Algebraic elliptic cohomology theory and flops I

Abstract

We define the algebraic elliptic cohomology theory coming from Krichever’s elliptic genus as an oriented cohomology theory on smooth varieties over an arbitrary perfect field. We show that in the algebraic cobordism ring with rational coefficients, the ideal generated by differences of classical flops coincides with the kernel of Krichever’s elliptic genus. This generalizes a theorem of B. Totaro in the complex analytic setting.

Appendices

Appendix A: The ideal generated by differences of flops

Let $\mathcal {I}_\text {clfl}$ be the ideal in ${{\,\mathrm{MGL}\,}}^*_\mathbb {Q}(k)$ generated by those $[X_1]-[X_2]$ with $X_1$ and $X_2$ related by a classical flop.

Proposition A.1

The ideal $\mathcal {I}_\text {clfl}$ in ${{\,\mathrm{MGL}\,}}^*_\mathbb {Q}(k)$ contains a system of polynomial generators $x_n$ of ${{\,\mathrm{MGL}\,}}^*_\mathbb {Q}(k)$ in all degrees $n \le -5$.

This proposition was originally proved in Section 5 of [37], using some non-algebraic constructions. Totaro’s proof is based on explicit calculations that lend themself to our setting after a slight adjustment.

For a smooth irreducible projective variety X over k, we have the characteristic class $s^n(X)=\langle \xi _1^n+\cdots +\xi _n^n,[X]\rangle$, with $\xi _i$ being the Chern roots of the tangent bundle of X (in the Chow ring ${{\,\mathrm{CH}\,}}^*$). We will use the fact that an element x of ${{\,\mathrm{MGL}\,}}^{-n}_\mathbb {Q}(k)$ is a polynomial generator of the ring ${{\,\mathrm{MGL}\,}}^*_\mathbb {Q}(k)$ if and only if the Chern number $s^n$ is not zero on x (see e.g., [1]).

Following Fulton, we have the ith Segré class of a rank r vector bundle V bundle over a smooth k-scheme X, defined as

$$\begin{aligned} s_i(V)=\pi _*(u^{i+r-1}),\quad i=0, 1, \ldots , \end{aligned}$$

where $u=c_1(\mathscr {O}(1))\in {{\,\mathrm{CH}\,}}^1(\mathbb {P}(V))$ and $\pi :\mathbb {P}(V)\rightarrow X$ is the structure morphism. From [5, Thereom 3.2], we have

Lemma A.2

Let $V\rightarrow X$ be a vector bundle over a smooth k-scheme X, let $s(V)=\sum _is_i(V)$ be the total Segré class, $c(V)=\sum _ic_i(V)$ the total Chern class. Then $s(V)=c(V)^{-1}$ in ${{\,\mathrm{CH}\,}}^*(X)$.

Recall that we have shown that, in ${{\,\mathrm{MGL}\,}}^*(k)$,

$$\begin{aligned}{}[X_1]-[X_2]=[\mathbb {P}_{\mathbb {P}(A)} (B\otimes \mathscr {O}_{\mathbb {P}(A)}(-1) \oplus \mathscr {O})]-[\mathbb {P}_{\mathbb {P}(B)} (A\otimes \mathscr {O}_{\mathbb {P}(B)}(-1) \oplus \mathscr {O})]. \end{aligned}$$

For each smooth Z and rank-2 vector bundles A and B, there is a pair $X_1$ and $X_2$, related by a classical flop and with exceptional fibers equal to $\mathbb {P}(A)$ and $\mathbb {P}(B)$ respectively. Indeed, consider the $\mathbb {P}^1\times \mathbb {P}^1$ bundle $q:\mathbb {P}(A)\times _Z\mathbb {P}(B)\rightarrow Z$, which we embed in the $\mathbb {P}^3$ bundle $\mathbb {P}(A\oplus B)\rightarrow Z$ via the line bundle $p_1^*\mathscr {O}_A(1)\otimes p_2^*\mathscr {O}_B(1)$. We then take $Y^0$ to be the affine Z cone in $A\oplus B$ associated to $\mathbb {P}(A)\times _Z\mathbb {P}(B)\subset \mathbb {P}(A\oplus B)$, and Y the closure of $Y^0$ in $\mathbb {P}(A\oplus B\oplus \mathscr {O}_Z)$. Y thus contains the $\mathbb {P}^2$ bundles $P_1:=\mathbb {P}(A\oplus \mathscr {O}_Z)$ and $P_2:=\mathbb {P}(B\oplus \mathscr {O}_Z)$; we take $X_i\rightarrow Y$ to be the blow-up of Y along $P_i$, $i=1, 2$ and ${\tilde{X}}$ the blow-up of Y along Z.

