhere the right vertical arrow becomes an isomorphism if m is
suﬃciently large, while so is the left vertical arrow if cb = p ∗ c.
14 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

Wall-crossing. In order to compare things on Mcbm with those on
Mcbm+1 , one may embed them in an intermediate space, the moduli
space of (m, m + 1)-semistable sheaves.
But, the moduli space of them has highly positive-dimensional
automorphism groups, so we use Mochizuki’s technique, theenhanced moduli stack N ℓ , which is a Deligne–Mumford stack,together with the intermediate Artin stack N ℓ,ℓ+1 , which has only
one-dimensional stabiliser group, satisfying the following diagram:
N 0,1
N0

···
N1

···

N N−1

NN
Mcbm+1 .

Mcbm
15 / 18

N N−1,N

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

We then construct the Kiem–Li style master space Z equippedwith a C∗ -action, and its set-theoretical C∗ -fixed loci are∗|Z|C = |N ℓ | + |N ℓ+1 | + the rest.
The stack-theoretical version of this with virtual integrations overthe fibres of N 0 and N N leads to a wall-crossing formula we desire.
Blowup formula for Donaldson–Mochizuki invariants. One
application of our wall-crossing formula is the Friedman–Morgan
style blowup formula for the virtual invariants:
DXb ,p∗ c1 (σ1 . . . σn [C ]4 ) = −2DX ,c1 (σ1 , . . . , σn ).
This leads to a direct proof of the equivalence between Mochizuki’s
virtual Donaldson invariants and the classical ones in our setting.

16 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

Nakajima–Yoshioka style structure theorem. Let Φ(E) be a
Chow cohomology class depending on the universal sheaf on
X × M and cohomology classes of X such as the insertions forvirtual Euler characteristics, virtual χy -genera, virtual Segre andVerlinde numbers of the moduli spaces of semistable sheaves on X ,or the Donaldson–Mochizuki invariants. We then obtain:
Theorem (Kuhn–T) Fix a Chern class cb = p ∗ c − ke for somek ≥ 0 on Xb , where c = r + c1 + c2 is that on X . Then there exist
universal power series Ωn ∈ Q[[ν2 , . . . , νr ]] depending only on r , Φand k, which
satisfy:∫∞ ∫∑Φ(E) =
Φ(E)Ωn (E).
[MXb (bc )]vir

n=0[MX (c+n[pt])]vir

From the construction, these Ωn can be determined by calculatingthem on the moduli spaces of framed sheaves on P2 .
This may resolve conjectures by Göttsche–Kool and Göttsche on
the structures of virtual Segre and Verlinde numbers of projective
surfaces, and so on.
17 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants

Donaldson invariants on closed four-manifolds
Mochizuki’s Donaldson invariants on projective surfaces
A blowup formula for virtual invariants and applications

For example, we obtain the following blowup formula for thegenerating series of the virtual χy -genera of the moduli spaces:
Theorem (Kuhn–Leigh–T)∑∑
vd MX (c)vd MX̂ (ĉ)χvir
= Yk (q, y )χvir,−y (MX (c))q−y (MX̂ (ĉ))q
ch2
ch2


2r
r
r
/24
∑
t
t


with Yk (q, y ) := (q y )(q 2r y r )v Av y v AI  ,
2rrr
η(q y )
kv ∈Zr −1 + r I

where A = (aij ) is the (r − 1) × (r − 1)-matrix with entries aij = 1
for i ≤ j and aij = 0 otherwise, and I is the column vector of lengthr − 1 with all entries equal to one, η is the Dedekind eta function.
This coincides with Göttsche’s conjecture on the generating seriesof topological χy -genera of the moduli spaces (hence it confirms
the conjecture for the unobstructed case).
18 / 18

Yuuji Tanaka, Kyoto University

On a blowup formula for virtual enumerative invariants