For each $n\ge 5$, we will find an $(n-3)$-fold Z and rank two vector bundles A and B over Z, such that $s^n([\mathbb {P}_{\mathbb {P}(A)} (B\otimes \mathscr {O}_{\mathbb {P}(A)}(-1) \oplus \mathscr {O})])\ne s^n([\mathbb {P}_{\mathbb {P}(B)} (A\otimes \mathscr {O}_{\mathbb {P}(B)}(-1) \oplus \mathscr {O})])$. In fact, we take $Z=\mathbb {P}^{n-3}$, $A=\mathscr {O}_Z(1)\oplus \mathscr {O}_Z$ and $B=\mathscr {O}_Z^2$.

Set $h=c_1(\mathscr {O}_Z(1))$, $v_A:=c_1(\mathscr {O}_{\mathbb {P}(A)}(1))$, $v_B=c_1(\mathscr {O}_{\mathbb {P}(B)}(1))$, $w_A:=c_1(\mathscr {O}_{\mathbb {P}(A\otimes \mathscr {O}(-1)\oplus \mathscr {O})}(1))$, $w_B=c_1(\mathscr {O}_{\mathbb {P}(B\otimes \mathscr {O}(-1)\oplus 1)}(1))$, and let $z_1\cdots ,z_{n-3}$ be the Chern roots of the tangent bundle of $Z=\mathbb {P}^{n-3}$. Then the Chern roots of the tangent bundle of $\mathbb {P}_{\mathbb {P}(A)} (B\otimes \mathscr {O}_{\mathbb {P}(A)}(-1) \oplus \mathscr {O})$ are $-v_A+w_B, -v_A+w_B, w_B, h+v_A, v_A, z_1, \dots , z_{n-3}$.

For the bundle $\pi _1: \mathbb {P}_{\mathbb {P}(A)} (B\otimes \mathscr {O}_{\mathbb {P}(A)}(-1) \oplus \mathscr {O}) \rightarrow \mathbb {P}(A)$, Lemma A.2 yields

$$\begin{aligned} \pi _{1*}(w_B^i)=s_{i-2}(B\otimes \mathscr {O}_{\mathbb {P}(A)}(-1) \oplus \mathscr {O})=(i-1)v_A^{i-2}. \end{aligned}$$

Similarly, for $\pi _2: \mathbb {P}(A) \rightarrow Z$, we have

$$\begin{aligned} \pi _{2*}(v_A^i)=s_{i-1}(A)=\sum _{j=0}^{i-1}(-h)^j. \end{aligned}$$

Using this and the projection formula, we find

$$\begin{aligned}&\pi _{1*}s^n\big ([\mathbb {P}_{\mathbb {P}(A)} (B\otimes \mathscr {O}_{\mathbb {P}(A)}(-1) \oplus \mathscr {O})]\big )\\&\quad =\pi _{1*}\left[ 2(-v_A+w_B)^n+w_B^n+(h+v_A)^n+(v_A)^n+\sum _{i=1}^{n-3}z_i^n\right] \\&\quad =2\sum _{i=2}^n{\left( {\begin{array}{c}n\\ i\end{array}}\right) }(-v_A)^{n-i}(i-1)(v_A)^{i-2}+(n-1)(v_A)^{n-2}\\&\quad =(v_A)^{n-2}(2(-1)^n+n-1). \end{aligned}$$

According to Lemma A.2, $\pi _{2*}(v_A)^{n-2}=(-h)^{n-3}$. Therefore,

$$\begin{aligned} \pi _{2*}\pi _{1*}s^n\big (\mathbb {P}_{\mathbb {P}(A)} (B\otimes \mathscr {O}_{\mathbb {P}(A)}(-1) \oplus \mathscr {O}) \big ) =h^{n-3}(-2+(-1)^{n-3}(n-1)). \end{aligned}$$

We do the same for the projection $\pi '_1: \mathbb {P}_{\mathbb {P}(B)} (A\otimes \mathscr {O}_{\mathbb {P}(B)}(-1) \oplus 1) \rightarrow \mathbb {P}(B)$, and $\pi '_2: \mathbb {P}(B) \rightarrow Z$. Using Lemma A.2, a calculation similar to the one above gives

$$\begin{aligned}&\pi '_{2*}\pi '_{1*}s^n \big (\mathbb {P}_{\mathbb {P}(B)} (A\otimes \mathscr {O}_{\mathbb {P}(B)}(-1) \oplus 1)\big ) \\&\quad =h^{n-3}(-(n-1)^2+{\left( {\begin{array}{c}n\\ 2\end{array}}\right) }+(n-2)(-1)^{n-3}) \end{aligned}$$

This gives

$$\begin{aligned} s^n([X_1]-[X_2]) =\frac{n^2-3n-2+2(-1)^{n-1}}{2}, \end{aligned}$$

so for $n\ge 5$, $s^n([X_1]-[X_2])\ne 0$, as desired.

Appendix B: $\ell '$-alterations and dualisability (by Joël Riou)

Proposition B.1

Let k be a perfect field. Let $\ell$ be a prime number different from the characteristic of k. Then, for any smooth finite type k-scheme U, the suspension spectrum $\varSigma ^\infty _T U_+$ belongs to the pseudo-abelian triangulated subcategory of ${{\,\mathrm{SH}\,}}(k)_{\mathbb {Z}_{(\ell )}}$ generated by the objects $\varSigma ^\infty _T X_+$ where X is projective and smooth over k.

Corollary B.2

Let k be a perfect field. Let p denote the caracteristic exponent of k (i.e., $p>0$ or $p=1$ if the characteristic of k is zero). Then, for any smooth finite type k-scheme U, the suspension spectrum $\varSigma ^\infty _T U_+$ is strongly dualisable in ${{\,\mathrm{SH}\,}}(k)_{\mathbb {Z}[\frac{1}{p}]}$.

First, we shall see how the corollary follows from the proposition. The strong dualisability can be formulated using the internal Hom, which shall be denoted ${{\,\mathrm{\mathbf {Hom}}\,}}$ here. An object $A\in {{\,\mathrm{SH}\,}}(k)_{\mathbb {Z}[\frac{1}{p}]}$ is strongly dualisable if and only if for any object $M\in {{\,\mathrm{SH}\,}}(k)_{\mathbb {Z}[\frac{1}{p}]}$, the canonical morphism ${{\,\mathrm{\mathbf {Hom}}\,}}(A,\mathbb {S}_k)\wedge M\rightarrow {{\,\mathrm{\mathbf {Hom}}\,}}(A,M)$ is an isomorphism. At the level of stable motivic homotopy sheaves, the localisation functor ${{\,\mathrm{SH}\,}}(k)_{\mathbb {Z}[\frac{1}{p}]}\rightarrow {{\,\mathrm{SH}\,}}(k)_{\mathbb {Z}_{(\ell )}}$ (for any prime number $\ell$ not dividing p) corresponds to the tensor product with $\mathbb {Z}_{(\ell )}$ over $\mathbb {Z}[\frac{1}{p}]$. Then, we see that in order to prove that an object $A\in {{\,\mathrm{SH}\,}}(k)_{\mathbb {Z}[\frac{1}{p}]}$ is strongly dualisable, it suffices to prove that its images in all the categories ${{\,\mathrm{SH}\,}}(k)_{\mathbb {Z}_{(\ell )}}$ (for $\ell$ not dividing p) are strongly dualisable. Finally, the corollary follows from the proposition and the fact that the objects $\varSigma ^\infty _T X_+$ are strongly dualisable in ${{\,\mathrm{SH}\,}}(k)$ if X is projective and smooth (see [32, §2] which relies on the work by J. Ayoub [2] and V. Voevodsky).

In order to prove the proposition, we shall need the following lemma:

Lemma B.3

Let k be a field of characteristic $p>0$. Let $\pi :V\rightarrow U$ be a finite and étale morphism between connected and smooth k-schemes. Let d be the degree of $\pi$. Then, there exists a dense open subset $U'\subset U$ and a morphism $s:\varSigma ^\infty _T U'_+ \rightarrow \varSigma ^\infty _T V'_+$ in ${{\,\mathrm{SH}\,}}(k)$ (where $V':=\pi ^{-1}(U')$), such that if we denote $\pi ':\varSigma ^\infty _T V'_+ \rightarrow \varSigma ^\infty _T U'_+$ the induced morphism, the composition $\pi '\circ s\in {{\,\mathrm{End}\,}}_{{{\,\mathrm{SH}\,}}(k)}(\varSigma ^\infty _T U'_+)$ can be written as $\pi '\circ s=d+\alpha$, where $\alpha$ is a nilpotent endomorphism of $\varSigma ^\infty _T U'_+$ in ${{\,\mathrm{SH}\,}}(k)$.

Proof

We may observe that the map s of the lemma can be obtained by application of the functor $a_\sharp :{{\,\mathrm{SH}\,}}(U')\rightarrow {{\,\mathrm{SH}\,}}(k)$ (see [23, p. 104]) where $a:U'\rightarrow {{\,\mathrm{Spec}\,}}(k)$ is the obvious morphism. Indeed, passing to the generic point of U, we see that in order to prove the lemma we may assume that $U={{\,\mathrm{Spec}\,}}(k)$ and that V is the spectrum of a finite separable field extension L of k. The map s is constructed in [32, Lemme 1.9] and what we know is that the composition $\pi '\circ s\in {{\,\mathrm{End}\,}}_{SH(k)}(\mathbb {S}_k)$ is of the form $d+\alpha$ where $\alpha$ is an endomorphism that vanishes after base change to a big enough extension of k. We now use the isomorphism ${{\,\mathrm{End}\,}}_{SH(k)}(\mathbb {S}_k)\simeq GW(k)$ from [22, Corollary 1.24 and Remark 1.26]. Using this identification, we see that $\alpha$ belongs to the kernel of the rank morphism $GW(k)\rightarrow \mathbb {Z}$. Then, we can conclude using the following lemma: $\square$

Lemma B.4

Let k be a field of characteristic $p>0$. Let $\alpha \in GW(k)$ be an element in the kernel of the rank morphism $GW(k)\rightarrow \mathbb {Z}$. Then, $\alpha$ is nilpotent in GW(k).

Proof

As the set of nilpotent elements in the commutative ring GW(k) is an ideal, we may assume $\alpha =\langle t\rangle -1$ where $t\in k^\times$. We have $(1+\alpha )^2=\langle t^2\rangle =1$, so that $\alpha ^2=-2\alpha$. By induction, we get $\alpha ^n=(-2)^{n-1}\alpha$ for $n\ge 1$: we have to show that $\alpha$ is annihilated by a power of two. If $p=2$, $2\alpha =0$ holds (see [22, Lemma 3.9]), i.e. $\alpha ^2=0$. Now we assume $p\ge 3$ so that there is no danger thinking in terms of usual quadratic forms. We first consider $\gamma :=\langle -1\rangle -1\in GW(\mathbf {F}_p)$. The quadratic form $-x^2-y^2$ over $\mathbf {F}_p$ represents 1 (see [35, Proposition 4, §IV.1.7]) so that $\langle -1\rangle +\langle -1\rangle =\langle 1\rangle +\langle 1\rangle \in GW(\mathbf {F}_p)$, i.e. $2\gamma =0\in GW(\mathbf {F}_p)$, which gives $\gamma ^2=0$. Let $t\in k^\times$ be any nonzero element in an extension k of $\mathbf {F}_p$. The quadratic form $q(x,y):=x^2-y^2=(x+y)(x-y)$ represents t (this is $q(\frac{1+t}{2},\frac{1-t}{2})$), which easily implies that $\langle 1\rangle +\langle -1\rangle =\langle t\rangle +\langle -t\rangle$. This is equivalent to saying $(2+\gamma )\alpha =0\in GW(k)$. It follows that $4\alpha =(2-\gamma )(2+\gamma )\alpha =0$, and then $\alpha ^3=0$.

We can now prove the proposition. It was already proven in the case k is of characteristic 0 in [32] using Hironaka’s resolution of singularities. In characteristic $p>0$, an argument using de Jong’s alterations also led to the same result with rational coefficients. The idea was to do an induction on the dimension of the variety U, which can be assumed connected. Then, the property we want to prove becomes a birational property of U, so that we can shrink U if needed. Using the Lemma B.3, we obtain that $\varSigma ^\infty _T U'_+$ is a direct factor of $\varSigma ^\infty _T V'_+$ if the degree d has been inverted in the coefficient ring. If d is invertible in the coefficient ring and if $V'$ is an open subset of a projective and smooth variety, we can deduce the expected property for U. The theorems by de Jong on alterations were sufficient to conclude in the case of rational coefficients. Here, for $\mathbb {Z}_{(\ell )}$-coefficients, we have to ensure that an appropriate alteration can be found with a prime-to-$\ell$ degree d: this is possible thanks to Gabber’s improvement [6, X 3.5] of de Jong’s results.